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Sample records for chemotaxis diffusion equations

  1. Fractional Chemotaxis Diffusion Equations

    CERN Document Server

    Langlands, T A M

    2010-01-01

    We introduce mesoscopic and macroscopic model equations of chemotaxis with anomalous subdiffusion for modelling chemically directed transport of biological organisms in changing chemical environments with diffusion hindered by traps or macro-molecular crowding. The mesoscopic models are formulated using Continuous Time Random Walk master equations and the macroscopic models are formulated with fractional order differential equations. Different models are proposed depending on the timing of the chemotactic forcing. Generalizations of the models to include linear reaction dynamics are also derived. Finally a Monte Carlo method for simulating anomalous subdiffusion with chemotaxis is introduced and simulation results are compared with numerical solutions of the model equations. The model equations developed here could be used to replace Keller-Segel type equations in biological systems with transport hindered by traps, macro-molecular crowding or other obstacles.

  2. Exact solutions of certain nonlinear chemotaxis diffusion reaction equations

    Indian Academy of Sciences (India)

    MISHRA AJAY; KAUSHAL R S; PRASAD AWADHESH

    2016-05-01

    Using the auxiliary equation method, we obtain exact solutions of certain nonlinear chemotaxis diffusion reaction equations in the presence of a stimulant. In particular, we account for the nonlinearities arising not only from the density-dependent source terms contributed by the particles and the stimulant but also from the coupling term of the stimulant. In addition to this, the diffusion of the stimulant and the effect of long-range interactions are also accounted for in theconstructed coupled differential equations. The results obtained here could be useful in the studies of several biological systems and processes, e.g., in bacterial infection, chemotherapy, etc.

  3. Generalized Keller-Segel models of chemotaxis. Analogy with nonlinear mean field Fokker-Planck equations

    CERN Document Server

    Chavanis, Pierre-Henri

    2008-01-01

    We consider a generalized class of Keller-Segel models describing the chemotaxis of biological populations (bacteria, amoebae, endothelial cells, social insects,...). We show the analogy with nonlinear mean field Fokker-Planck equations and generalized thermodynamics. As an illustration, we introduce a new model of chemotaxis incorporating both effects of anomalous diffusion and exclusion principle (volume filling). We also discuss the analogy between biological populations described by the Keller-Segel model and self-gravitating Brownian particles described by the Smoluchowski-Poisson system.

  4. Effective Medium Equations for Chemotaxis in Porous Media

    Science.gov (United States)

    Valdes-Parada, F.; Porter, M.; Wood, B. D.; Narayanaswamy, K.; Ford, R.

    2008-12-01

    Biodegradation is an important mechanism for contaminant reduction in groundwater environments; in fact, in-situ bioremediation and bioaugmentation methods represent alternatives to traditional methods such as pump-and-treat. Chemotaxis has been shown to enhance bacterial transport toward or away from concentration gradients of chemical species in laboratory experiments and may signifficantly increase contaminant flux undergoing degradation at the interfaces of low- and high-permeability regions. In this work, the method of volume averaging is used to upscale the microscale description of chemotactic microbial transport in order to obtain the corresponding macroscale equations for bacteria and the chemoattractant. As a first apprach, cellular growth/death and consumption of the attractant by chemical reaction are assumed negligible with respect to convective and diffusive transport, in both levels of scale. For bacteria, two effective coefficients are introduced, namely a total motility tensor and an effective chemotactic sensitivity tensor. Both coefficients are computed by solving the associated closure problems in a capillary tube. Analysis of breakthrough curves resulting from numerical experiments is also presented.

  5. Global Solutions to the Coupled Chemotaxis-Fluid Equations

    KAUST Repository

    Duan, Renjun

    2010-08-10

    In this paper, we are concerned with a model arising from biology, which is a coupled system of the chemotaxis equations and the viscous incompressible fluid equations through transport and external forcing. The global existence of solutions to the Cauchy problem is investigated under certain conditions. Precisely, for the Chemotaxis-Navier-Stokes system over three space dimensions, we obtain global existence and rates of convergence on classical solutions near constant states. When the fluid motion is described by the simpler Stokes equations, we prove global existence of weak solutions in two space dimensions for cell density with finite mass, first-order spatial moment and entropy provided that the external forcing is weak or the substrate concentration is small. © Taylor & Francis Group, LLC.

  6. Travelling Waves in Hyperbolic Chemotaxis Equations

    KAUST Repository

    Xue, Chuan

    2010-10-16

    Mathematical models of bacterial populations are often written as systems of partial differential equations for the densities of bacteria and concentrations of extracellular (signal) chemicals. This approach has been employed since the seminal work of Keller and Segel in the 1970s (Keller and Segel, J. Theor. Biol. 30:235-248, 1971). The system has been shown to permit travelling wave solutions which correspond to travelling band formation in bacterial colonies, yet only under specific criteria, such as a singularity in the chemotactic sensitivity function as the signal approaches zero. Such a singularity generates infinite macroscopic velocities which are biologically unrealistic. In this paper, we formulate a model that takes into consideration relevant details of the intracellular processes while avoiding the singularity in the chemotactic sensitivity. We prove the global existence of solutions and then show the existence of travelling wave solutions both numerically and analytically. © 2010 Society for Mathematical Biology.

  7. Speed ot travelling waves in reaction-diffusion equations

    CERN Document Server

    Benguria, R D; Méndez, V

    2002-01-01

    Reaction diffusion equations arise in several problems of population dynamics, flame propagation and others. In one dimensional cases the systems may evolve into travelling fronts. Here we concentrate on a reaction diffusion equation which arises as a simple model for chemotaxis and present results for the speed of the travelling fronts. (Author)

  8. Drift-Diffusion Equation

    Directory of Open Access Journals (Sweden)

    K. Banoo

    1998-01-01

    equation in the discrete momentum space. This is shown to be similar to the conventional drift-diffusion equation except that it is a more rigorous solution to the Boltzmann equation because the current and carrier densities are resolved into M×1 vectors, where M is the number of modes in the discrete momentum space. The mobility and diffusion coefficient become M×M matrices which connect the M momentum space modes. This approach is demonstrated by simulating electron transport in bulk silicon.

  9. Nonlocal electrical diffusion equation

    Science.gov (United States)

    Gómez-Aguilar, J. F.; Escobar-Jiménez, R. F.; Olivares-Peregrino, V. H.; Benavides-Cruz, M.; Calderón-Ramón, C.

    2016-07-01

    In this paper, we present an analysis and modeling of the electrical diffusion equation using the fractional calculus approach. This alternative representation for the current density is expressed in terms of the Caputo derivatives, the order for the space domain is 0solar panels, electrochemical phenomena and the description of anomalous complex processes.

  10. Numerical study of plume patterns in the chemotaxis-diffusion-convection coupling system

    CERN Document Server

    Deleuze, Yannick; Thiriet, Marc; Sheu, Tony W H

    2015-01-01

    A chemotaxis-diffusion-convection coupling system for describing a form of buoyant convection in which the fluid develops convection cells and plume patterns will be investigated numerically in this study. Based on the two-dimensional convective chemotaxis-fluid model proposed in the literature, we developed an upwind finite element method to investigate the pattern formation and the hydrodynamical stability of the system. The numerical simulations illustrate different predicted physical regimes in the system. In the convective regime, the predicted plumes resemble B\\'enard instabilities. Our numerical results show how structured layers of bacteria are formed before bacterium rich plumes fall in the fluid. The plumes have a well defined spectrum of wavelengths and have an exponential growth rate, yet their position can only be predicted in very simple examples. In the chemotactic and diffusive regimes, the effects of chemotaxis are investigated. Our results indicate that the chemotaxis can stabilize the overa...

  11. Different Behaviour for the Solutions of Some Chemotaxis Equations

    Institute of Scientific and Technical Information of China (English)

    2000-01-01

    @@  In biology, it is very important to investigate the movement of some cells ororganisms in some given biological system (cf. [1, 2]). In order to understand howthe movement rules are affected by the effect of the chemo-attractant, Othmer and Stevens introduced in [1] several general classes of partial differential equations. In one of their models, they considered a master equation, i.e. barrier and nearestneighbor lattice model. Following a limiting process the model is described by the following system of partial differential equations:

  12. Anomalous Fractional Diffusion Equation for Transport Phenomena

    Institute of Scientific and Technical Information of China (English)

    QiuhuaZENG; HouqiangLI; 等

    1999-01-01

    We derive the standard diffusion equation from the continuity equation and by discussing the defectiveness of earlier proposed equations,we get the generalized fractional diffusion equation for anomalous diffusion.

  13. Reaction Diffusion and Chemotaxis for Decentralized Gathering on FPGAs

    Directory of Open Access Journals (Sweden)

    Bernard Girau

    2009-01-01

    and rapid simulations of the complex dynamics of this reaction-diffusion model. Then we describe the FPGA implementation of the environment together with the agents, to study the major challenges that must be solved when designing a fast embedded implementation of the decentralized gathering model. We analyze the results according to the different goals of these hardware implementations.

  14. Marangoni-driven chemotaxis, chemotactic collapse, and the Keller-Segel equation

    Science.gov (United States)

    Shelley, Michael; Masoud, Hassan

    2013-11-01

    Almost by definition, chemotaxis involves the biased motion of motile particles along gradients of a chemical concentration field. Perhaps the most famous model for collective chemotaxis in mathematical biology is the Keller-Segel model, conceived to describe collective aggregation of slime mold colonies in response to an intrinsically produced, and diffusing, chemo-attractant. Heavily studied, particularly in 2D where the system is ``super-critical'', it has been proved that the KS model can develop finite-time singularities - so-called chemotactic collapse - of delta-function type. Here, we study the collective dynamics of immotile particles bound to a 2D interface above a 3D fluid. These particles are chemically active and produce a diffusing field that creates surface-tension gradients along the surface. The resultant Marangoni stresses create flows that carry the particles, possibly concentrating them. Remarkably, we show that this system involving 3D diffusion and fluid dynamics, exactly yields the 2D Keller-Segel model for the surface-flow of active particles. We discuss the consequences of collapse on the 3D fluid dynamics, and generalizations of the fluid-dynamical model.

  15. A diffusion based long-range and steady chemical gradient generator on a microfluidic device for studying bacterial chemotaxis

    Science.gov (United States)

    Murugesan, Nithya; Singha, Siddhartha; Panda, Tapobrata; Das, Sarit K.

    2016-03-01

    Studies on chemotaxis in microfluidics device have become a major area of research to generate physiologically similar environment in vitro. In this work, a novel micro-fluidic device has been developed to study chemo-taxis of cells in near physiological condition which can create controllable, steady and long-range chemical gradients using various chemo-effectors in a micro-channel. Hydrogels like agarose, collagen, etc, can be used in the device to maintain exclusive diffusive flux of various chemical species into the micro-channel under study. Variations of concentrations and flow rates of Texas Red dextran in the device revealed that an increase in the concentration of the dye in the feed from 6 to 18 μg ml-1, causes a steeper chemical gradient in the device, whereas the flow rate of the dye has practically no effect on the chemical gradient in the device. This observation confirms that a diffusion controlled chemical gradient is generated in the micro-channel. Chemo-taxis of E. coli cells were studied under the steady gradient of a chemo-attractant and a chemo-repellent separately in the same chemical gradient generator. For sorbitol and NiSO4·6H2O, the bacterial cells exhibit a steady distribution in the micro channel after 1 h and 30 min, respectively. From the distribution of bacterial population chemo-tactic strength of the chemo-effectors was estimated for E. coli. In a long microfluidic channel, migration behavior of bacterial cells under diffusion controlled chemical gradient showed chemotaxis, random movement, aggregation, and concentration dependent reverse chemotaxis.

  16. Behavior of the Solutions for the Chemotaxis Equations with Saturation Growth%具饱和增长的Chemotaxis方程组解的性质

    Institute of Scientific and Technical Information of China (English)

    杨茵; 刘伟安

    2003-01-01

    @@ In this paper, we study the behavior of the solutions to a kind of chemotaxis equations that describe the mechanism of communication between cells or organisms in some biology systems, which is introduced by [1] and [2].

  17. Parabolic equations in biology growth, reaction, movement and diffusion

    CERN Document Server

    Perthame, Benoît

    2015-01-01

    This book presents several fundamental questions in mathematical biology such as Turing instability, pattern formation, reaction-diffusion systems, invasion waves and Fokker-Planck equations. These are classical modeling tools for mathematical biology with applications to ecology and population dynamics, the neurosciences, enzymatic reactions, chemotaxis, invasion waves etc. The book presents these aspects from a mathematical perspective, with the aim of identifying those qualitative properties of the models that are relevant for biological applications. To do so, it uncovers the mechanisms at work behind Turing instability, pattern formation and invasion waves. This involves several mathematical tools, such as stability and instability analysis, blow-up in finite time, asymptotic methods and relative entropy properties. Given the content presented, the book is well suited as a textbook for master-level coursework.

  18. Fractional diffusion equations coupled by reaction terms

    Science.gov (United States)

    Lenzi, E. K.; Menechini Neto, R.; Tateishi, A. A.; Lenzi, M. K.; Ribeiro, H. V.

    2016-09-01

    We investigate the behavior for a set of fractional reaction-diffusion equations that extend the usual ones by the presence of spatial fractional derivatives of distributed order in the diffusive term. These equations are coupled via the reaction terms which may represent reversible or irreversible processes. For these equations, we find exact solutions and show that the spreading of the distributions is asymptotically governed by the same the long-tailed distribution. Furthermore, we observe that the coupling introduced by reaction terms creates an interplay between different diffusive regimes leading us to a rich class of behaviors related to anomalous diffusion.

  19. A coupled chemotaxis-fluid model: Global existence

    KAUST Repository

    Liu, Jian-Guo

    2011-09-01

    We consider a model arising from biology, consisting of chemotaxis equations coupled to viscous incompressible fluid equations through transport and external forcing. Global existence of solutions to the Cauchy problem is investigated under certain conditions. Precisely, for the chemotaxis-Navier- Stokes system in two space dimensions, we obtain global existence for large data. In three space dimensions, we prove global existence of weak solutions for the chemotaxis-Stokes system with nonlinear diffusion for the cell density.© 2011 Elsevier Masson SAS. All rights reserved.

  20. The Nonlinear Convection—Reaction—Diffusion Equation

    Institute of Scientific and Technical Information of China (English)

    ShiminTANG; MaochangCUI; 等

    1996-01-01

    A nonlinear convection-reaction-diffusion equation is used as a model equation of the El Nino events.In this model,the effects of convection,turbulent diffusion,linear feed-back and nolinear radiation on the anomaly of Sea Surface Temperature(SST) are considered.In the case of constant convection,this equation has exact kink-like travelling wave solutions,which can be used to explain the history of an El Nino event.

  1. Symmetry properties of fractional diffusion equations

    Energy Technology Data Exchange (ETDEWEB)

    Gazizov, R K; Kasatkin, A A; Lukashchuk, S Yu [Ufa State Aviation Technical University, Karl Marx strausse 12, Ufa (Russian Federation)], E-mail: gazizov@mail.rb.ru, E-mail: alexei_kasatkin@mail.ru, E-mail: lsu@mail.rb.ru

    2009-10-15

    In this paper, nonlinear anomalous diffusion equations with time fractional derivatives (Riemann-Liouville and Caputo) of the order of 0-2 are considered. Lie point symmetries of these equations are investigated and compared. Examples of using the obtained symmetries for constructing exact solutions of the equations under consideration are presented.

  2. Fractional Diffusion Limit for Collisional Kinetic Equations

    KAUST Repository

    Mellet, Antoine

    2010-08-20

    This paper is devoted to diffusion limits of linear Boltzmann equations. When the equilibrium distribution function is a Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffusion equation. In this paper, we consider situations in which the equilibrium distribution function is a heavy-tailed distribution with infinite variance. We then show that for an appropriate time scale, the small mean free path limit gives rise to a fractional diffusion equation. © 2010 Springer-Verlag.

  3. Global Attractors for a Nonclassical Diffusion Equation

    Institute of Scientific and Technical Information of China (English)

    Chun You SUN; Su Yun WANG; Cheng Kui ZHONG

    2007-01-01

    We prove the existence of global attractors in H10 (Ω) for a nonclassical diffusion equation.Two types of nonlinearity f are considered: one is the critical exponent, and the other is the polynomial growth of arbitrary order.

  4. The Riccati Differential Equation and a Diffusion-Type Equation

    CERN Document Server

    Suazo, Erwin; Vega-Guzman, Jose M

    2008-01-01

    We construct an explicit solution of the Cauchy initial value problem for certain diffusion-type equation with variable coefficients on the entire real line. The corresponding Green function (heat kernel) is given in terms of elementary functions and certain integrals involving a characteristic function, which should be found as an analytic or numerical solution of the second order linear differential equation with time-dependent coefficients. Some special and limiting cases are outlined. Solution of the corresponding nonhomogeneous equation is also found.

  5. Linearization of Systems of Nonlinear Diffusion Equations

    Institute of Scientific and Technical Information of China (English)

    KANG Jing; QU Chang-Zheng

    2007-01-01

    We investigate the linearization of systems of n-component nonlinear diffusion equations; such systems have physical applications in soil science, mathematical biology and invariant curve flows. Equivalence transformations of their auxiliary systems are used to identify the systems that can be linearized. We also provide several examples of systems with two-component equations, and show how to linearize them by nonlocal mappings.

  6. Diffusion phenomenon for linear dissipative wave equations

    KAUST Repository

    Said-Houari, Belkacem

    2012-01-01

    In this paper we prove the diffusion phenomenon for the linear wave equation. To derive the diffusion phenomenon, a new method is used. In fact, for initial data in some weighted spaces, we prove that for {equation presented} decays with the rate {equation presented} [0,1] faster than that of either u or v, where u is the solution of the linear wave equation with initial data {equation presented} [0,1], and v is the solution of the related heat equation with initial data v 0 = u 0 + u 1. This result improves the result in H. Yang and A. Milani [Bull. Sci. Math. 124 (2000), 415-433] in the sense that, under the above restriction on the initial data, the decay rate given in that paper can be improved by t -γ/2. © European Mathematical Society.

  7. Reaction diffusion equations with boundary degeneracy

    Directory of Open Access Journals (Sweden)

    Huashui Zhan

    2016-03-01

    Full Text Available In this article, we consider the reaction diffusion equation $$ \\frac{\\partial u}{\\partial t} = \\Delta A(u,\\quad (x,t\\in \\Omega \\times (0,T, $$ with the homogeneous boundary condition. Inspired by the Fichera-Oleinik theory, if the equation is not only strongly degenerate in the interior of $\\Omega$, but also degenerate on the boundary, we show that the solution of the equation is free from any limitation of the boundary condition.

  8. A Diffusion Equation on Fractals in Random Media

    Institute of Scientific and Technical Information of China (English)

    DeLIU; HouqiangLI; 等

    1997-01-01

    The proper transformation method is used in this paper to extend the diffusion equation on Euclidean space to the standard diffusion equation on fractals,By this standard diffusion equation,it is proved that the fractal Brownian particle moving belongs to the anomalous diffusion.At the same time,the generalized diffusion equation and its asymptotic solution is discussed.

  9. Langevin Equations for Reaction-Diffusion Processes

    Science.gov (United States)

    Benitez, Federico; Duclut, Charlie; Chaté, Hugues; Delamotte, Bertrand; Dornic, Ivan; Muñoz, Miguel A.

    2016-09-01

    For reaction-diffusion processes with at most bimolecular reactants, we derive well-behaved, numerically tractable, exact Langevin equations that govern a stochastic variable related to the response field in field theory. Using duality relations, we show how the particle number and other quantities of interest can be computed. Our work clarifies long-standing conceptual issues encountered in field-theoretical approaches and paves the way for systematic numerical and theoretical analyses of reaction-diffusion problems.

  10. Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior

    KAUST Repository

    Markowich, Peter

    2010-06-01

    We study the system ct + u · ∇c = ∇c -nf(c) nt + u · ∇n = ∇n m - ∇ · (n×(c) ∇c) ut + u·∇u + ∇P - η∇u + n∇φ/ = 0 ∇·u = 0. arising in the modelling of the motion of swimming bacteria under the effect of diffusion, oxygen-taxis and transport through an incompressible fluid. The novelty with respect to previous papers in the literature lies in the presence of nonlinear porous-medium-like diffusion in the equation for the density n of the bacteria, motivated by a finite size effect. We prove that, under the constraint m ε (3/2, 2] for the adiabatic exponent, such system features global in time solutions in two space dimensions for large data. Moreover, in the case m = 2 we prove that solutions converge to constant states in the large-time limit. The proofs rely on standard energy methods and on a basic entropy estimate which cannot be achieved in the case m = 1. The case m = 2 is very special as we can provide a Lyapounov functional. We generalize our results to the three-dimensional case and obtain a smaller range of exponents m ε (m*, 2] with m* > 3/2, due to the use of classical Sobolev inequalities.

  11. Lotka-Volterra equations with chemotaxis: walls, barriers and travelling waves.

    Science.gov (United States)

    Pettet, G J; McElwain, D L; Norbury, J

    2000-12-01

    In this paper we consider a simple two species model for the growth of new blood vessels. The model is based upon the Lotka-Volterra system of predator and prey interaction, where we identify newly developed capillary tips as the predator species and a chemoattractant which directs their motion as the prey. We extend the Lotka-Volterra system to include a one-dimensional spatial dependence, by allowing the predators to migrate in a manner modelled on the phenomenon of chemotaxis. A feature of this model is its potential to support travelling wave solutions. We emphasize that in order to determine the existence of such travelling waves it is essential that the global relationships of a number of phase plane features other than the equilibria be investigated.

  12. Relativistic diffusion equation from stochastic quantization

    CERN Document Server

    Kazinski, P O

    2007-01-01

    The new scheme of stochastic quantization is proposed. This quantization procedure is equivalent to the deformation of an algebra of observables in the manner of deformation quantization with an imaginary deformation parameter (the Planck constant). We apply this method to the models of nonrelativistic and relativistic particles interacting with an electromagnetic field. In the first case we establish the equivalence of such a quantization to the Fokker-Planck equation with a special force. The application of the proposed quantization procedure to the model of a relativistic particle results in a relativistic generalization of the Fokker-Planck equation in the coordinate space, which in the absence of the electromagnetic field reduces to the relativistic diffusion (heat) equation. The stationary probability distribution functions for a stochastically quantized particle diffusing under a barrier and a particle in the potential of a harmonic oscillator are derived.

  13. Logarithmic diffusion and porous media equations: a unified description

    OpenAIRE

    Pedron, I. T.; Mendes, R. S.; Buratta, T. J.; L. C. Malacarne; Lenzi, E. K.

    2005-01-01

    In this work we present the logarithmic diffusion equation as a limit case when the index that characterizes a nonlinear Fokker-Planck equation, in its diffusive term, goes to zero. A linear drift and a source term are considered in this equation. Its solution has a lorentzian form, consequently this equation characterizes a super diffusion like a L\\'evy kind. In addition is obtained an equation that unifies the porous media and the logarithmic diffusion equations, including a generalized dif...

  14. Analysis of discrete reaction-diffusion equations for autocatalysis and continuum diffusion equations for transport

    Energy Technology Data Exchange (ETDEWEB)

    Wang, Chi-Jen [Iowa State Univ., Ames, IA (United States)

    2013-01-01

    In this thesis, we analyze both the spatiotemporal behavior of: (A) non-linear “reaction” models utilizing (discrete) reaction-diffusion equations; and (B) spatial transport problems on surfaces and in nanopores utilizing the relevant (continuum) diffusion or Fokker-Planck equations. Thus, there are some common themes in these studies, as they all involve partial differential equations or their discrete analogues which incorporate a description of diffusion-type processes. However, there are also some qualitative differences, as shall be discussed below.

  15. Entropy methods for diffusive partial differential equations

    CERN Document Server

    Jüngel, Ansgar

    2016-01-01

    This book presents a range of entropy methods for diffusive PDEs devised by many researchers in the course of the past few decades, which allow us to understand the qualitative behavior of solutions to diffusive equations (and Markov diffusion processes). Applications include the large-time asymptotics of solutions, the derivation of convex Sobolev inequalities, the existence and uniqueness of weak solutions, and the analysis of discrete and geometric structures of the PDEs. The purpose of the book is to provide readers an introduction to selected entropy methods that can be found in the research literature. In order to highlight the core concepts, the results are not stated in the widest generality and most of the arguments are only formal (in the sense that the functional setting is not specified or sufficient regularity is supposed). The text is also suitable for advanced master and PhD students and could serve as a textbook for special courses and seminars.

  16. Simplification and Application of Boyd Membrane-diffusion Equation

    Institute of Scientific and Technical Information of China (English)

    甄捷

    2004-01-01

    The havirng-been-used-for-50-year Boyd membrane diffusion Equation - In( 1 - F) = R t can be deduced into F = kt through using Maclanrin expansion equation and the Lagerange remainders. The latter is a simple membrane diffusion equation, which is available to judge if the exchanging course of the resin obeys the rules of membrane-diffusion mechanism more conveniently.

  17. Turbulence and diffusion scaling versus equations

    CERN Document Server

    Bakunin, Oleg G

    2008-01-01

    This book is an introduction to the multidisciplinary field of anomalous diffusion in complex systems, with emphasis on the scaling approach as opposed to techniques based on the quantitative analysis of underlying transport equations. Typical examples of such systems are turbulent plasmas, convective rolls, zonal flow systems and stochastic magnetic fields. From the more methodological point of view, the approach relies on the general use of correlations estimates, quasilinear equations and continuous time random walk techniques. Yet, the mathematical descriptions are not meant to become a fixed set of recipes but rather develop and strengthen the reader's physical intuition and understanding on the underlying mechanisms involved. Most of the material stems from class-tested lectures, where graduate students where assumed to have a working knowledge of classical physics, fluid dynamics and plasma physics but otherwise no prior knowledge of the subject matter is assumed from the side of the reader.

  18. Differential constraints and exact solutions of nonlinear diffusion equations

    OpenAIRE

    Kaptsov, Oleg V.; Verevkin, Igor V.

    2002-01-01

    The differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the determining equations used in the search for classical Lie symmetries.

  19. FRACTIONAL DIFFUSION EQUATIONS WITH INTERNAL DEGREES OF FREEDOM

    Institute of Scientific and Technical Information of China (English)

    Luis Vázquez

    2003-01-01

    We present a generalization of the linear one-dimensional diffusion equation by combining the fractional derivatives and the internal degrees of freedom. The solutions are constructed from those of the scalar fractional diffusion equation. We analyze the interpolation between the standard diffusion and wave equations defined by the fractional derivatives. Our main result is that we can define a diffusion process depending on the internal degrees of freedom associated to the system.

  20. The solution of integral equation of plasma diffusion problems

    International Nuclear Information System (INIS)

    Two topics which concern with the solution of one dimensional diffusion equation are discussed in this paper. The solution of the one dimensional diffusion equation with one initial condition is discussed in the first topic, and the solution of the same equation with two initial conditions is discussed in the second topic. Both solutions are presented in integral forms. (author)

  1. Solution spectrum of nonlinear diffusion equations

    Science.gov (United States)

    Ulmer, W.

    1992-08-01

    The stationary version of the nonlinear diffusion equation -∂c/∂t + D Δc = λ1c - λ2c2 can be solved with the ansatz c = ∑{/p = 1/∞} Ap(cosh kx) -p , inducing a band structure with regard to the ratio λ1/λ2. The resulting solution manifold can be related to an equilibrium of fluxes of nonequilibrium thermodynamics. The modification of this ansatz yielding the expansion c = ∑ skp = 1/∞ A pq (cosh kx)- p [(cosh αt) -q-1 sinh αt + b(cosh αt)-q] represents a solution spectrum of the time-dependent nonlinear equation, and the stationary version can be found from the asymptotic behavior of the expansion. The solutions can be associated with reactive processes propagating along molecular chains, and their applicability to biophysical processes such as active transport phenomena and control circuit problems is discussed. There are also applications to cellular kinetics of clonogenic cell assays and spheroids.

  2. Neutron transport equation - indications on homogenization and neutron diffusion

    International Nuclear Information System (INIS)

    In PWR nuclear reactor, the practical study of the neutrons in the core uses diffusion equation to describe the problem. On the other hand, the most correct method to describe these neutrons is to use the Boltzmann equation, or neutron transport equation. In this paper, we give some theoretical indications to obtain a diffusion equation from the general transport equation, with some simplifying hypothesis. The work is organised as follows: (a) the most general formulations of the transport equation are presented: integro-differential equation and integral equation; (b) the theoretical approximation of this Boltzmann equation by a diffusion equation is introduced, by the way of asymptotic developments; (c) practical homogenization methods of transport equation is then presented. In particular, the relationships with some general and useful methods in neutronic are shown, and some homogenization methods in energy and space are indicated. A lot of other points of view or complements are detailed in the text or the remarks

  3. Distributed-order diffusion equations and multifractality: Models and solutions

    Science.gov (United States)

    Sandev, Trifce; Chechkin, Aleksei V.; Korabel, Nickolay; Kantz, Holger; Sokolov, Igor M.; Metzler, Ralf

    2015-10-01

    We study distributed-order time fractional diffusion equations characterized by multifractal memory kernels, in contrast to the simple power-law kernel of common time fractional diffusion equations. Based on the physical approach to anomalous diffusion provided by the seminal Scher-Montroll-Weiss continuous time random walk, we analyze both natural and modified-form distributed-order time fractional diffusion equations and compare the two approaches. The mean squared displacement is obtained and its limiting behavior analyzed. We derive the connection between the Wiener process, described by the conventional Langevin equation and the dynamics encoded by the distributed-order time fractional diffusion equation in terms of a generalized subordination of time. A detailed analysis of the multifractal properties of distributed-order diffusion equations is provided.

  4. Finite formulation of one-group neutron diffusion equation

    International Nuclear Information System (INIS)

    The objective of this paper is to introduce a new numerical method for neutron diffusion equation. This method produces the final algebraic form of the neutron diffusion equation by classifying the neutronic variables and using two kinds of cell complexes (primal and dual cells) without using the conventional differential form of the neutron diffusion equation. The convergence of the method is exactly same as finite element method with linear interpolation. (author)

  5. On an explicit finite difference method for fractional diffusion equations

    OpenAIRE

    S. B. Yuste; Acedo, L.

    2003-01-01

    A numerical method to solve the fractional diffusion equation, which could also be easily extended to many other fractional dynamics equations, is considered. These fractional equations have been proposed in order to describe anomalous transport characterized by non-Markovian kinetics and the breakdown of Fick's law. In this paper we combine the forward time centered space (FTCS) method, well known for the numerical integration of ordinary diffusion equations, with the Grunwald-Letnikov defin...

  6. Voter Model Perturbations and Reaction Diffusion Equations

    CERN Document Server

    Cox, J Theodore; Perkins, Edwin

    2011-01-01

    We consider particle systems that are perturbations of the voter model and show that when space and time are rescaled the system converges to a solution of a reaction diffusion equation in dimensions $d \\ge 3$. Combining this result with properties of the PDE, some methods arising from a low density super-Brownian limit theorem, and a block construction, we give general, and often asymptotically sharp, conditions for the existence of non-trivial stationary distributions, and for extinction of one type. As applications, we describe the phase diagrams of three systems when the parameters are close to the voter model: (i) a stochastic spatial Lotka-Volterra model of Neuhauser and Pacala, (ii) a model of the evolution of cooperation of Ohtsuki, Hauert, Lieberman, and Nowak, and (iii) a continuous time version of the non-linear voter model of Molofsky, Durrett, Dushoff, Griffeath, and Levin. The first application confirms a conjecture of Cox and Perkins and the second confirms a conjecture of Ohtsuki et al in the ...

  7. QUENCHING PROBLEMS OF DEGENERATE FUNCTIONAL REACTION-DIFFUSION EQUATION

    Institute of Scientific and Technical Information of China (English)

    2006-01-01

    This paper is concerned with the quenching problem of a degenerate functional reaction-diffusion equation. The quenching problem and global existence of solution for the reaction-diffusion equation are derived and, some results of the positive steady state solutions for functional elliptic boundary value are also presented.

  8. Diffusion equations and the time evolution of foreign exchange rates

    International Nuclear Information System (INIS)

    We investigate which type of diffusion equation is most appropriate to describe the time evolution of foreign exchange rates. We modify the geometric diffusion model assuming a non-exponential time evolution and the stochastic term is the sum of a Wiener noise and a jump process. We find the resulting diffusion equation to obey the Kramers–Moyal equation. Analytical solutions are obtained using the characteristic function formalism and compared with empirical data. The analysis focus on the first four central moments considering the returns of foreign exchange rate. It is shown that the proposed model offers a good improvement over the classical geometric diffusion model.

  9. The Riccati System and a Diffusion-Type Equation

    Directory of Open Access Journals (Sweden)

    Erwin Suazo

    2014-05-01

    Full Text Available We discuss a method of constructing solutions of the initial value problem for diffusion-type equations in terms of solutions of certain Riccati and Ermakov-type systems. A nonautonomous Burgers-type equation is also considered. Examples include, but are not limited to the Fokker-Planck equation in physics, the Black-Scholes equation and the Hull-White model in finance.

  10. Numerical solution of the diffusion equation. Part 1

    International Nuclear Information System (INIS)

    The finite difference method as an approximation to the analytic solution of the diffusion equation is described briefly. For higher dimension this finite difference method will result in linear equation system which can be further solved using various techniques. Some techniques of solving this linear equation are also discussed. The main problem encountered is not that of replacing the governing partial differential equation by the finite difference equations, but that of finding a solution to the system of linear equation formed by the approximation. (author)

  11. LAGRANGE STABILITY IN MEAN SQUARE OF STOCHASTIC REACTION DIFFUSION EQUATIONS

    Institute of Scientific and Technical Information of China (English)

    2006-01-01

    This work is devoted to the discussion of stochastic reaction diffusion equations and some new theorems on Lagrange stability in mean square of the solution are established via Lyapunov method which is nothing to be done in the past.

  12. Solutions of fractional diffusion equations by variation of parameters method

    Directory of Open Access Journals (Sweden)

    Mohyud-Din Syed Tauseef

    2015-01-01

    Full Text Available This article is devoted to establish a novel analytical solution scheme for the fractional diffusion equations. Caputo’s formulation followed by the variation of parameters method has been employed to obtain the analytical solutions. Following the derived analytical scheme, solution of the fractional diffusion equation for several initial functions has been obtained. Graphs are plotted to see the physical behavior of obtained solutions.

  13. Notes on Stefan-Maxwell Equation versus Grahan's Diffusion Law

    Institute of Scientific and Technical Information of China (English)

    2000-01-01

    Certain prerequisite information on the component fluxes is necessary for solution of the Stefan-Maxwell equation in multicomponent diffusion systems and the Graham's law of diffusion and effusion is often resorted for this purpose. This article addresses solution of the Stefan-Maxwell equation in binary gas systems and explores the necessary conditions for definite solution of concentration profiles and pertinent component fluxes. It is found that there are multiple solutions for component fluxes in contradiction to what specified by the Graham's law of diffusion. The theorem of minimum entropy production in the non-equilibrium thermodynamics is believed instructive in determining the stable steady state solution out of infinite multiple solutions possible under the specified conditions. It is suggested that only when the boundary condition of component concentration is symmetrical in an isothermal binary system, the counter-diffusion becomes equimolar. The Graham's law of diffusion seems not generally valid for the case of isothermal ordinary diffusion.

  14. Some Remarks on Some Strongly Coupled Reaction-Diffusion Equations

    OpenAIRE

    Diagana, Toka

    2003-01-01

    The primary goal of this paper is to characterize solutions to coupled reaction-diffusion systems. Indeed, we use operators theory to show that under suitable assumptions, then the solutions to the reaction-diffusion equations exist. As applications, we consider a mathematical model arising in Biology and in Chemistry.

  15. BOUNDARY LAYER AND VANISHING DIFFUSION LIMIT FOR NONLINEAR EVOLUTION EQUATIONS

    Institute of Scientific and Technical Information of China (English)

    彭艳

    2014-01-01

    In this paper, we consider an initial-boundary value problem for some nonlinear evolution equations with damping and diffusion. The main purpose is to investigate the boundary layer effect and the convergence rates as the diffusion parameterαgoes to zero.

  16. A Fluctuating Lattice Boltzmann Method for the Diffusion Equation

    CERN Document Server

    Wagner, Alexander J

    2016-01-01

    We derive a fluctuating lattice Boltzmann method for the diffusion equation. The derivation removes several shortcomings of previous derivations for fluctuating lattice Boltzmann methods for hydrodynamic systems. The comparative simplicity of this diffusive system highlights the basic features of this first exact derivation of a fluctuating lattice Boltzmann method.

  17. Retarding Sub- and Accelerating Super-Diffusion Governed by Distributed Order Fractional Diffusion Equations

    OpenAIRE

    Chechkin, A. V.; Gorenflo, R.; I. M. Sokolov

    2002-01-01

    We propose diffusion-like equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena, whose diffusion exponent varies with time and which, correspondingly, can not be viewed as self-affine random processes possessing a unique Hurst exponent. We prove the positivity of the solutions of the proposed equations and establish the relation to the Continuous Time Random Walk theory. We show that the distri...

  18. Exact solutions of some coupled nonlinear diffusion-reaction equations using auxiliary equation method

    Indian Academy of Sciences (India)

    Ranjit Kumar

    2012-09-01

    Travelling and solitary wave solutions of certain coupled nonlinear diffusion-reaction equations have been constructed using the auxiliary equation method. These equations arise in a variety of contexts not only in biological, chemical and physical sciences but also in ecological and social sciences.

  19. EXPONENTIAL ATTRACTOR FOR A CLASS OF NONCLASSICAL DIFFUSION EQUATION

    Institute of Scientific and Technical Information of China (English)

    尚亚东; 郭柏灵

    2003-01-01

    In this paper,we consider the asymptotic behavior of solutions for a class of nonclassical diffusion equation.We show the squeezing property and the existence of exponential attractor for this equation.We also make the estimates on its fractal dimension and exponential attraction.

  20. TRAJECTORY ATTRACTORS FOR NONCLASSICAL DIFFUSION EQUATIONS WITH FADING MEMORY

    Institute of Scientific and Technical Information of China (English)

    Yonghai WANG; Lingzhi WANG

    2013-01-01

    In this article,we consider the existence of trajectory and global attractors for nonclassical diffusion equations with linear fading memory.For this purpose,we will apply the method presented by Chepyzhov and Miranville [7,8],in which the authors provide some new ideas in describing the trajectory attractors for evolution equations with memory.

  1. The numerical simulation of convection delayed dominated diffusion equation

    Directory of Open Access Journals (Sweden)

    Mohan Kumar P. Murali

    2016-01-01

    Full Text Available In this paper, we propose a fitted numerical method for solving convection delayed dominated diffusion equation. A fitting factor is introduced and the model equation is discretized by cubic spline method. The error analysis is analyzed for the consider problem. The numerical examples are solved using the present method and compared the result with the exact solution.

  2. Higher Order and Fractional Diffusive Equations

    Directory of Open Access Journals (Sweden)

    D. Assante

    2015-07-01

    Full Text Available We discuss the solution of various generalized forms of the Heat Equation, by means of different tools ranging from the use of Hermite-Kampé de Fériet polynomials of higher and fractional order to operational techniques. We show that these methods are useful to obtain either numerical or analytical solutions.

  3. From Newton's Equation to Fractional Diffusion and Wave Equations

    Directory of Open Access Journals (Sweden)

    Vázquez Luis

    2011-01-01

    Full Text Available Fractional calculus represents a natural instrument to model nonlocal (or long-range dependence phenomena either in space or time. The processes that involve different space and time scales appear in a wide range of contexts, from physics and chemistry to biology and engineering. In many of these problems, the dynamics of the system can be formulated in terms of fractional differential equations which include the nonlocal effects either in space or time. We give a brief, nonexhaustive, panoramic view of the mathematical tools associated with fractional calculus as well as a description of some fields where either it is applied or could be potentially applied.

  4. Semianalytic Solution of Space-Time Fractional Diffusion Equation

    Directory of Open Access Journals (Sweden)

    A. Elsaid

    2016-01-01

    Full Text Available We study the space-time fractional diffusion equation with spatial Riesz-Feller fractional derivative and Caputo fractional time derivative. The continuation of the solution of this fractional equation to the solution of the corresponding integer order equation is proved. The series solution of this problem is obtained via the optimal homotopy analysis method (OHAM. Numerical simulations are presented to validate the method and to show the effect of changing the fractional derivative parameters on the solution behavior.

  5. Exact solutions for logistic reaction-diffusion equations in biology

    Science.gov (United States)

    Broadbridge, P.; Bradshaw-Hajek, B. H.

    2016-08-01

    Reaction-diffusion equations with a nonlinear source have been widely used to model various systems, with particular application to biology. Here, we provide a solution technique for these types of equations in N-dimensions. The nonclassical symmetry method leads to a single relationship between the nonlinear diffusion coefficient and the nonlinear reaction term; the subsequent solutions for the Kirchhoff variable are exponential in time (either growth or decay) and satisfy the linear Helmholtz equation in space. Example solutions are given in two dimensions for particular parameter sets for both quadratic and cubic reaction terms.

  6. Linear fractional diffusion-wave equation for scientists and engineers

    CERN Document Server

    Povstenko, Yuriy

    2015-01-01

    This book systematically presents solutions to the linear time-fractional diffusion-wave equation. It introduces the integral transform technique and discusses the properties of the Mittag-Leffler, Wright, and Mainardi functions that appear in the solutions. The time-nonlocal dependence between the flux and the gradient of the transported quantity with the “long-tail” power kernel results in the time-fractional diffusion-wave equation with the Caputo fractional derivative. Time-nonlocal generalizations of classical Fourier’s, Fick’s and Darcy’s laws are considered and different kinds of boundary conditions for this equation are discussed (Dirichlet, Neumann, Robin, perfect contact). The book provides solutions to the fractional diffusion-wave equation with one, two and three space variables in Cartesian, cylindrical and spherical coordinates. The respective sections of the book can be used for university courses on fractional calculus, heat and mass transfer, transport processes in porous media and ...

  7. Lattice Boltzmann method for the fractional advection-diffusion equation.

    Science.gov (United States)

    Zhou, J G; Haygarth, P M; Withers, P J A; Macleod, C J A; Falloon, P D; Beven, K J; Ockenden, M C; Forber, K J; Hollaway, M J; Evans, R; Collins, A L; Hiscock, K M; Wearing, C; Kahana, R; Villamizar Velez, M L

    2016-04-01

    Mass transport, such as movement of phosphorus in soils and solutes in rivers, is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or superdiffusion, and it is well described using a fractional advection-diffusion equation (FADE). The FADE finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems. However, solution to the FADE is difficult, and the existing numerical methods are complicated and inefficient. In this study, a fresh lattice Boltzmann method is developed for solving the fractional advection-diffusion equation (LabFADE). The FADE is transformed into an equation similar to an advection-diffusion equation and solved using the lattice Boltzmann method. The LabFADE has all the advantages of the conventional lattice Boltzmann method and avoids a complex solution procedure, unlike other existing numerical methods. The method has been validated through simulations of several benchmark tests: a point-source diffusion, a boundary-value problem of steady diffusion, and an initial-boundary-value problem of unsteady diffusion with the coexistence of source and sink terms. In addition, by including the effects of the skewness β, the fractional order α, and the single relaxation time τ, the accuracy and convergence of the method have been assessed. The numerical predictions are compared with the analytical solutions, and they indicate that the method is second-order accurate. The method presented will allow the FADE to be more widely applied to complex mass transport problems in science and engineering.

  8. Lattice Boltzmann method for the fractional advection-diffusion equation

    Science.gov (United States)

    Zhou, J. G.; Haygarth, P. M.; Withers, P. J. A.; Macleod, C. J. A.; Falloon, P. D.; Beven, K. J.; Ockenden, M. C.; Forber, K. J.; Hollaway, M. J.; Evans, R.; Collins, A. L.; Hiscock, K. M.; Wearing, C.; Kahana, R.; Villamizar Velez, M. L.

    2016-04-01

    Mass transport, such as movement of phosphorus in soils and solutes in rivers, is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or superdiffusion, and it is well described using a fractional advection-diffusion equation (FADE). The FADE finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems. However, solution to the FADE is difficult, and the existing numerical methods are complicated and inefficient. In this study, a fresh lattice Boltzmann method is developed for solving the fractional advection-diffusion equation (LabFADE). The FADE is transformed into an equation similar to an advection-diffusion equation and solved using the lattice Boltzmann method. The LabFADE has all the advantages of the conventional lattice Boltzmann method and avoids a complex solution procedure, unlike other existing numerical methods. The method has been validated through simulations of several benchmark tests: a point-source diffusion, a boundary-value problem of steady diffusion, and an initial-boundary-value problem of unsteady diffusion with the coexistence of source and sink terms. In addition, by including the effects of the skewness β , the fractional order α , and the single relaxation time τ , the accuracy and convergence of the method have been assessed. The numerical predictions are compared with the analytical solutions, and they indicate that the method is second-order accurate. The method presented will allow the FADE to be more widely applied to complex mass transport problems in science and engineering.

  9. Group classification of systems of non-linear reaction-diffusion equations with general diffusion matrix. II. Generalized Turing systems

    OpenAIRE

    Nikitin, A. G.

    2004-01-01

    Group classification of systems of two coupled nonlinear reaction-diffusion equation with a diagonal diffusion matrix is carried out. Symmetries of diffusion systems with singular diffusion matrix and additional first order derivative terms are described.

  10. Stochastic differential equations and diffusion processes

    CERN Document Server

    Ikeda, N

    1989-01-01

    Being a systematic treatment of the modern theory of stochastic integrals and stochastic differential equations, the theory is developed within the martingale framework, which was developed by J.L. Doob and which plays an indispensable role in the modern theory of stochastic analysis.A considerable number of corrections and improvements have been made for the second edition of this classic work. In particular, major and substantial changes are in Chapter III and Chapter V where the sections treating excursions of Brownian Motion and the Malliavin Calculus have been expanded and refined. Sectio

  11. From State Dependent Diffusion to Constant Diffusion in Stochastic Differential Equations by the Lamperti Transform

    DEFF Research Database (Denmark)

    Møller, Jan Kloppenborg; Madsen, Henrik

    This report describes methods to eliminate state dependent diffusion terms in Stochastic Differential Equations (SDEs). Transformations that leave the diffusion term of SDEs constant is important for simulation, and estimation. It is important for simulation because the Euler approximation...... convergence rate is faster, and for estimation because the Extended Kalman Filter equations are easier to implement than higher order filters needed in the case of state dependent diffusion terms. The general class of transformations which leaves the diffusion term independent of the state is called...

  12. Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation

    Science.gov (United States)

    Johnson, Mathew A.; Noble, Pascal; Rodrigues, L. Miguel; Zumbrun, Kevin

    2013-02-01

    In a companion paper, we established nonlinear stability with detailed diffusive rates of decay of spectrally stable periodic traveling-wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized modulation plus a localized ( L 1) perturbation. Here, we determine time-asymptotic behavior under such perturbations, showing that solutions consist of a leading order of a modulation whose parameter evolution is governed by an associated Whitham averaged equation.

  13. Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation

    CERN Document Server

    Johnson, Mathew; Rodrigues, L Miguel; Zumbrun, Kevin

    2011-01-01

    In a companion paper, we established nonlinear stability with detailed diffusive rates of decay of spectrally stable periodic traveling-wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized modulation plus a localized perturbation. Here, we determine time-asymptotic behavior under such perturbations, showing that solutions consist to leading order of a modulation whose parameter evolution is governed by an associated Whitham averaged equation.

  14. FINITE PROXIMATE METHOD FOR CONVECTION-DIFFUSION EQUATION

    Institute of Scientific and Technical Information of China (English)

    ZHAO Ming-deng; LI Tai-ru; HUAI Wen-xin; LI Liang-liang

    2008-01-01

    A finite proximate method was presented to solve the convection-diffusion equation in curvilinear grids. The method has characteristics of automatic upwind effect and the good stability. It was verified through exact solution and other calculation results of two-dimensional dam-break flow in a frictionless, horizontal channel. The calculation results are in good agreement with the exact solution and other calculation results, which show that the finite proximate method can be applied to solve the convection-diffusion equation directly not only in the rectangular grids, but also in the curvilinear grids.

  15. Traveling waves for a boundary reaction-diffusion equation

    CERN Document Server

    Caffarelli, L; Sire, Y

    2011-01-01

    We prove the existence of a traveling wave solution for a boundary reaction diffusion equation when the reaction term is the combustion nonlinearity with ignition temperature. A key role in the proof is plaid by an explicit formula for traveling wave solutions of a free boundary problem obtained as singular limit for the reaction-diffusion equation (the so-called high energy activation energy limit). This explicit formula, which is interesting in itself, also allows us to get an estimate on the decay at infinity of the traveling wave (which turns out to be faster than the usual exponential decay).

  16. Continuity, the Bloch-Torrey equation, and Diffusion MRI

    CERN Document Server

    Hall, Matt G

    2016-01-01

    The Bloch equation describes the evolution of classical particles tagged with a magnetisation vector in a strong magnetic field and is fundamental to many NMR and MRI contrast methods. The equation can be generalised to include the effects of spin motion by including a spin flux, which typically contains a Fickian diffusive term and/or a coherent velocity term. This form is known as the Bloch-Torrey equation, and is fundamental to MR modalities which are sensitive to spin dynamics such as diffusion MRI. Such modalities have received a great deal of interest in the research literature over the last few years, resulting in a huge range of models and methods. In this work we make make use of a more general Bloch-Torrey equation with a generalised flux term. We show that many commonly employed approaches in Diffusion MRI may be viewed as different choices for the flux terms in this equation. This viewpoint, although obvious theoretically, is not usually emphasised in the diffusion MR literature and points to inte...

  17. The Nonclassical Diffusion Approximation to the Nonclassical Linear Boltzmann Equation

    CERN Document Server

    Vasques, Richard

    2015-01-01

    We show that, by correctly selecting the probability distribution function $p(s)$ for a particle's distance-to-collision, the nonclassical diffusion equation can be represented exactly by the nonclassical linear Boltzmann equation for an infinite homogeneous medium. This choice of $p(s)$ preserves the $true$ mean-squared free path of the system, which sheds new light on the results obtained in previous work.

  18. Group classification of systems of non-linear reaction-diffusion equations with general diffusion matrix. I. Generalized Landau-Ginzburg equations

    OpenAIRE

    Nikitin, A. G.

    2004-01-01

    Group classification of the generalized complex Ginzburg-Landau equations is presented. An approach to group classification of systems of reaction-diffusion equations with general diffusion matrix is developed.

  19. Biomixing by chemotaxis and enhancement of biological reactions

    CERN Document Server

    Kiselev, Alexander

    2011-01-01

    Many processes in biology involve both reactions and chemotaxis. However, to the best of our knowledge, the question of interaction between chemotaxis and reactions has not yet been addressed either analytically or numerically. We consider a model with a single density function involving diffusion, advection, chemotaxis, and absorbing reaction. The model is motivated, in particular, by studies of coral broadcast spawning, where experimental observations of the efficiency of fertilization rates significantly exceed the data obtained from numerical models that do not take chemotaxis (attraction of sperm gametes by a chemical secreted by egg gametes) into account. We prove that in the framework of our model, chemotaxis plays a crucial role. There is a rigid limit to how much the fertilization efficiency can be enhanced if there is no chemotaxis but only advection and diffusion. On the other hand, when chemotaxis is present, the fertilization rate can be arbitrarily close to being complete provided that the chemo...

  20. High order backward discretization of the neutron diffusion equation

    International Nuclear Information System (INIS)

    Fast codes capable of dealing with three-dimensional geometries, are needed to be able to simulate spatially complicated transients in a nuclear reactor. We propose a new discretization technique for the time integration of the neutron diffusion equation, based on the backward difference formulas for systems of stiff ordinary differential equations. This method needs to solve a system of linear equations for each integration step, and for this purpose, we have developed an iterative block algorithm combined with a variational acceleration technique. We tested the algorithm with two benchmark problems, and compared the results with those provided by other codes, concluding that the performance and overall agreement are very good. (author)

  1. Kinetic methods for the diffusion equations in nuclear PWR

    International Nuclear Information System (INIS)

    The neutronic diffusion equations are used to calculate the flux in the core of a nuclear reactor. We can solve these equations with direct methods, but these methods are too expensive for slow transient phenomenons. So, according to a method developed by Ott and Meneley, we split the flux into a product of two functions: the first one is only time dependent and is supposed to change rapidly. The second one is time and space dependent but is supposed to vary slowly with time. Numerical results are given and compared to those obtained by a direct resolution of the equations and by experiments

  2. Gradient estimates for a nonlinear diffusion equation on complete manifolds

    OpenAIRE

    Wu, Jia-Yong

    2010-01-01

    Let $(M,g)$ be a complete non-compact Riemannian manifold with the $m$-dimensional Bakry-\\'{E}mery Ricci curvature bounded below by a non-positive constant. In this paper, we give a localized Hamilton-type gradient estimate for the positive smooth bounded solutions to the following nonlinear diffusion equation \\[ u_t=\\Delta u-\

  3. THE INTERIOR LAYER SOLUTION TO NONLOCAL REACTION DIFFUSION EQUATIONS

    Institute of Scientific and Technical Information of China (English)

    2011-01-01

    An initial boundary value problem of semilinear nonlocal reaction diffusion equations is considered.Under some suitable conditions,using the asymptotic theory,the existence and asymptotic behavior of the interior layer solution to the initial boundary value problem are studied.

  4. Target Pattern Waves in Specific Reaction-Diffusion Equation

    Institute of Scientific and Technical Information of China (English)

    ZHOU Tian-Shou; TANG Yun

    2002-01-01

    This paper uses two-timing to solve a class of reaction-diffusion equations of Oregonator and constructsformally stable target pattern solutions to leading term in small parameter, which can be used to account for someobserved experimental facts. In addition, it also gives the explicit expression of the period corresponding to the targetwaves.

  5. Explicit solutions of fractional diffusion equations via Generalized Gamma Convolution

    CERN Document Server

    D'Ovidio, Mirko

    2010-01-01

    In this paper we deal with Mellin convolution of generalized Gamma densities which brings to integrals of modified Bessel functions of the second kind. Such convolutions allow us to write explicitly the solutions of the time-fractional diffusion equations involving the adjoint operators of a square Bessel process and a Bessel process.

  6. DISCONTINUOUS FINITE ELEMENT METHOD FOR CONVECTION-DIFFUSION EQUATIONS

    Institute of Scientific and Technical Information of China (English)

    Abdellatif Agouzal

    2000-01-01

    A discontinuous finite element method for convection-diffusion equations is proposed and analyzed. This scheme is designed to produce an approximate solution which is completely discontinuous. Optimal order of convergence is obtained for model problem. This is the same convergence rate known for the classical methods.

  7. Diffusive Limits of the Master Equation in Inhomogeneous Media

    CERN Document Server

    Sattin, F; Salasnich, L

    2015-01-01

    In inhomogeneous environments several expressions for the flux of a diffusing quantity may apply--from Fick-Fourier's to Fokker-Planck's--depending upon the system studied. The integro-differential Master Equation (ME) provides a fairly generic framework for describing the dynamics of arbitrary systems driven by stochastic rules. Diffusive dynamics does arise as long-wavelength limit of the ME. However, while it is straightforward to obtain a diffusion equation with Fokker-Planck flux, its Fick-Fourier counterpart has never been worked out from the ME. In this work we show under which hypothesis the Fick's flux can actually be recovered from the ME. Analytical considerations are supported by explicit computer models.

  8. Langevin and diffusion equation of turbulent fluid flow

    Science.gov (United States)

    Brouwers, J. J. H.

    2010-08-01

    A derivation of the Langevin and diffusion equations describing the statistics of fluid particle displacement and passive admixture in turbulent flow is presented. Use is made of perturbation expansions. The small parameter is the inverse of the Kolmogorov constant C 0 , which arises from Lagrangian similarity theory. The value of C 0 in high Reynolds number turbulence is 5-6. To achieve sufficient accuracy, formulations are not limited to terms of leading order in C0 - 1 including terms next to leading order in C0 - 1 as well. Results of turbulence theory and statistical mechanics are invoked to arrive at the descriptions of the Langevin and diffusion equations, which are unique up to truncated terms of O ( C0 - 2 ) in displacement statistics. Errors due to truncation are indicated to amount to a few percent. The coefficients of the presented Langevin and diffusion equations are specified by fixed-point averages of the Eulerian velocity field. The equations apply to general turbulent flow in which fixed-point Eulerian velocity statistics are non-Gaussian to a degree of O ( C0 - 1 ) . The equations provide the means to calculate and analyze turbulent dispersion of passive or almost passive admixture such as fumes, smoke, and aerosols in areas ranging from atmospheric fluid motion to flows in engineering devices.

  9. Kinetic equations for diffusion in the presence of entropic barriers.

    Science.gov (United States)

    Reguera, D; Rubí, J M

    2001-12-01

    We use the mesoscopic nonequilibrium thermodynamics theory to derive the general kinetic equation of a system in the presence of potential barriers. The result is applied to a description of the evolution of systems whose dynamics is influenced by entropic barriers. We analyze in detail the case of diffusion in a domain of irregular geometry in which the presence of the boundaries induces an entropy barrier when approaching the exact dynamics by a coarsening of the description. The corresponding kinetic equation, named the Fick-Jacobs equation, is obtained, and its validity is generalized through the formulation of a scaling law for the diffusion coefficient which depends on the shape of the boundaries. The method we propose can be useful to analyze the dynamics of systems at the nanoscale where the presence of entropy barriers is a common feature. PMID:11736170

  10. Unstructured Grids and the Multigroup Neutron Diffusion Equation

    Directory of Open Access Journals (Sweden)

    German Theler

    2013-01-01

    Full Text Available The neutron diffusion equation is often used to perform core-level neutronic calculations. It consists of a set of second-order partial differential equations over the spatial coordinates that are, both in the academia and in the industry, usually solved by discretizing the neutron leakage term using a structured grid. This work introduces the alternatives that unstructured grids can provide to aid the engineers to solve the neutron diffusion problem and gives a brief overview of the variety of possibilities they offer. It is by understanding the basic mathematics that lie beneath the equations that model real physical systems; better technical decisions can be made. It is in this spirit that this paper is written, giving a first introduction to the basic concepts which can be incorporated into core-level neutron flux computations. A simple two-dimensional homogeneous circular reactor is solved using a coarse unstructured grid in order to illustrate some basic differences between the finite volumes and the finite elements method. Also, the classic 2D IAEA PWR benchmark problem is solved for eighty combinations of symmetries, meshing algorithms, basic geometric entities, discretization schemes, and characteristic grid lengths, giving even more insight into the peculiarities that arise when solving the neutron diffusion equation using unstructured grids.

  11. Optimal prediction for moment models: crescendo diffusion and reordered equations

    Science.gov (United States)

    Seibold, Benjamin; Frank, Martin

    2009-12-01

    A direct numerical solution of the radiative transfer equation or any kinetic equation is typically expensive, since the radiative intensity depends on time, space and direction. An expansion in the direction variables yields an equivalent system of infinitely many moments. A fundamental problem is how to truncate the system. Various closures have been presented in the literature. We want to generally study the moment closure within the framework of optimal prediction, a strategy to approximate the mean solution of a large system by a smaller system, for radiation moment systems. We apply this strategy to radiative transfer and show that several closures can be re-derived within this framework, such as P N , diffusion, and diffusion correction closures. In addition, the formalism gives rise to new parabolic systems, the reordered P N equations, that are similar to the simplified P N equations. Furthermore, we propose a modification to existing closures. Although simple and with no extra cost, this newly derived crescendo diffusion yields better approximations in numerical tests.

  12. Image restoration using total variation and anisotropic diffusion equation

    Institute of Scientific and Technical Information of China (English)

    LI Min; FENG Xiangchu

    2007-01-01

    This Paper proposes a new model for the image restoration which combines the total variation minimization with the"pure"anisotropic diffusion equation of Alvarez and Morel.According to the introduction of new diffusion term,this model can not only remove noise but also enhance edges and keep their locality.And it can also keeD textures and large-scale fine features that are not characterized by edges.Due to these favorable characteristics,the processed images turn much clearer and smoother,meanwhile,their significant details are kept,which results in appealing vision.

  13. Langevin equation with fluctuating diffusivity: A two-state model.

    Science.gov (United States)

    Miyaguchi, Tomoshige; Akimoto, Takuma; Yamamoto, Eiji

    2016-07-01

    Recently, anomalous subdiffusion, aging, and scatter of the diffusion coefficient have been reported in many single-particle-tracking experiments, though the origins of these behaviors are still elusive. Here, as a model to describe such phenomena, we investigate a Langevin equation with diffusivity fluctuating between a fast and a slow state. Namely, the diffusivity follows a dichotomous stochastic process. We assume that the sojourn time distributions of these two states are given by power laws. It is shown that, for a nonequilibrium ensemble, the ensemble-averaged mean-square displacement (MSD) shows transient subdiffusion. In contrast, the time-averaged MSD shows normal diffusion, but an effective diffusion coefficient transiently shows aging behavior. The propagator is non-Gaussian for short time and converges to a Gaussian distribution in a long-time limit; this convergence to Gaussian is extremely slow for some parameter values. For equilibrium ensembles, both ensemble-averaged and time-averaged MSDs show only normal diffusion and thus we cannot detect any traces of the fluctuating diffusivity with these MSDs. Therefore, as an alternative approach to characterizing the fluctuating diffusivity, the relative standard deviation (RSD) of the time-averaged MSD is utilized and it is shown that the RSD exhibits slow relaxation as a signature of the long-time correlation in the fluctuating diffusivity. Furthermore, it is shown that the RSD is related to a non-Gaussian parameter of the propagator. To obtain these theoretical results, we develop a two-state renewal theory as an analytical tool. PMID:27575079

  14. Langevin equation with fluctuating diffusivity: A two-state model

    Science.gov (United States)

    Miyaguchi, Tomoshige; Akimoto, Takuma; Yamamoto, Eiji

    2016-07-01

    Recently, anomalous subdiffusion, aging, and scatter of the diffusion coefficient have been reported in many single-particle-tracking experiments, though the origins of these behaviors are still elusive. Here, as a model to describe such phenomena, we investigate a Langevin equation with diffusivity fluctuating between a fast and a slow state. Namely, the diffusivity follows a dichotomous stochastic process. We assume that the sojourn time distributions of these two states are given by power laws. It is shown that, for a nonequilibrium ensemble, the ensemble-averaged mean-square displacement (MSD) shows transient subdiffusion. In contrast, the time-averaged MSD shows normal diffusion, but an effective diffusion coefficient transiently shows aging behavior. The propagator is non-Gaussian for short time and converges to a Gaussian distribution in a long-time limit; this convergence to Gaussian is extremely slow for some parameter values. For equilibrium ensembles, both ensemble-averaged and time-averaged MSDs show only normal diffusion and thus we cannot detect any traces of the fluctuating diffusivity with these MSDs. Therefore, as an alternative approach to characterizing the fluctuating diffusivity, the relative standard deviation (RSD) of the time-averaged MSD is utilized and it is shown that the RSD exhibits slow relaxation as a signature of the long-time correlation in the fluctuating diffusivity. Furthermore, it is shown that the RSD is related to a non-Gaussian parameter of the propagator. To obtain these theoretical results, we develop a two-state renewal theory as an analytical tool.

  15. A Continuum of Extinction Rates for the Fast Diffusion Equation

    OpenAIRE

    Fila, Marek; Vazquez, Juan Luis; Winkler, Michael

    2010-01-01

    We find a continuum of extinction rates for solutions $u(y,\\tau)\\ge 0$ of the fast diffusion equation $u_\\tau=\\Delta u^m$ in a subrange of exponents $m\\in (0,1)$. The equation is posed in $\\ren$ for times up to the extinction time $T>0$. The rates take the form $\\|u(\\cdot,\\tau)\\|_\\infty\\sim (T-\\tau)^\\theta$ \\ for a whole interval of $\\theta>0$. These extinction rates depend explicitly on the spatial decay rates of initial data.

  16. Simple jumping process with memory: Transport equation and diffusion

    Science.gov (United States)

    Kamińska, A.; Srokowski, T.

    2004-06-01

    We present a stochastic jumping process, defined in terms of jump-size probability density and jumping rate, which is a generalization of the well-known kangaroo process. The definition takes into account two process values: after and before the jump. Therefore, the process is able to preserve memory about its previous values. It possesses a simple stationary limit. Its master equation is interpreted as the kinetic equation with variable collision rate. The process can be easily applied to model systems which relax to distributions other than Maxwellian. The case of a constant jumping rate corresponds to the diffusion process, either normal or ballistic.

  17. Algorithm refinement for stochastic partial differential equations I. linear diffusion

    CERN Document Server

    Alexander, F J; Tartakovsky, D M

    2002-01-01

    A hybrid particle/continuum algorithm is formulated for Fickian diffusion in the fluctuating hydrodynamic limit. The particles are taken as independent random walkers; the fluctuating diffusion equation is solved by finite differences with deterministic and white-noise fluxes. At the interface between the particle and continuum computations the coupling is by flux matching, giving exact mass conservation. This methodology is an extension of Adaptive Mesh and Algorithm Refinement to stochastic partial differential equations. Results from a variety of numerical experiments are presented for both steady and time-dependent scenarios. In all cases the mean and variance of density are captured correctly by the stochastic hybrid algorithm. For a nonstochastic version (i.e., using only deterministic continuum fluxes) the mean density is correct, but the variance is reduced except in particle regions away from the interface. Extensions of the methodology to fluid mechanics applications are discussed.

  18. Support Operators Method for the Diffusion Equation in Multiple Materials

    Energy Technology Data Exchange (ETDEWEB)

    Winters, Andrew R. [Los Alamos National Laboratory; Shashkov, Mikhail J. [Los Alamos National Laboratory

    2012-08-14

    A second-order finite difference scheme for the solution of the diffusion equation on non-uniform meshes is implemented. The method allows the heat conductivity to be discontinuous. The algorithm is formulated on a one dimensional mesh and is derived using the support operators method. A key component of the derivation is that the discrete analog of the flux operator is constructed to be the negative adjoint of the discrete divergence, in an inner product that is a discrete analog of the continuum inner product. The resultant discrete operators in the fully discretized diffusion equation are symmetric and positive definite. The algorithm is generalized to operate on meshes with cells which have mixed material properties. A mechanism to recover intermediate temperature values in mixed cells using a limited linear reconstruction is introduced. The implementation of the algorithm is verified and the linear reconstruction mechanism is compared to previous results for obtaining new material temperatures.

  19. Wavelets method for the time fractional diffusion-wave equation

    Energy Technology Data Exchange (ETDEWEB)

    Heydari, M.H., E-mail: heydari@stu.yazd.ac.ir [Faculty of Mathematics, Yazd University, Yazd (Iran, Islamic Republic of); The Laboratory of Quantum Information Processing, Yazd University, Yazd (Iran, Islamic Republic of); Hooshmandasl, M.R., E-mail: hooshmandasl@yazd.ac.ir [Faculty of Mathematics, Yazd University, Yazd (Iran, Islamic Republic of); The Laboratory of Quantum Information Processing, Yazd University, Yazd (Iran, Islamic Republic of); Maalek Ghaini, F.M., E-mail: maalek@yazd.ac.ir [Faculty of Mathematics, Yazd University, Yazd (Iran, Islamic Republic of); The Laboratory of Quantum Information Processing, Yazd University, Yazd (Iran, Islamic Republic of); Cattani, C., E-mail: ccattani@unisa.it [Department of Mathematics, University of Salerno, Fisciano (Italy)

    2015-01-23

    In this paper, an efficient and accurate computational method based on the Legendre wavelets (LWs) is proposed for solving the time fractional diffusion-wave equation (FDWE). To this end, a new fractional operational matrix (FOM) of integration for the LWs is derived. The LWs and their FOM of integration are used to transform the problem under consideration into a linear system of algebraic equations, which can be simply solved to achieve the solution of the problem. The proposed method is very convenient for solving such problems, since the initial and boundary conditions are taken into account automatically. - Highlights: • A new operational matrix of fractional integration for the LWs is derived. • A new method based on the LWs is proposed for the time FDWE. • The paper contains some useful properties of the LWs. • The proposed method can be applied for fractional sub-diffusion systems. • The proposed method can be extended for fourth-order FDWE.

  20. Reaction diffusion equation with spatio-temporal delay

    Science.gov (United States)

    Zhao, Zhihong; Rong, Erhua

    2014-07-01

    We investigate reaction-diffusion equation with spatio-temporal delays, the global existence, uniqueness and asymptotic behavior of solutions for which in relation to constant steady-state solution, included in the region of attraction of a stable steady solution. It is shown that if the delay reaction function satisfies some conditions and the system possesses a pair of upper and lower solutions then there exists a unique global solution. In terms of the maximal and minimal constant solutions of the corresponding steady-state problem, we get the asymptotic stability of reaction-diffusion equation with spatio-temporal delay. Applying this theory to Lotka-Volterra model with spatio-temporal delay, we get the global solution asymptotically tend to the steady-state problem's steady-state solution.

  1. Maximum Principles for Discrete and Semidiscrete Reaction-Diffusion Equation

    Directory of Open Access Journals (Sweden)

    Petr Stehlík

    2015-01-01

    Full Text Available We study reaction-diffusion equations with a general reaction function f on one-dimensional lattices with continuous or discrete time ux′  (or  Δtux=k(ux-1-2ux+ux+1+f(ux, x∈Z. We prove weak and strong maximum and minimum principles for corresponding initial-boundary value problems. Whereas the maximum principles in the semidiscrete case (continuous time exhibit similar features to those of fully continuous reaction-diffusion model, in the discrete case the weak maximum principle holds for a smaller class of functions and the strong maximum principle is valid in a weaker sense. We describe in detail how the validity of maximum principles depends on the nonlinearity and the time step. We illustrate our results on the Nagumo equation with the bistable nonlinearity.

  2. UPWIND SPLITTING SCHEME FOR CONVECTION-DIFFUSION EQUATIONS

    Institute of Scientific and Technical Information of China (English)

    2000-01-01

    A new class of numerical schemes is proposed to solve convection-diffusion equations by combining the upwind technique and the method of operator splitting. For every time step, the multi-dimensional approximation is performed in several independent directions alternatively, while the upwind technique is applied to treat the convection term in every individual direction. This scheme possesses maximum principle. Stability and convergence are analysed by energy method.

  3. Nonlocal Symmetries to Systems of Nonlinear Diffusion Equations

    Institute of Scientific and Technical Information of China (English)

    DING Lu; QU Chang-Zheng; ZHENG Zhu-Jun; KANG Jing

    2008-01-01

    In this paper, we study potential symmetries to certain systems of nonlinear diffusion equations. Those systems have physical applications in soil science, mathematical biology, and invariant curve flows in R3. Lie point symmetries of the potential system, which cannot be projected to vector fields of the given dependent and independent variables, yield potential symmetries. The class of the system that admits potential symmetries is expanded.

  4. HEXAN - a hexagonal nodal code for solving the diffusion equation

    International Nuclear Information System (INIS)

    This report describes the theory of and provides a user's manual for the HEXAN program, which is a nodal program for the solution of the few-group diffusion equation in hexagonal geometry. Based upon symmetry considerations, the theory provides an analytical solution in a homogeneous node. WWER and HTGR test problem solutions are presented. The equivalence of the finite-difference scheme and the response matrix method is proven. The properties of a symmetric node's response matrix are investigated. (author)

  5. Stability of planar diffusion wave for nonlinear evolution equation

    Institute of Scientific and Technical Information of China (English)

    2012-01-01

    It is known that the one-dimensional nonlinear heat equation ut = f(u)x1x1,f'(u) 0,u(±∞,t) = u±,u+ = u_ has a unique self-similar solution u(x1/1+t).In multi-dimensional space,u(x1/1+t) is called a planar diffusion wave.In the first part of the present paper,it is shown that under some smallness conditions,such a planar diffusion wave is nonlinearly stable for the nonlinear heat equation:ut-△f(u) = 0,x ∈ Rn.The optimal time decay rate is obtained.In the second part of this paper,it is further shown that this planar diffusion wave is still nonlinearly stable for the quasilinear wave equation with damping:utt + utt+ △f(u) = 0,x ∈ Rn.The time decay rate is also obtained.The proofs are given by an elementary energy method.

  6. Reaction rates for a generalized reaction-diffusion master equation.

    Science.gov (United States)

    Hellander, Stefan; Petzold, Linda

    2016-01-01

    It has been established that there is an inherent limit to the accuracy of the reaction-diffusion master equation. Specifically, there exists a fundamental lower bound on the mesh size, below which the accuracy deteriorates as the mesh is refined further. In this paper we extend the standard reaction-diffusion master equation to allow molecules occupying neighboring voxels to react, in contrast to the traditional approach, in which molecules react only when occupying the same voxel. We derive reaction rates, in two dimensions as well as three dimensions, to obtain an optimal match to the more fine-grained Smoluchowski model and show in two numerical examples that the extended algorithm is accurate for a wide range of mesh sizes, allowing us to simulate systems that are intractable with the standard reaction-diffusion master equation. In addition, we show that for mesh sizes above the fundamental lower limit of the standard algorithm, the generalized algorithm reduces to the standard algorithm. We derive a lower limit for the generalized algorithm which, in both two dimensions and three dimensions, is of the order of the reaction radius of a reacting pair of molecules.

  7. Reaction rates for a generalized reaction-diffusion master equation

    Science.gov (United States)

    Hellander, Stefan; Petzold, Linda

    2016-01-01

    It has been established that there is an inherent limit to the accuracy of the reaction-diffusion master equation. Specifically, there exists a fundamental lower bound on the mesh size, below which the accuracy deteriorates as the mesh is refined further. In this paper we extend the standard reaction-diffusion master equation to allow molecules occupying neighboring voxels to react, in contrast to the traditional approach, in which molecules react only when occupying the same voxel. We derive reaction rates, in two dimensions as well as three dimensions, to obtain an optimal match to the more fine-grained Smoluchowski model and show in two numerical examples that the extended algorithm is accurate for a wide range of mesh sizes, allowing us to simulate systems that are intractable with the standard reaction-diffusion master equation. In addition, we show that for mesh sizes above the fundamental lower limit of the standard algorithm, the generalized algorithm reduces to the standard algorithm. We derive a lower limit for the generalized algorithm which, in both two dimensions and three dimensions, is of the order of the reaction radius of a reacting pair of molecules.

  8. Optimal prediction for moment models: Crescendo diffusion and reordered equations

    CERN Document Server

    Seibold, Benjamin

    2009-01-01

    A direct numerical solution of the radiative transfer equation or any kinetic equation is typically expensive, since the radiative intensity depends on time, space and direction. An expansion in the direction variables yields an equivalent system of infinitely many moments. A fundamental problem is how to truncate the system. Various closures have been presented in the literature. We want to study moment closure generally within the framework of optimal prediction, a strategy to approximate the mean solution of a large system by a smaller system, for radiation moment systems. We apply this strategy to radiative transfer and show that several closures can be re-derived within this framework, e.g. $P_N$, diffusion, and diffusion correction closures. In addition, the formalism gives rise to new parabolic systems, the reordered $P_N$ equations, that are similar to the simplified $P_N$ equations. Furthermore, we propose a modification to existing closures. Although simple and with no extra cost, this newly derived...

  9. Analysis of a mixed space-time diffusion equation

    Science.gov (United States)

    Momoniat, Ebrahim

    2015-06-01

    An energy method is used to analyze the stability of solutions of a mixed space-time diffusion equation that has application in the unidirectional flow of a second-grade fluid and the distribution of a compound Poisson process. Solutions to the model equation satisfying Dirichlet boundary conditions are proven to dissipate total energy and are hence stable. The stability of asymptotic solutions satisfying Neumann boundary conditions coincides with the condition for the positivity of numerical solutions of the model equation from a Crank-Nicolson scheme. The Crank-Nicolson scheme is proven to yield stable numerical solutions for both Dirichlet and Neumann boundary conditions for positive values of the critical parameter. Numerical solutions are compared to analytical solutions that are valid on a finite domain.

  10. A mixed finite element method for nonlinear diffusion equations

    KAUST Repository

    Burger, Martin

    2010-01-01

    We propose a mixed finite element method for a class of nonlinear diffusion equations, which is based on their interpretation as gradient flows in optimal transportation metrics. We introduce an appropriate linearization of the optimal transport problem, which leads to a mixed symmetric formulation. This formulation preserves the maximum principle in case of the semi-discrete scheme as well as the fully discrete scheme for a certain class of problems. In addition solutions of the mixed formulation maintain exponential convergence in the relative entropy towards the steady state in case of a nonlinear Fokker-Planck equation with uniformly convex potential. We demonstrate the behavior of the proposed scheme with 2D simulations of the porous medium equations and blow-up questions in the Patlak-Keller-Segel model. © American Institute of Mathematical Sciences.

  11. A Nonlinear Singularly Perturbed Problem for Reaction Diffusion Equations with Boundary Perturbation

    Institute of Scientific and Technical Information of China (English)

    Jia-qi Mo; Wan-tao Lin

    2005-01-01

    A nonlinear singularly perturbed problems for reaction diffusion equation with boundary perturbation is considered. Under suitable conditions, the asymptotic behavior of solution for the initial boundary value problems of reaction diffusion equations is studied using the theory of differential inequalities.

  12. Group Foliation Method and Functional Separation of Variables to Nonlinear Diffusion Equations

    Institute of Scientific and Technical Information of China (English)

    QU Chang-Zheng; ZHANG Shun-Li

    2005-01-01

    @@ Generalized functional separation of variables to nonlinear diffusion equations is studied in terms of the extended group foliation method. A complete classification for the nonlinear diffusion equation with source term which admits functional separable solutions is presented.

  13. COUPLED CHEMOTAXIS FLUID MODEL

    KAUST Repository

    LORZ, ALEXANDER

    2010-06-01

    We consider a model system for the collective behavior of oxygen-driven swimming bacteria in an aquatic fluid. In certain parameter regimes, such suspensions of bacteria feature large-scale convection patterns as a result of the hydrodynamic interaction between bacteria. The presented model consist of a parabolicparabolic chemotaxis system for the oxygen concentration and the bacteria density coupled to an incompressible Stokes equation for the fluid driven by a gravitational force of the heavier bacteria. We show local existence of weak solutions in a bounded domain in d, d = 2, 3 with no-flux boundary condition and in 2 in the case of inhomogeneous Dirichlet conditions for the oxygen. © 2010 World Scientific Publishing Company.

  14. The Singular Limit of a Chemotaxis-Growth System with General Initial Data

    CERN Document Server

    Alfaro, Matthieu

    2009-01-01

    We study the singular limit of a system of partial differential equations which is a model for an aggregation of amoebae subjected to three effects: diffusion, growth and chemotaxis. The limit problem involves motion by mean curvature together with a nonlocal drift term. We consider rather general initial data. We prove a generation of interface property and study the motion of interfaces. We also obtain an optimal estimate of the thickness and the location of the transition layer that develops.

  15. Interior corner problem for the neutron diffusion equation

    International Nuclear Information System (INIS)

    The qualitative and quantitative behavior of the solution of the neutron diffusion equation at interior corners of two different materials is investigated. This is done by assuming regionwise constant coefficients and sources, reducing the problem to dimensionless form and solving the Neumann problem over a circular disk. The general analytic solution is given, as well as special solutions. A series of parameter studies have been made which include the extreme cases that can occur in neutron diffusion. The results including eigenvalues, eigenfunctions, Fourier coefficients and flux profiles are tabulated and presented in graphical form. It is shown how to select appropriate function spaces to obtain good numerical methods for coarse network calculations. (33 figures, 25 tables) (U.S.)

  16. Time Fractional Diffusion Equations and Analytical Solvable Models

    Science.gov (United States)

    Bakalis, Evangelos; Zerbetto, Francesco

    2016-08-01

    The anomalous diffusion of a particle that moves in complex environments is analytically studied by means of the time fractional diffusion equation. The influence on the dynamics of a random moving particle caused by a uniform external field is taken into account. We extract analytical solutions in terms either of the Mittag-Leffler functions or of the M- Wright function for the probability distribution, for the velocity autocorrelation function as well as for the mean and the mean square displacement. Discussion of the applicability of the model to real systems is made in order to provide new insight of the medium from the analysis of the motion of a particle embedded in it.

  17. On the Reaction Diffusion Master Equation in the Microscopic Limit

    CERN Document Server

    Hellander, Stefan; Petzold, Linda

    2011-01-01

    Stochastic modeling of reaction-diffusion kinetics has emerged as a powerful theoretical tool in the study of biochemical reaction networks. Two frequently employed models are the particle-tracking Smoluchowski framework and the on-lattice Reaction-Diffusion Master Equation (RDME) framework. As the mesh size goes from coarse to fine, the RDME initially becomes more accurate. However, recent developments have shown that it will become increasingly inaccurate compared to the Smoluchowski model as the lattice spacing becomes very fine. In this paper we give a new, general and simple argument for why the RDME breaks down. Our analysis reveals a hard limit on the voxel size for which no local RDME can agree with the Smoluchowski model.

  18. Reaction-diffusion master equation in the microscopic limit

    Science.gov (United States)

    Hellander, Stefan; Hellander, Andreas; Petzold, Linda

    2012-04-01

    Stochastic modeling of reaction-diffusion kinetics has emerged as a powerful theoretical tool in the study of biochemical reaction networks. Two frequently employed models are the particle-tracking Smoluchowski framework and the on-lattice reaction-diffusion master equation (RDME) framework. As the mesh size goes from coarse to fine, the RDME initially becomes more accurate. However, recent developments have shown that it will become increasingly inaccurate compared to the Smoluchowski model as the lattice spacing becomes very fine. Here we give a general and simple argument for why the RDME breaks down. Our analysis reveals a hard limit on the voxel size for which no local RDME can agree with the Smoluchowski model and lets us quantify this limit in two and three dimensions. In this light we review and discuss recent work in which the RDME has been modified in different ways in order to better agree with the microscale model for very small voxel sizes.

  19. THE NONLINEAR NONLOCAL SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS

    Institute of Scientific and Technical Information of China (English)

    莫嘉琪; 朱江

    2003-01-01

    A class of nonlinear nonlocal for singularly perturbed Robin initial boundaryvalue problems for reaction diffusion equations is considered. Under suitable conditions,firstly, the outer solution of the original problem is obtained, secondly, using the stretchedvariable, the composing expansion method and the expanding theory of power series theinitial layer is constructed, finally, using the theory of differential inequalities theasymptotic behavior of solution for the initial boundary value problems are studied andeducing some relational inequalities the existence and uniqueness of solution for the originalproblem and the uniformly valid asymptotic estimation is discussed.

  20. IDENTIFYING AN UNKNOWN SOURCE IN SPACE-FRACTIONAL DIFFUSION EQUATION

    Institute of Scientific and Technical Information of China (English)

    杨帆; 傅初黎; 李晓晓

    2014-01-01

    In this paper, we identify a space-dependent source for a fractional diffusion equation. This problem is ill-posed, i.e., the solution (if it exists) does not depend continu-ously on the data. The generalized Tikhonov regularization method is proposed to solve this problem. An a priori error estimate between the exact solution and its regularized approxi-mation is obtained. Moreover, an a posteriori parameter choice rule is proposed and a stable error estimate is also obtained. Numerical examples are presented to illustrate the validity and effectiveness of this method.

  1. Dynamic hysteresis modeling including skin effect using diffusion equation model

    Science.gov (United States)

    Hamada, Souad; Louai, Fatima Zohra; Nait-Said, Nasreddine; Benabou, Abdelkader

    2016-07-01

    An improved dynamic hysteresis model is proposed for the prediction of hysteresis loop of electrical steel up to mean frequencies, taking into account the skin effect. In previous works, the analytical solution of the diffusion equation for low frequency (DELF) was coupled with the inverse static Jiles-Atherton (JA) model in order to represent the hysteresis behavior for a lamination. In the present paper, this approach is improved to ensure the reproducibility of measured hysteresis loops at mean frequency. The results of simulation are compared with the experimental ones. The selected results for frequencies 50 Hz, 100 Hz, 200 Hz and 400 Hz are presented and discussed.

  2. Diffusive Wave Approximation to the Shallow Water Equations: Computational Approach

    KAUST Repository

    Collier, Nathan

    2011-05-14

    We discuss the use of time adaptivity applied to the one dimensional diffusive wave approximation to the shallow water equations. A simple and computationally economical error estimator is discussed which enables time-step size adaptivity. This robust adaptive time discretization corrects the initial time step size to achieve a user specified bound on the discretization error and allows time step size variations of several orders of magnitude. In particular, in the one dimensional results presented in this work feature a change of four orders of magnitudes for the time step over the entire simulation.

  3. ENERGY ESTIMATES FOR DELAY DIFFUSION-REACTION EQUATIONS

    Institute of Scientific and Technical Information of China (English)

    J.A.Ferreira; P.M.da Silva

    2008-01-01

    In this paper we consider nonlinear delay diffusion-reaction equations with initial and Dirichlet boundary conditions.The behaviour and the stability of the solution of such initial boundary value problems(IBVPs)are studied using the energy method.Simple numerical methods are considered for the computation of numerical approximations to the solution of the nonlinear IBVPs.Using the discrete energy method we study the stability and convergence of the numerical approximations.Numerical experiments are carried out to illustrate our theoretical results.

  4. Mimetic discretization of two-dimensional magnetic diffusion equations

    Science.gov (United States)

    Lipnikov, Konstantin; Reynolds, James; Nelson, Eric

    2013-08-01

    In case of non-constant resistivity, cylindrical coordinates, and highly distorted polygonal meshes, a consistent discretization of the magnetic diffusion equations requires new discretization tools based on a discrete vector and tensor calculus. We developed a new discretization method using the mimetic finite difference framework. It is second-order accurate on arbitrary polygonal meshes and a consistent calculation of the Joule heating is intrinsic within it. The second-order convergence rates in L2 and L1 norms were verified with numerical experiments.

  5. New variable separation solutions for the generalized nonlinear diffusion equations

    Science.gov (United States)

    Fei-Yu, Ji; Shun-Li, Zhang

    2016-03-01

    The functionally generalized variable separation of the generalized nonlinear diffusion equations ut = A(u,ux)uxx + B(u,ux) is studied by using the conditional Lie-Bäcklund symmetry method. The variant forms of the considered equations, which admit the corresponding conditional Lie-Bäcklund symmetries, are characterized. To construct functionally generalized separable solutions, several concrete examples defined on the exponential and trigonometric invariant subspaces are provided. Project supported by the National Natural Science Foundation of China (Grant Nos. 11371293, 11401458, and 11501438), the National Natural Science Foundation of China, Tian Yuan Special Foundation (Grant No. 11426169), and the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2015JQ1014).

  6. Chaotic dynamics and diffusion in a piecewise linear equation

    Science.gov (United States)

    Shahrear, Pabel; Glass, Leon; Edwards, Rod

    2015-03-01

    Genetic interactions are often modeled by logical networks in which time is discrete and all gene activity states update simultaneously. However, there is no synchronizing clock in organisms. An alternative model assumes that the logical network is preserved and plays a key role in driving the dynamics in piecewise nonlinear differential equations. We examine dynamics in a particular 4-dimensional equation of this class. In the equation, two of the variables form a negative feedback loop that drives a second negative feedback loop. By modifying the original equations by eliminating exponential decay, we generate a modified system that is amenable to detailed analysis. In the modified system, we can determine in detail the Poincaré (return) map on a cross section to the flow. By analyzing the eigenvalues of the map for the different trajectories, we are able to show that except for a set of measure 0, the flow must necessarily have an eigenvalue greater than 1 and hence there is sensitive dependence on initial conditions. Further, there is an irregular oscillation whose amplitude is described by a diffusive process that is well-modeled by the Irwin-Hall distribution. There is a large class of other piecewise-linear networks that might be analyzed using similar methods. The analysis gives insight into possible origins of chaotic dynamics in periodically forced dynamical systems.

  7. Chaotic dynamics and diffusion in a piecewise linear equation

    Energy Technology Data Exchange (ETDEWEB)

    Shahrear, Pabel, E-mail: pabelshahrear@yahoo.com [Department of Mathematics, Shah Jalal University of Science and Technology, Sylhet–3114 (Bangladesh); Glass, Leon, E-mail: glass@cnd.mcgill.ca [Department of Physiology, 3655 Promenade Sir William Osler, McGill University, Montreal, Quebec H3G 1Y6 (Canada); Edwards, Rod, E-mail: edwards@uvic.ca [Department of Mathematics and Statistics, University of Victoria, P.O. Box 1700 STN CSC, Victoria, British Columbia V8W 2Y2 (Canada)

    2015-03-15

    Genetic interactions are often modeled by logical networks in which time is discrete and all gene activity states update simultaneously. However, there is no synchronizing clock in organisms. An alternative model assumes that the logical network is preserved and plays a key role in driving the dynamics in piecewise nonlinear differential equations. We examine dynamics in a particular 4-dimensional equation of this class. In the equation, two of the variables form a negative feedback loop that drives a second negative feedback loop. By modifying the original equations by eliminating exponential decay, we generate a modified system that is amenable to detailed analysis. In the modified system, we can determine in detail the Poincaré (return) map on a cross section to the flow. By analyzing the eigenvalues of the map for the different trajectories, we are able to show that except for a set of measure 0, the flow must necessarily have an eigenvalue greater than 1 and hence there is sensitive dependence on initial conditions. Further, there is an irregular oscillation whose amplitude is described by a diffusive process that is well-modeled by the Irwin-Hall distribution. There is a large class of other piecewise-linear networks that might be analyzed using similar methods. The analysis gives insight into possible origins of chaotic dynamics in periodically forced dynamical systems.

  8. Localization of chemical sources using e. coli chemotaxis

    Science.gov (United States)

    Davison, Timothy; Nguyen, Hoa; Nickels, Kevin; Frasch, Duncan; Basagaoglu, Hakan

    2016-04-01

    This paper furthers the application of chemotaxis to small-scale robots by simulating a system that localizes a chemical source in a dynamic fluid environment. This type of system responds to a chemical stimulus by mimicking, for example, the way that E. Coli bacteria move toward attractants (nutrients) and away from repellents. E. Coli use the intracellular signaling pathway to process the temporal change in the chemical concentration to determine if the cells should run or tumble. Previous work has shown that this process can be simulated with robots and used to localize chemical sources based upon a fixed nutrient gradient. Our work furthers this study by simulating the injection of an effluent of chemical at a specified location in an environment and uses computational fluid dynamics to model the interactions of the robot with the fluid while performing chemotaxis. The interactions between the chemical and fluid are also modelled with the advection diffusion equation to determine the concentration gradient. This method allows us to compute, over a lattice, the chemical concentration at all points and feed these results into an existing E. Coli controller for the robot, which results in the robot executing a tumble or a run according to a probabilistic formula. By simulating the robot in this complex environment, our work facilitates refinement of the chemotaxis controller while proving the ability of chemotactic robots to localize specific chemicals in environments that more closely resemble those encountered in the wide-ranging types of locations in which this robotic system might be deployed.

  9. PERTURBATION FINITE VOLUME METHOD FOR CONVECTIVE-DIFFUSION INTEGRAL EQUATION

    Institute of Scientific and Technical Information of China (English)

    GAO Zhi; YANG Guowei

    2004-01-01

    A perturbation finite volume (PFV) method for the convective-diffusion integral equation is developed in this paper. The PFV scheme is an upwind and mixed scheme using any higher-order interpolation and second-order integration approximations, with the least nodes similar to the standard three-point schemes, that is, the number of the nodes needed is equal to unity plus the face-number of the control volume. For instance, in the two-dimensional (2-D) case, only four nodes for the triangle grids and five nodes for the Cartesian grids are utilized, respectively. The PFV scheme is applied on a number of 1-D linear and nonlinear problems, 2-D and 3-D flow model equations. Comparing with other standard three-point schemes, the PFV scheme has much smaller numerical diffusion than the first-order upwind scheme (UDS). Its numerical accuracies are also higher than the second-order central scheme (CDS), the power-law scheme (PLS) and QUICK scheme.

  10. On the supercritically diffusive magneto-geostrophic equations

    CERN Document Server

    Friedlander, Susan; Vicol, Vlad

    2011-01-01

    We address the well-posedness theory for the magento-geostrophic equation, namely an active scalar equation in which the divergence-free drift velocity is one derivative more singular than the active scalar. In the presence of supercritical fractional diffusion given by (-\\Delta)^\\gamma, where 01/2 the equations are locally well-posed, while for \\gamma<1/2 they are ill-posed, in the sense that there is no Lipschitz solution map. The main reason for the striking loss of regularity when \\gamma goes below 1/2 is that the constitutive law used to obtain the velocity from the active scalar is given by an unbounded Fourier multiplier which is both even and anisotropic. Lastly, we note that the anisotropy of the constitutive law for the velocity may be explored in order to obtain an improvement in the regularity of the solutions when the initial data and the force have thin Fourier support, i.e. they are supported on a plane in frequency space. In particular, for such well-prepared data one may prove the local ex...

  11. Traveling wave solutions of density-dependent nonlinear reaction-diffusion equation via the extended generalized Riccati equation mapping method

    Science.gov (United States)

    Kengne, Emmanuel; Saydé, Michel; Ben Hamouda, Fathi; Lakhssassi, Ahmed

    2013-11-01

    Analytical entire traveling wave solutions to the 1+1 density-dependent nonlinear reaction-diffusion equation via the extended generalized Riccati equation mapping method are presented in this paper. This equation can be regarded as an extension case of the Fisher-Kolmogoroff equation, which is used for studying insect and animal dispersal with growth dynamics. The analytical solutions are then used to investigate the effect of equation parameters on the population distribution.

  12. Derivation of a volume-averaged neutron diffusion equation; Atomos para el desarrollo de Mexico

    Energy Technology Data Exchange (ETDEWEB)

    Vazquez R, R.; Espinosa P, G. [UAM-Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, Mexico D.F. 09340 (Mexico); Morales S, Jaime B. [UNAM, Laboratorio de Analisis en Ingenieria de Reactores Nucleares, Paseo Cuauhnahuac 8532, Jiutepec, Morelos 62550 (Mexico)]. e-mail: rvr@xanum.uam.mx

    2008-07-01

    This paper presents a general theoretical analysis of the problem of neutron motion in a nuclear reactor, where large variations on neutron cross sections normally preclude the use of the classical neutron diffusion equation. A volume-averaged neutron diffusion equation is derived which includes correction terms to diffusion and nuclear reaction effects. A method is presented to determine closure-relationships for the volume-averaged neutron diffusion equation (e.g., effective neutron diffusivity). In order to describe the distribution of neutrons in a highly heterogeneous configuration, it was necessary to extend the classical neutron diffusion equation. Thus, the volume averaged diffusion equation include two corrections factor: the first correction is related with the absorption process of the neutron and the second correction is a contribution to the neutron diffusion, both parameters are related to neutron effects on the interface of a heterogeneous configuration. (Author)

  13. Diffusive Boltzmann equation, its fluid dynamics, Couette flow and Knudsen layers

    CERN Document Server

    Abramov, Rafail V

    2016-01-01

    In the current work we propose a diffusive modification of the Boltzmann equation. This naturally leads to the corresponding diffusive fluid dynamics equations, which we numerically investigate in a simple Couette flow setting. This diffusive modification is based on the assumption of the "imperfect" model collision term, which is unable to track all collisions in the corresponding real gas particle system. The effect of missed collisions is then modeled by an appropriately scaled long-term homogenization process of the particle dynamics. The corresponding diffusive fluid dynamics equations are produced in a standard way by closing the hierarchy of the moment equations using either the Euler or the Grad closure. In the numerical experiments with the Couette flow, we discover that the diffusive Euler equations behave similarly to the conventional Navier-Stokes equations, while the diffusive Grad equations additionally exhibit Knudsen-like velocity boundary layers. We compare the simulations with the correspond...

  14. Resolution of the time dependent Pn equations by a Godunov type scheme having the diffusion limit

    International Nuclear Information System (INIS)

    We consider the Pn model to approximate the transport equation in one dimension of space. In a diffusive regime, the solution of this system is solution of a diffusion equation. We are looking for a numerical scheme having the diffusion limit property: in a diffusive regime, it gives the solution of the limiting diffusion equation on a mesh at the diffusion scale. The numerical scheme proposed is an extension of the Godunov type scheme proposed by L. Gosse to solve the P1 model without absorption term. Moreover, it has the well-balanced property: it preserves the steady solutions of the system. (authors)

  15. A granular computing method for nonlinear convection-diffusion equation

    Directory of Open Access Journals (Sweden)

    Tian Ya Lan

    2016-01-01

    Full Text Available This paper introduces a method of solving nonlinear convection-diffusion equation (NCDE, based on the combination of granular computing (GrC and characteristics finite element method (CFEM. The key idea of the proposed method (denoted as GrC-CFEM is to reconstruct the solution from coarse-grained layer to fine-grained layer. It first gets the nonlinear solution on the coarse-grained layer, and then the function (Taylor expansion is applied to linearize the NCDE on the fine-grained layer. Switch to the fine-grained layer, the linear solution is directly derived from the nonlinear solution. The full nonlinear problem is solved only on the coarse-grained layer. Numerical experiments show that the GrC-CFEM can accelerate the convergence and improve the computational efficiency without sacrificing the accuracy.

  16. From baking a cake to solving the diffusion equation

    Science.gov (United States)

    Olszewski, Edward A.

    2006-06-01

    We explain how modifying a cake recipe by changing either the dimensions of the cake or the amount of cake batter alters the baking time. We restrict our consideration to the génoise and obtain a semiempirical relation for the baking time as a function of oven temperature, initial temperature of the cake batter, and dimensions of the unbaked cake. The relation, which is based on the diffusion equation, has three parameters whose values are estimated from data obtained by baking cakes in cylindrical pans of various diameters. The relation takes into account the evaporation of moisture at the top surface of the cake, which is the dominant factor affecting the baking time of a cake.

  17. Guiding brine shrimp through mazes by solving reaction diffusion equations

    Science.gov (United States)

    Singal, Krishma; Fenton, Flavio

    Excitable systems driven by reaction diffusion equations have been shown to not only find solutions to mazes but to also to find the shortest path between the beginning and the end of the maze. In this talk we describe how we can use the Fitzhugh-Nagumo model, a generic model for excitable media, to solve a maze by varying the basin of attraction of its two fixed points. We demonstrate how two dimensional mazes are solved numerically using a Java Applet and then accelerated to run in real time by using graphic processors (GPUs). An application of this work is shown by guiding phototactic brine shrimp through a maze solved by the algorithm. Once the path is obtained, an Arduino directs the shrimp through the maze using lights from LEDs placed at the floor of the Maze. This method running in real time could be eventually used for guiding robots and cars through traffic.

  18. Precipitation of Phase Using General Diffusion Equation with Comparison to Vitek Diffusion Model in Dissimilar Stainless Steels

    OpenAIRE

    Chih-Chun Hsieh; Weite Wu

    2012-01-01

    This study performs a precipitation examination of the phase using the general diffusion equation with comparison to the Vitek model in dissimilar stainless steels during multipass welding. Experimental results demonstrate that the diffusivities (, , and ) of Cr, Ni, and Si are higher in -ferrite than (, , and ) in the phase, and that they facilitate the precipitation of the σ phase in the third pass fusion zone. The Vitek diffusion equation can be modified as follows: .

  19. Precipitation of Phase Using General Diffusion Equation with Comparison to Vitek Diffusion Model in Dissimilar Stainless Steels

    Directory of Open Access Journals (Sweden)

    Chih-Chun Hsieh

    2012-01-01

    Full Text Available This study performs a precipitation examination of the phase using the general diffusion equation with comparison to the Vitek model in dissimilar stainless steels during multipass welding. Experimental results demonstrate that the diffusivities (, , and of Cr, Ni, and Si are higher in -ferrite than (, , and in the phase, and that they facilitate the precipitation of the σ phase in the third pass fusion zone. The Vitek diffusion equation can be modified as follows: .

  20. On the uniqueness of semi-wavefronts for non-local delayed reaction-diffusion equations

    OpenAIRE

    Aguerrea, Maitere

    2013-01-01

    We establish the uniqueness of semi-wavefront solution for a non-local delayed reaction-diffusion equation. This result is obtained by using a generalization of the Diekman-Kaper theory for a nonlinear convolution equation. Several applications to the systems of non-local reaction-diffusion equations with distributed time delay are also considered.

  1. Diffusion-equation representations of landform evolution in the simplest circumstances: Appendix C

    Science.gov (United States)

    Hanks, Thomas C.

    2009-01-01

    The diffusion equation is one of the three great partial differential equations of classical physics. It describes the flow or diffusion of heat in the presence of temperature gradients, fluid flow in porous media in the presence of pressure gradients, and the diffusion of molecules in the presence of chemical gradients. [The other two equations are the wave equation, which describes the propagation of electromagnetic waves (including light), acoustic (sound) waves, and elastic (seismic) waves radiated from earthquakes; and LaPlace’s equation, which describes the behavior of electric, gravitational, and fluid potentials, all part of potential field theory. The diffusion equation reduces to LaPlace’s equation at steady state, when the field of interest does not depend on t. Poisson’s equation is LaPlace’s equation with a source term.

  2. Conditional Symmetry Groups of Nonlinear Diffusion Equations with x-Dependent Convection and Absorption

    Institute of Scientific and Technical Information of China (English)

    QU Chang-Zheng; ZHANG Shun-Li

    2004-01-01

    The generalized conditionalsymmetry and sign-invariant approaches are developed to study the nonlinear diffusion equations with x-dependent convection and source terms. We obtain conditions under which the equations admit the second-order generalized conditional symmetries and the first-order sign-invariants on the solutions. Several types of different generalized conditional symmetries and first-order sign-invariants for the equations with diffusion of power law are obtained. Exact solutions to the resulting equations are constructed.

  3. New Nonlocal Symmetries of Diffusion-Convection Equations and Their Connection with Generalized Hodograph Transformation

    OpenAIRE

    Valentyn Tychynin

    2015-01-01

    Additional nonlocal symmetries of diffusion-convection equations and the Burgers equation are obtained. It is shown that these equations are connected via a generalized hodograph transformation and appropriate nonlocal symmetries arise from additional Lie symmetries of intermediate equations. Two entirely different techniques are used to search nonlocal symmetry of a given equation: the first is based on usage of the characteristic equations generated by additional operators, another techniqu...

  4. The constructive technique and its application in solving a nonlinear reaction diffusion equation

    Institute of Scientific and Technical Information of China (English)

    Lai Shao-Yong; Guo Yun-Xi; Qing Yin; Wu Yong-Hong

    2009-01-01

    A mathematical technique based on the consideration of a nonlinear partial differential equation together with an additional condition in the form of an ordinary differential equation is employed to study a nonlinear reaction diffusion equation which describes a real process in physics and in chemistry. Several exact solutions for the equation are acquired under certain circumstances.

  5. Regularized lattice Boltzmann model for a class of convection-diffusion equations.

    Science.gov (United States)

    Wang, Lei; Shi, Baochang; Chai, Zhenhua

    2015-10-01

    In this paper, a regularized lattice Boltzmann model for a class of nonlinear convection-diffusion equations with variable coefficients is proposed. The main idea of the present model is to introduce a set of precollision distribution functions that are defined only in terms of macroscopic moments. The Chapman-Enskog analysis shows that the nonlinear convection-diffusion equations can be recovered correctly. Numerical tests, including Fokker-Planck equations, Buckley-Leverett equation with discontinuous initial function, nonlinear convection-diffusion equation with anisotropic diffusion, are carried out to validate the present model, and the results show that the present model is more accurate than some available lattice Boltzmann models. It is also demonstrated that the present model is more stable than the traditional single-relaxation-time model for the nonlinear convection-diffusion equations. PMID:26565368

  6. A cladding oxidation model based on diffusion equations

    International Nuclear Information System (INIS)

    During severe accident in PWRs, the cladding oxidation with steam in the core is very important to the accident process. When oxidation time is long, or oxidation occurs in steam starvation conditions, the parabolic rate correlations based on experiments are restricted, which impacts the prediction of cladding failure, hydrogen production, and temperature. According to Fick's laws, a cladding oxidation model in a wide temperature range based on diffusion equations is developed. The developed oxidation model has a wider applicability than those parabolic rate correlations, and can simulate long-term experiments well. The restricted assumptions of short term oxidation time and enough steam environment in the core implemented by those parabolic rate correlations are removed in the model, therefore this model perfectly fit for long-term and steam starvation conditions which are more realistic during a severe accident. This model also can obtain detailed oxygen distribution in the cladding, which is helpful to simulate the cladding failure in detail and develop advanced cladding failure criteria. (authors)

  7. L2-stability independent of diffusion for a finite element -- finite volume discretization of a linear convection-diffusion equation

    OpenAIRE

    Deuring, Paul; Eymard,, Robert; Mildner, Marcus

    2014-01-01

    We consider a time-dependent and a steady linear convection-diffusion equation. These equations are approximately solved by a combined finite element -- finite volume method: the diffusion term is discretized by Crouzeix-Raviart piecewise linear finite elements on a triangular grid, and the convection term by upwind barycentric finite volumes. In the unsteady case, the implicit Euler method is used as time discretization. This scheme is shown to be unconditionally L2-stable, uniformly with re...

  8. Innovation diffusion equations on correlated scale-free networks

    OpenAIRE

    Bertotti, M. L.; Brunner, J; Modanese, G.

    2016-01-01

    We introduce a heterogeneous network structure into the Bass diffusion model, in order to study the diffusion times of innovation or information in networks with a scale-free structure, typical of regions where diffusion is sensitive to geographic and logistic influences (like for instance Alpine regions). We consider both the diffusion peak times of the total population and of the link classes. In the familiar trickle-down processes the adoption curve of the hubs is found to anticipate the t...

  9. The Semiclassical Limit in the Quantum Drift-Diffusion Equations with Isentropic Pressure

    Institute of Scientific and Technical Information of China (English)

    Li CHEN; Qiangchang JU

    2008-01-01

    The semiclassical limit in the transient quantum drift-diffusion equations with isentropic pressure in one space dimension is rigorously proved. The equations are supple- mented with homogeneous Neumann boundary conditions. It is shown that the semiclas- sical limit of this solution solves the classical drift-diffusion model. In the meanwhile, the global existence of weak solutions is proved.

  10. NUMERICAL SIMULATION OF SINGLE-GROUP, STEADY STATE AND ISOTROPIC NEUTRON TRANSPORT EQUATION IN DIFFUSIVE REGIMES

    Institute of Scientific and Technical Information of China (English)

    Ying Genjun; Fu Ying; Ma Yichen; Zhang Zhipeng

    2006-01-01

    We present an algorithm for numerical solution of transport equation in diffusive regimes, in which the transport equation is nearly singular and its solution becomes a solution of a diffusion equation. This algorithm, which is based on the Least-squares FEM in combination with a scaling transformation, presents a good approximation of a diffusion operator in diffusive regimes and guarantees an accurate discrete solution. The numerical experiments in 2D and 3D case are given, and the numerical results show that this algorithm is correct and efficient.

  11. A Fractional Diffusion Equation for an n-Dimensional Correlated Levy Walk

    CERN Document Server

    Taylor-King, J P; Fedotov, S; Van Gorder, R A

    2016-01-01

    Levy walks define a fundamental concept in random walk theory which allows one to model diffusive spreading that is faster than Brownian motion. They have many applications across different disciplines. However, so far the derivation of a diffusion equation for an n-dimensional correlated Levy walk remained elusive. Starting from a fractional Klein-Kramers equation here we use a moment method combined with a Cattaneo approximation to derive a fractional diffusion equation for superdiffusive short range auto-correlated Levy walks in the large time limit, and solve it. Our derivation discloses different dynamical mechanisms leading to correlated Levy walk diffusion in terms of quantities that can be measured experimentally.

  12. Travelling Waves in Hybrid Chemotaxis Models

    KAUST Repository

    Franz, Benjamin

    2013-12-18

    Hybrid models of chemotaxis combine agent-based models of cells with partial differential equation models of extracellular chemical signals. In this paper, travelling wave properties of hybrid models of bacterial chemotaxis are investigated. Bacteria are modelled using an agent-based (individual-based) approach with internal dynamics describing signal transduction. In addition to the chemotactic behaviour of the bacteria, the individual-based model also includes cell proliferation and death. Cells consume the extracellular nutrient field (chemoattractant), which is modelled using a partial differential equation. Mesoscopic and macroscopic equations representing the behaviour of the hybrid model are derived and the existence of travelling wave solutions for these models is established. It is shown that cell proliferation is necessary for the existence of non-transient (stationary) travelling waves in hybrid models. Additionally, a numerical comparison between the wave speeds of the continuum models and the hybrid models shows good agreement in the case of weak chemotaxis and qualitative agreement for the strong chemotaxis case. In the case of slow cell adaptation, we detect oscillating behaviour of the wave, which cannot be explained by mean-field approximations. © 2013 Society for Mathematical Biology.

  13. Green functions and Langevin equations for nonlinear diffusion equations: A comment on ‘Markov processes, Hurst exponents, and nonlinear diffusion equations’ by Bassler et al.

    Science.gov (United States)

    Frank, T. D.

    2008-02-01

    We discuss two central claims made in the study by Bassler et al. [K.E. Bassler, G.H. Gunaratne, J.L. McCauley, Physica A 369 (2006) 343]. Bassler et al. claimed that Green functions and Langevin equations cannot be defined for nonlinear diffusion equations. In addition, they claimed that nonlinear diffusion equations are linear partial differential equations disguised as nonlinear ones. We review bottom-up and top-down approaches that have been used in the literature to derive Green functions for nonlinear diffusion equations and, in doing so, show that the first claim needs to be revised. We show that the second claim as well needs to be revised. To this end, we point out similarities and differences between non-autonomous linear Fokker-Planck equations and autonomous nonlinear Fokker-Planck equations. In this context, we raise the question whether Bassler et al.’s approach to financial markets is physically plausible because it necessitates the introduction of external traders and causes. Such external entities can easily be eliminated when taking self-organization principles and concepts of nonextensive thermostatistics into account and modeling financial processes by means of nonlinear Fokker-Planck equations.

  14. Asymptotic Speed of Wave Propagation for A Discrete Reaction-Diffusion Equation

    Institute of Scientific and Technical Information of China (English)

    Xiu-xiang Liu; Pei-xuan Weng

    2006-01-01

    We deal with asymptotic speed of wave propagation for a discrete reaction-diffusion equation. We find the minimal wave speed c* from the characteristic equation and show that c* is just the asymptotic speed of wave propagation. The isotropic property and the existence of solution of the initial value problem for the given equation are also discussed.

  15. Classification and Approximate Functional Separable Solutions to the Generalized Diffusion Equations with Perturbation

    Science.gov (United States)

    Ji, Fei-Yu; Zhang, Shun-Li

    2013-11-01

    In this paper, the generalized diffusion equation with perturbation ut = A(u;ux)uII+eB(u;ux) is studied in terms of the approximate functional variable separation approach. A complete classification of these perturbed equations which admit approximate functional separable solutions is presented. Some approximate solutions to the resulting perturbed equations are obtained by examples.

  16. The second boundary value problem for equations of viscoelastic diffusion in polymers

    OpenAIRE

    Vorotnikov, Dmitry A.

    2009-01-01

    The classical approach to diffusion processes is based on Fick's law that the flux is proportional to the concentration gradient. Various phenomena occurring during propagation of penetrating liquids in polymers show that this type of diffusion exhibits anomalous behavior and contradicts the just mentioned law. However, they can be explained in the framework of non-Fickian diffusion theories based on viscoelasticity of polymers. Initial-boundary value problems for viscoelastic diffusion equat...

  17. Travelling waves in hybrid chemotaxis models

    CERN Document Server

    Franz, Benjamin; Painter, Kevin J; Erban, Radek

    2013-01-01

    Hybrid models of chemotaxis combine agent-based models of cells with partial differential equation models of extracellular chemical signals. In this paper, travelling wave properties of hybrid models of bacterial chemotaxis are investigated. Bacteria are modelled using an agent-based (individual-based) approach with internal dynamics describing signal transduction. In addition to the chemotactic behaviour of the bacteria, the individual-based model also includes cell proliferation and death. Cells consume the extracellular nutrient field (chemoattractant) which is modelled using a partial differential equation. Mesoscopic and macroscopic equations representing the behaviour of the hybrid model are derived and the existence of travelling wave solutions for these models is established. It is shown that cell proliferation is necessary for the existence of non-transient (stationary) travelling waves in hybrid models. Additionally, a numerical comparison between the wave speeds of the continuum models and the hybr...

  18. Numerical Solution of Fractional Diffusion Equation Model for Freezing in Finite Media

    Directory of Open Access Journals (Sweden)

    R. S. Damor

    2013-01-01

    Full Text Available Phase change problems play very important role in engineering sciences including casting of nuclear waste materials, vivo freezing of biological tissues, solar collectors and so forth. In present paper, we propose fractional diffusion equation model for alloy solidification. A transient heat transfer analysis is carried out to study the anomalous diffusion. Finite difference method is used to solve the fractional differential equation model. The temperature profiles, the motion of interface, and interface velocity have been evaluated for space fractional diffusion equation.

  19. A high-order splitting scheme for the advection-diffusion equation of pollutants

    Institute of Scientific and Technical Information of China (English)

    2001-01-01

    A high-order splitting scheme for the advection-diffusion equation of pollutants is proposed in this paper. The multidimensional advection-diffusion equation is splitted into several one-dimensional equations that are solved by the scheme. Only three spatial grid points are needed in each direction and the scheme has fourth-order spatial accuracy. Several typically pure advection and advection-diffusion problems are simulated. Numerical results show that the accuracy of the scheme is much higher than that of the classical schemes and the scheme can be efficiently solved with little programming effort.

  20. A semi-analytical finite element method for a class of time-fractional diffusion equations

    CERN Document Server

    Sun, HongGuang; Sze, K Y

    2011-01-01

    As fractional diffusion equations can describe the early breakthrough and the heavy-tail decay features observed in anomalous transport of contaminants in groundwater and porous soil, they have been commonly employed in the related mathematical descriptions. These models usually involve long-time range computation, which is a critical obstacle for its application, improvement of the computational efficiency is of great significance. In this paper, a semi-analytical method is presented for solving a class of time-fractional diffusion equations which overcomes the critical long-time range computation problem of time fractional differential equations. In the procedure, the spatial domain is discretized by the finite element method which reduces the fractional diffusion equations into approximate fractional relaxation equations. As analytical solutions exist for the latter equations, the burden arising from long-time range computation can effectively be minimized. To illustrate its efficiency and simplicity, four...

  1. Variable order differential equations with piecewise constant order-function and diffusion with changing modes

    CERN Document Server

    Umarov, Sabir

    2009-01-01

    In this paper diffusion processes with changing modes are studied involving the variable order partial differential equations. We prove the existence and uniqueness theorem of a solution of the Cauchy problem for fractional variable order (with respect to the time derivative) pseudo-differential equations. Depending on the parameters of variable order derivatives short or long range memories may appear when diffusion modes change. These memory effects are classified and studied in detail. Processes that have distinctive regimes of different types of diffusion depending on time are ubiquitous in the nature. Examples include diffusion in a heterogeneous media and protein movement in cell biology.

  2. SOME RESULTS ON A DEGENERATE AND SINGULAR DIFFUSION EQUATION

    Institute of Scientific and Technical Information of China (English)

    Yao Zhengan; Zhou Wenshu

    2007-01-01

    In this article, a degenerate and singular diffusion problem is studied. The existence of solutions is established by parabolic regularization. Some properties of solutions, for instance, asymptotic behavior, are also discussed.

  3. Innovation diffusion equations on correlated scale-free networks

    Science.gov (United States)

    Bertotti, M. L.; Brunner, J.; Modanese, G.

    2016-07-01

    We introduce a heterogeneous network structure into the Bass diffusion model, in order to study the diffusion times of innovation or information in networks with a scale-free structure, typical of regions where diffusion is sensitive to geographic and logistic influences (like for instance Alpine regions). We consider both the diffusion peak times of the total population and of the link classes. In the familiar trickle-down processes the adoption curve of the hubs is found to anticipate the total adoption in a predictable way. In a major departure from the standard model, we model a trickle-up process by introducing heterogeneous publicity coefficients (which can also be negative for the hubs, thus turning them into stiflers) and a stochastic term which represents the erratic generation of innovation at the periphery of the network. The results confirm the robustness of the Bass model and expand considerably its range of applicability.

  4. Innovation diffusion equations on correlated scale-free networks

    CERN Document Server

    Bertotti, M L; Modanese, G

    2016-01-01

    We introduce a heterogeneous network structure into the Bass diffusion model, in order to study the diffusion times of innovation or information in networks with a scale-free structure, typical of regions where diffusion is sensitive to geographic and logistic influences (like for instance Alpine regions). We consider both the diffusion peak times of the total population and of the link classes. In the familiar trickle-down processes the adoption curve of the hubs is found to anticipate the total adoption in a predictable way. In a major departure from the standard model, we model a trickle-up process by introducing heterogeneous publicity coefficients (which can also be negative for the hubs, thus turning them into stiflers) and a stochastic term which represents the erratic generation of innovation at the periphery of the network. The results confirm the robustness of the Bass model and expand considerably its range of applicability.

  5. Transition probabilities for diffusion equations by means of path integrals

    OpenAIRE

    Goovaerts, Marc; DE SCHEPPER, Ann; Decamps, Marc

    2002-01-01

    In this paper, we investigate the transition probabilities for diffusion processes. In a first part, we show how transition probabilities for rather general diffusion processes can always be expressed by means of a path integral. For several classical models, an exact calculation is possible, leading to analytical expressions for the transition probabilities and for the maximum probability paths. A second part consists of the derivation of an analytical approximation for the transition probab...

  6. Transition probabilities for diffusion equations by means of path integrals.

    OpenAIRE

    Goovaerts, Marc; De Schepper, A; Decamps, M.

    2002-01-01

    In this paper, we investigate the transition probabilities for diffusion processes. In a first part, we show how transition probabilities for rather general diffusion processes can always be expressed by means of a path integral. For several classical models, an exact calculation is possible, leading to analytical expressions for the transition probabilities and for the maximum probability paths. A second part consists of the derivation of an analytical approximation for the transition probab...

  7. Application of The Full-Sweep AOR Iteration Concept for Space-Fractional Diffusion Equation

    Science.gov (United States)

    Sunarto, A.; Sulaiman, J.; Saudi, A.

    2016-04-01

    The aim of this paper is to investigate the effectiveness of the Full-Sweep AOR Iterative method by using Full-Sweep Caputo’s approximation equation to solve space-fractional diffusion equations. The governing space-fractional diffusion equations were discretized by using Full-Sweep Caputo’s implicit finite difference scheme to generate a system of linear equations. Then, the Full-Sweep AOR iterative method is applied to solve the generated linear system To examine the application of FSAOR method two numerical tests are conducted to show that the FSAOR method is superior to the FSSOR and FSGS methods.

  8. Exact solutions of multi-term fractional diffusion-wave equations with Robin type boundary conditions

    Institute of Scientific and Technical Information of China (English)

    Xiao-jing LIU; Ji-zeng WANG; Xiao-min WANG; You-he ZHOU

    2014-01-01

    General exact solutions in terms of wavelet expansion are obtained for multi-term time-fractional diffusion-wave equations with Robin type boundary conditions. By proposing a new method of integral transform for solving boundary value problems, such fractional partial differential equations are converted into time-fractional ordinary differ-ential equations, which are further reduced to algebraic equations by using the Laplace transform. Then, with a wavelet-based exact formula of Laplace inversion, the resulting exact solutions in the Laplace transform domain are reversed to the time-space domain. Three examples of wave-diffusion problems are given to validate the proposed analytical method.

  9. Numerical schemes for kinetic equations in the diffusion and anomalous diffusion limits. Part I: the case of heavy-tailed equilibrium

    OpenAIRE

    Crouseilles, Nicolas; Hivert, Hélène; Lemou, Mohammed

    2015-01-01

    In this work, we propose some numerical schemes for linear kinetic equations in the diffusion and anomalous diffusion limit. When the equilibrium distribution function is a Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffusion type equation. However, when a heavy-tailed distribution is considered, another time scale is required and the small mean free path limit leads to a fractional anomalous diffusion equation....

  10. A CLASS OF SINGULARLY PERTURBED INITIAL BOUNDARY PROBLEM FOR REACTION DIFFUSION EQUATION

    Institute of Scientific and Technical Information of China (English)

    Xie Feng

    2003-01-01

    The singularly perturbed initial boundary value problem for a class of reaction diffusion equation isconsidered. Under appropriate conditions, the existence-uniqueness and the asymptotic behavior of the solu-tion are showed by using the fixed-point theorem.

  11. A Class of Nonlinear Singularly Perturbed Problems for Reaction Diffusion Equations with Boundary Perturbation

    Institute of Scientific and Technical Information of China (English)

    Jia-qi Mo; Wan-tao Lin

    2006-01-01

    A class of nonlinear singularly perturbed problems for reaction diffusion equations with boundary perturbation are considered. Under suitable conditions, the asymptotic behavior of solution for the initial boundary value problems is studied using the theory of differential inequalities.

  12. THE NONLINEAR SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS WITH BOUNDARY PERTURBATION

    Institute of Scientific and Technical Information of China (English)

    OuyangCheng; MoJiaqi

    2005-01-01

    The nonlinear singularly perturbed problems for reaction diffusion equations with boundary perturbation are considered. Under suitable conditions, using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems is studied.

  13. THE CORNER LAYER SOLUTION TO ROBIN PROBLEM FOR REACTION DIFFUSION EQUATION

    Institute of Scientific and Technical Information of China (English)

    2012-01-01

    A class of Robin boundary value problem for reaction diffusion equation is considered. Under suitable conditions, using the theory of differential inequalities the existence and asymptotic behavior of the corner layer solution to the initial boundary value problem are studied.

  14. ASYMPTOTIC BEHAVIOR OF SOLUTION FOR A CLASS OF REACTION DIFFUSION EQUATIONS

    Institute of Scientific and Technical Information of China (English)

    MoJiaqi; LinWantao; ZhuJiang

    2004-01-01

    A class of initial boundary value problems for the reaction diffusion equations are considered. The asymptotic behavior of solution for the problem is obtained using the theory of differential inequality.

  15. A CLASS OF REACTION-DIFFUSION EQUATIONS WITH HYSTERESIS DIFFERENTIAL OPERATOR

    Institute of Scientific and Technical Information of China (English)

    XuLongfeng

    2002-01-01

    In this paper, the classical and weak derivatives with respect to spatial variable of a class of hysteresis functional are discussed. Some conclusions about solutions of a class of reaction-diffusion equations with hysteresis differential operator are given.

  16. THE NONLINEAR SINGULARLY PERTURBEDPROBLEMS FOR REACTION DIFFUSION EQUATIONS WITH TIME DELAY

    Institute of Scientific and Technical Information of China (English)

    莫嘉琪; 冯茂春

    2001-01-01

    A class of nonlinear for singularly perturbed problems for reaction diffusion equations with time delays are considered. Under suitable conditions, using theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems are studied.

  17. Exact solutions to nonlinear delay reaction-diffusion equations of hyperbolic type

    International Nuclear Information System (INIS)

    Authors present periodic and antiperiodic solutions, composite solutions resulting from a nonlinear superposition of generalized separable and traveling wave solutions, and others. Some results are extended to nonlinear delay reaction-diffusion equations with time-varying delay

  18. Application of linear-extended neutron diffusion equation in a semi-infinite homogeneous medium

    Energy Technology Data Exchange (ETDEWEB)

    Espinosa-Paredes, Gilberto, E-mail: gepe@xanum.uam.mx [Area de Ingenieria en Recursos Energeticos, Universidad Autonoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186 Col. Vicentina, Mexico 09340, D.F. (Mexico); Vazquez-Rodriguez, Rodolfo [Area de Ingenieria en Recursos Energeticos, Universidad Autonoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186 Col. Vicentina, Mexico 09340, D.F. (Mexico)

    2011-02-15

    The linear-extended neutron diffusion equation (LENDE) is the volume-averaged neutron diffusion equation (VANDE) which includes two correction terms: the first correction is related with the absorption process of the neutron and the second is a contribution to the neutron diffusion, both parameters are related to neutron effects on the interface of a heterogeneous configuration. In this work an analysis of a plane source in a semi-infinite homogeneous medium was considered to study the effects of the correction terms and the results obtained with the linear-extended neutron diffusion equation were compared against a semi-analytical benchmark for the same case. The comparison of the results demonstrate the excellent approach between the linear-extended diffusion theory and the selected benchmark, which means that the correction terms of the VANDE are physically acceptable.

  19. Mixed, Nonsplit, Extended Stability, Stiff Integration of Reaction Diffusion Equations

    KAUST Repository

    Alzahrani, Hasnaa H.

    2016-07-26

    A tailored integration scheme is developed to treat stiff reaction-diffusion prob- lems. The construction adapts a stiff solver, namely VODE, to treat reaction im- plicitly together with explicit treatment of diffusion. The second-order Runge-Kutta- Chebyshev (RKC) scheme is adjusted to integrate diffusion. Spatial operator is de- scretised by second-order finite differences on a uniform grid. The overall solution is advanced over S fractional stiff integrations, where S corresponds to the number of RKC stages. The behavior of the scheme is analyzed by applying it to three simple problems. The results show that it achieves second-order accuracy, thus, preserving the formal accuracy of the original RKC. The presented development sets the stage for future extensions, particularly, to multidimensional reacting flows with detailed chemistry.

  20. A modified diffusion equation for room-acoustic predication.

    Science.gov (United States)

    Jing, Yun; Xiang, Ning

    2007-06-01

    This letter presents a modified diffusion model using an Eyring absorption coefficient to predict the reverberation time and sound pressure distributions in enclosures. While the original diffusion model [Ollendorff, Acustica 21, 236-245 (1969); J. Picaut et al., Acustica 83, 614-621 (1997); Valeau et al., J. Acoust. Soc. Am. 119, 1504-1513 (2006)] usually has good performance for low absorption, the modified diffusion model yields more satisfactory results for both low and high absorption. Comparisons among the modified model, the original model, a geometrical-acoustics model, and several well-established theories in terms of reverberation times and sound pressure level distributions, indicate significantly improved prediction accuracy by the modification. PMID:17552680

  1. On Nonlinear Nonlocal Systems of Reaction Diffusion Equations

    Directory of Open Access Journals (Sweden)

    B. Ahmad

    2014-01-01

    Full Text Available The reaction diffusion system with anomalous diffusion and a balance law ut+-Δα/2u=-fu,v,   vt+-∆β/2v=fu,v, 0<α, β<2, is con sidered. The existence of global solutions is proved in two situations: (i a polynomial growth condition is imposed on the reaction term f when 0<α≤β≤2; (ii no growth condition is imposed on the reaction term f when 0<β≤α≤2.

  2. Numerical Solutions for Convection-Diffusion Equation through Non-Polynomial Spline

    Directory of Open Access Journals (Sweden)

    Ravi Kanth A.S.V.

    2016-01-01

    Full Text Available In this paper, numerical solutions for convection-diffusion equation via non-polynomial splines are studied. We purpose an implicit method based on non-polynomial spline functions for solving the convection-diffusion equation. The method is proven to be unconditionally stable by using Von Neumann technique. Numerical results are illustrated to demonstrate the efficiency and stability of the purposed method.

  3. Variable order differential equations with piecewise constant order-function and diffusion with changing modes

    OpenAIRE

    Umarov, Sabir; Steinberg, Stanly

    2009-01-01

    In this paper diffusion processes with changing modes are studied involving the variable order partial differential equations. We prove the existence and uniqueness theorem of a solution of the Cauchy problem for fractional variable order (with respect to the time derivative) pseudo-differential equations. Depending on the parameters of variable order derivatives short or long range memories may appear when diffusion modes change. These memory effects are classified and studied in detail. Pro...

  4. Non-negative mixed finite element formulations for a tensorial diffusion equation

    OpenAIRE

    Nakshatrala, K. B.; Valocchi, A. J.

    2008-01-01

    We consider the tensorial diffusion equation, and address the discrete maximum-minimum principle of mixed finite element formulations. In particular, we address non-negative solutions (which is a special case of the maximum-minimum principle) of mixed finite element formulations. The discrete maximum-minimum principle is the discrete version of the maximum-minimum principle. In this paper we present two non-negative mixed finite element formulations for tensorial diffusion equations based on ...

  5. Sound Field Modeling in Architectural Acoustics using a Diffusion Equation Based Model

    OpenAIRE

    Fortin, Nicolas; Picaut, Judicaël; Billon, Alexis; Valeau, Vincent; SAKOUT, Anas

    2009-01-01

    In this paper, an implementation of a model for room-acoustic predictions in COMSOL Multiphysics is presented. The model (called diffusion model) is based on the solving of diffusion equations instead of classical wave equations and allows simulating the sound propagation in complex geometries at high frequency. Instead of using COMSOL Multiplysics to solve directly the problem, a specific tool has been developed. It is composed of a user-friendly interface (I-Simpa) which manipulates all the...

  6. Sound Field Modeling in Architectural Acoustics using a Diffusion Equation Based Model

    OpenAIRE

    Fortin, Nicolas; Picaut, Judicaël; Billon, Alexis; Valeau, Vincent; SAKOUT, Anas

    2009-01-01

    In this paper, an implementation of a model for room-acoustic predictions in COMSOL Multiphysics is presented. The model (called diffusion model) is based on the solving of diffusion equations instead of classical wave equations and allows simulating the sound propagation in complex geometries at high frequency. Instead of using COMSOL Multiplysics to solve directly the problem, a specific tool has been developed. It is composed of a user-friendly interface (I-Simpa) which manipulates a...

  7. Diffusion approximations to the chemical master equation only have a consistent stochastic thermodynamics at chemical equilibrium

    International Nuclear Information System (INIS)

    The stochastic thermodynamics of a dilute, well-stirred mixture of chemically reacting species is built on the stochastic trajectories of reaction events obtained from the chemical master equation. However, when the molecular populations are large, the discrete chemical master equation can be approximated with a continuous diffusion process, like the chemical Langevin equation or low noise approximation. In this paper, we investigate to what extent these diffusion approximations inherit the stochastic thermodynamics of the chemical master equation. We find that a stochastic-thermodynamic description is only valid at a detailed-balanced, equilibrium steady state. Away from equilibrium, where there is no consistent stochastic thermodynamics, we show that one can still use the diffusive solutions to approximate the underlying thermodynamics of the chemical master equation

  8. Solving the Advection-Diffusion Equations in Biological Contexts using the Cellular Potts Model

    CERN Document Server

    Dan, D; Chen, K; Glazier, J A; Dan, Debasis; Mueller, Chris; Chen, Kun; Glazier, James A.

    2005-01-01

    The Cellular Potts Model (CPM) is a robust, cell-level methodology for simulation of biological tissues and morphogenesis. Both tissue physiology and morphogenesis depend on diffusion of chemical morphogens in the extra-cellular fluid or matrix (ECM). Standard diffusion solvers applied to the cellular potts model use finite difference methods on the underlying CPM lattice. However, these methods produce a diffusing field tied to the underlying lattice, which is inaccurate in many biological situations in which cell or ECM movement causes advection rapid compared to diffusion. Finite difference schemes suffer numerical instabilities solving the resulting advection-diffusion equations. To circumvent these problems we simulate advection-diffusion within the framework of the CPM using off-lattice finite-difference methods. We define a set of generalized fluid particles which detach advection and diffusion from the lattice. Diffusion occurs between neighboring fluid particles by local averaging rules which approxi...

  9. THE ANALYTICAL SOLUTION FOR SEDIMENT REACTION AND DIFFUSION EQUATION WITH GENERALIZED INITIAL-BOUNDARY CONDITIONS

    Institute of Scientific and Technical Information of China (English)

    熊岳山; 韦永康

    2001-01-01

    The sediment reaction and diffusion equation with generalized initial and boundary condition is studied. By using Laplace transform and Jordan lemma , an analytical solution is got, which is an extension of analytical solution provided by Cheng Kwokming James ( only diffusion was considered in analytical solution of Cheng ). Some problems arisen in the computation of analytical solution formula are also analysed.

  10. Variational iteration method for solving the time-fractional diffusion equations in porous medium

    Institute of Scientific and Technical Information of China (English)

    Wu Guo-Cheng

    2012-01-01

    The variational iteration method is successfully extended to the case of solving fractional differential equations,and the Lagrange multiplier of the method is identified in a more accurate way.Some diffusion models with fractional derivatives are investigated analytically,and the results show the efficiency of the new Lagrange multiplier for fractional differential equations of arbitrary order.

  11. New approach for analysing master equations of generalized phase diffusion models in the entangled state representation

    Institute of Scientific and Technical Information of China (English)

    Xu Xing-Lei; Li Hong-Qi; Fan Hong-Yi

    2009-01-01

    By virtue of the well-behaved properties of the bipartite entangled states representation, this paper analyse and solves some master equations for generalized phase diffusion models, which seems concise and effective. This method can also be applied to solve other master equations.

  12. On the numerical solution of the one-dimensional convection-diffusion equation

    Directory of Open Access Journals (Sweden)

    Mehdi Dehghan

    2005-01-01

    also used to develop new methods of high accuracy. This approach allows simple comparison of the errors associated with the partial differential equation. Various difference approximations are derived for the one-dimensional constant coefficient convection-diffusion equation. The results of a numerical experiment are provided, to verify the efficiency of the designed new algorithms. The paper ends with a concluding remark.

  13. L1 Error Estimates for Difference Approximations Of Degenerate Convection-Diffusion Equations

    OpenAIRE

    Karlsen, Kenneth H.; Risebro, Nils Henrik; Storrøsten, Erlend B.

    2012-01-01

    We analyze monotone difference schemes for strongly degenerate convection-diffusion equations in one spatial dimension. These nonlinear equations are well-posed within a class of (discontinuous) entropy solutions. We prove that the L1 difference between the approximate solutions and the unique entropy solution is bounded above by a constant times the cube root of the spatial discretization parameter.

  14. Approximate Generalized Conditional Symmetries for the Perturbed Nonlinear Diffusion-Convection Equations

    Institute of Scientific and Technical Information of China (English)

    ZHANG Shun-Li; QU Chang-Zheng

    2006-01-01

    @@ The concept of approximate generalized conditional symmetry (AGCS) as a generalization to both approximate Lie point symmetry and generalized conditional symmetry is introduced, and it is applied to study the perturbed nonlinear diffusion-convection equations. Complete classification of those perturbed equations which admit cer tain types of AGCSs is derived.

  15. Solitary wave solutions of selective nonlinear diffusion-reaction equations using homogeneous balance method

    Indian Academy of Sciences (India)

    Ranjit Kumar; R S Kaushal; Awadhesh Prasad

    2010-10-01

    An auto-Bäcklund transformation derived in the homogeneous balance method is employed to obtain several new exact solutions of certain kinds of nonlinear diffusion-reaction (D-R) equations. These equations arise in a variety of problems in physical, chemical, biological, social and ecological sciences.

  16. Strang-type preconditioners for solving fractional diffusion equations by boundary value methods

    NARCIS (Netherlands)

    Gu, Xian-Ming; Huang, Ting-Zhu; Zhao, Xi-Le; Li, Hou-Biao; Li, Liang

    2015-01-01

    The finite difference scheme with the shifted Grünwarld formula is employed to semi-discrete the fractional diffusion equations. This spatial discretization can reduce to the large system of ordinary differential equations (ODEs) with initial values. Recently, boundary value method (BVM) was develop

  17. Wave front propagation for a reaction–diffusion equation in narrow random channels

    International Nuclear Information System (INIS)

    We consider a reaction–diffusion equation in narrow random channels. We approximate the generalized solution to this equation by the corresponding one on a random graph. By making use of large deviation analysis we study the asymptotic wave front propagation. (paper)

  18. Global Null Controllability of the 1-Dimensional Nonlinear Slow Diffusion Equation

    Institute of Scientific and Technical Information of China (English)

    Jean-Michel CORON; Jesús Ildefonso D(I)AZ; Abdelmalek DRICI; Tommaso MINGAZZINI

    2013-01-01

    The authors prove the global null controllability for the 1-dimensional nonlinear slow diffusion equation by using both a boundary and an internal control.They assume that the internal control is only time dependent.The proof relies on the return method in combination with some local controllability results for nondegenerate equations and rescaling techniques.

  19. Solution of a Scalar Convection-diffusion Equation using FEMLAB

    CERN Document Server

    Yazdani, A

    2011-01-01

    A steady scalar convection-diffusion problem has been studied for one and two dimensional cases. The major problem of unrealistic oscillations of the convection dominated problems is relaxed thanks to the wide range of the elements FEMLAB 3.1 benefits. The FEMLAB 3.1 solution has been presented for the problems, unique features and illustrations of the software have been used and results have been tested against analytic solution.

  20. An Application of Equivalence Transformations to Reaction Diffusion Equations

    Directory of Open Access Journals (Sweden)

    Mariano Torrisi

    2015-10-01

    Full Text Available In this paper, we consider a quite general class of advection reaction diffusion systems. By using an equivalence generator, derived in a previous paper, the authors apply a projection theorem to determine some special forms of the constitutive functions that allow the extension by one of the two-dimensional principal Lie algebra. As an example, a special case is discussed at the end of the paper.

  1. Reaction-diffusion equation for quark-hadron transition in heavy-ion collisions

    CERN Document Server

    Bagchi, Partha; Sengupta, Srikumar; Srivastava, Ajit M

    2015-01-01

    Reaction-diffusion equations with suitable boundary conditions have special propagating solutions which very closely resemble the moving interfaces in a first order transition. We show that the dynamics of chiral order parameter for chiral symmetry breaking transition in heavy-ion collisions, with dissipative dynamics, is governed by one such equation, specifically, the Newell-Whitehead equation. Further, required boundary conditions are automatically satisfied due to the geometry of the collision. The chiral transition is, therefore, completed by a propagating interface, exactly as for a first order transition, even though the transition actually is a crossover for relativistic heavy-ion collisions. Same thing also happens when we consider the initial confinement-deconfinement transition with Polyakov loop order parameter. The resulting equation, again with dissipative dynamics, can then be identified with the reaction-diffusion equation known as the Fitzhugh-Nagumo equation which is used in population genet...

  2. Fractional Fokker-Planck Equation and Black-Scholes Formula in Composite-Diffusive Regime

    Science.gov (United States)

    Liang, Jin-Rong; Wang, Jun; Lǔ, Long-Jin; Gu, Hui; Qiu, Wei-Yuan; Ren, Fu-Yao

    2012-01-01

    In statistical physics, anomalous diffusion plays an important role, whose applications have been found in many areas. In this paper, we introduce a composite-diffusive fractional Brownian motion X α, H ( t)= X H ( S α ( t)), 0Black-Scholes formula. We obtain the fractional Fokker-Planck equation governing the dynamics of the probability density function of the composite-diffusive fractional Brownian motion and find the Black-Scholes differential equation driven by the stock asset X α, H ( t) and the corresponding Black-Scholes formula for the fair prices of European option.

  3. Boundary element method for the solution of the diffusion equation in cylindrical symmetry

    International Nuclear Information System (INIS)

    Equations for the solution of the diffusion equation in plane Cartesian geometry with the Boundary Element method was derived. The equation for the axi-symmetric case were set and included in the computer program. The results were compared to those obtained by the Finite Difference method. Comparing the results some advantages of the proposed method can be observed, with implications on the multidimensional problems. (author)

  4. Multi-dimensional traveling fronts in bistable reaction-diffusion equations

    OpenAIRE

    谷口 雅治; Masaharu Taniguchi

    2012-01-01

    Multi-dimensional traveling fronts have been studied in the Allen-Cahn equation (Nagumo equation) and also in multistable reaction-diffusion equations recently. Two-dimensional V-form fronts are studied by Ninomiya and myself (2005) and also by Hamel, Monneau and Roquejoffre (2005). Rotationally symmetric traveling fronts are studied by several authors.In this talk I will give a brief survey on multi-dimensional traveling fronts, and explain what is different and what gives the difficulties c...

  5. A New Approach and Solution Technique to Solve Time Fractional Nonlinear Reaction-Diffusion Equations

    OpenAIRE

    Inci Cilingir Sungu; Huseyin Demir

    2015-01-01

    A new application of the hybrid generalized differential transform and finite difference method is proposed by solving time fractional nonlinear reaction-diffusion equations. This method is a combination of the multi-time-stepping temporal generalized differential transform and the spatial finite difference methods. The procedure first converts the time-evolutionary equations into Poisson equations which are then solved using the central difference method. The temporal differential transform ...

  6. Numerical Solution of Advection-Diffusion Equation Using a Sixth-Order Compact Finite Difference Method

    OpenAIRE

    Gurhan Gurarslan; Halil Karahan; Devrim Alkaya; Murat Sari; Mutlu Yasar

    2013-01-01

    This study aims to produce numerical solutions of one-dimensional advection-diffusion equation using a sixth-order compact difference scheme in space and a fourth-order Runge-Kutta scheme in time. The suggested scheme here has been seen to be very accurate and a relatively flexible solution approach in solving the contaminant transport equation for Pe≤5. For the solution of the present equation, the combined technique has been used instead of conventional solution techniques. The accuracy and...

  7. Traveling Wave Solutions of Reaction-Diffusion Equations Arising in Atherosclerosis Models

    Directory of Open Access Journals (Sweden)

    Narcisa Apreutesei

    2014-05-01

    Full Text Available In this short review article, two atherosclerosis models are presented, one as a scalar equation and the other one as a system of two equations. They are given in terms of reaction-diffusion equations in an infinite strip with nonlinear boundary conditions. The existence of traveling wave solutions is studied for these models. The monostable and bistable cases are introduced and analyzed.

  8. Conservation Laws and Exact Solutions for a Reaction-Diffusion Equation with a Variable Coefficient

    Directory of Open Access Journals (Sweden)

    Zhijie Cao

    2014-01-01

    Full Text Available In this paper a variable-coefficient reaction-diffusion equation is studied. We classify the equation into three kinds by different restraints imposed on the variable coefficient b(x in the process of solving the determining equations of Lie groups. Then, for each kind, the conservation laws corresponding to the symmetries obtained are considered. Finally, some exact solutions are constructed.

  9. Fractional diffusion equation for an n -dimensional correlated Lévy walk

    Science.gov (United States)

    Taylor-King, Jake P.; Klages, Rainer; Fedotov, Sergei; Van Gorder, Robert A.

    2016-07-01

    Lévy walks define a fundamental concept in random walk theory that allows one to model diffusive spreading faster than Brownian motion. They have many applications across different disciplines. However, so far the derivation of a diffusion equation for an n -dimensional correlated Lévy walk remained elusive. Starting from a fractional Klein-Kramers equation here we use a moment method combined with a Cattaneo approximation to derive a fractional diffusion equation for superdiffusive short-range auto-correlated Lévy walks in the large time limit, and we solve it. Our derivation discloses different dynamical mechanisms leading to correlated Lévy walk diffusion in terms of quantities that can be measured experimentally.

  10. Nonlinear equation for anomalous diffusion: Unified power-law and stretched exponential exact solution.

    Science.gov (United States)

    Malacarne, L C; Mendes, R S; Pedron, I T; Lenzi, E K

    2001-03-01

    The nonlinear diffusion equation partial delta rho/delta t=D Delta rho(nu) is analyzed here, where Delta[triple bond](1/r(d-1))(delta/delta r)r(d-1-theta) delta/delta r, and d, theta, and nu are real parameters. This equation unifies the anomalous diffusion equation on fractals (nu=1) and the spherical anomalous diffusion for porous media (theta=0). An exact point-source solution is obtained, enabling us to describe a large class of subdiffusion [ theta>(1-nu)d], "normal" diffusion [theta=(1-nu)d] and superdiffusion [theta<(1-nu)d]. Furthermore, a thermostatistical basis for this solution is given from the maximum entropic principle applied to the Tsallis entropy.

  11. Fokker-Planck equations with memory: the cross over from ballistic to diffusive processes

    International Nuclear Information System (INIS)

    The unified description of diffusion processes that crosses over from a ballistic behavior at short times to a fractional diffusion (sub- or superdiffusion), as well as to the ordinary diffusion at longer times, is proposed on the basis of a non-Markovian generalization of the Fokker-Planck equation. The relations between the non-Markovian kinetic coefficients and observable quantities (mean- and mean square displacements) are established. The problem of calculations of the kinetic coefficients using the Langevin equations is discussed. Solutions of the non-Markovian equation describing diffusive processes in the real (coordinate) space are obtained. For long times, such a solution agrees with results obtained within the continuous random walk theory but is much superior to this solution at shorter times, where the effect of the ballistic region is crucial.

  12. Diffusion equation and spin drag in spin-polarized transport

    DEFF Research Database (Denmark)

    Flensberg, Karsten; Jensen, Thomas Stibius; Mortensen, Asger

    2001-01-01

    We study the role of electron-electron interactions for spin-polarized transport using the Boltzmann equation, and derive a set of coupled transport equations. For spin-polarized transport the electron-electron interactions are important, because they tend to equilibrate the momentum of the two-spin...... species. This "spin drag" effect enhances the resistivity of the system. The enhancement is stronger the lower the dimension is, and should be measurable in, for example, a two-dimensional electron gas with ferromagnetic contacts. We also include spin-flip scattering, which has two effects......: it equilibrates the spin density imbalance and, provided it has a non-s-wave component, also a current imbalance....

  13. Lie group invariant finite difference schemes for the neutron diffusion equation

    Energy Technology Data Exchange (ETDEWEB)

    Jaegers, P.J.

    1994-06-01

    Finite difference techniques are used to solve a variety of differential equations. For the neutron diffusion equation, the typical local truncation error for standard finite difference approximation is on the order of the mesh spacing squared. To improve the accuracy of the finite difference approximation of the diffusion equation, the invariance properties of the original differential equation have been incorporated into the finite difference equations. Using the concept of an invariant difference operator, the invariant difference approximations of the multi-group neutron diffusion equation were determined in one-dimensional slab and two-dimensional Cartesian coordinates, for multiple region problems. These invariant difference equations were defined to lie upon a cell edged mesh as opposed to the standard difference equations, which lie upon a cell centered mesh. Results for a variety of source approximations showed that the invariant difference equations were able to determine the eigenvalue with greater accuracy, for a given mesh spacing, than the standard difference approximation. The local truncation errors for these invariant difference schemes were found to be highly dependent upon the source approximation used, and the type of source distribution played a greater role in determining the accuracy of the invariant difference scheme than the local truncation error.

  14. Inverse Lax-Wendroff procedure for numerical boundary conditions of convection-diffusion equations

    Science.gov (United States)

    Lu, Jianfang; Fang, Jinwei; Tan, Sirui; Shu, Chi-Wang; Zhang, Mengping

    2016-07-01

    We consider numerical boundary conditions for high order finite difference schemes for solving convection-diffusion equations on arbitrary geometry. The two main difficulties for numerical boundary conditions in such situations are: (1) the wide stencil of the high order finite difference operator requires special treatment for a few ghost points near the boundary; (2) the physical boundary may not coincide with grid points in a Cartesian mesh and may intersect with the mesh in an arbitrary fashion. For purely convection equations, the so-called inverse Lax-Wendroff procedure [28], in which we convert the normal derivatives into the time derivatives and tangential derivatives along the physical boundary by using the equations, has been quite successful. In this paper, we extend this methodology to convection-diffusion equations. It turns out that this extension is non-trivial, because totally different boundary treatments are needed for the diffusion-dominated and the convection-dominated regimes. We design a careful combination of the boundary treatments for the two regimes and obtain a stable and accurate boundary condition for general convection-diffusion equations. We provide extensive numerical tests for one- and two-dimensional problems involving both scalar equations and systems, including the compressible Navier-Stokes equations, to demonstrate the good performance of our numerical boundary conditions.

  15. Coupled Oscillators with Chemotaxis

    CERN Document Server

    Sawai, S; Sawai, Satoshi; Aizawa, Yoji

    1998-01-01

    A simple coupled oscillator system with chemotaxis is introduced to study morphogenesis of cellular slime molds. The model successfuly explains the migration of pseudoplasmodium which has been experimentally predicted to be lead by cells with higher intrinsic frequencies. Results obtained predict that its velocity attains its maximum value in the interface region between total locking and partial locking and also suggest possible roles played by partial synchrony during multicellular development.

  16. Approximation of The Neutron Diffusion Equation on Hexagonal Geometries Using a h-p finite element method

    OpenAIRE

    FAYEZ MOUSTAFA MOAWAD, RAGAB

    2016-01-01

    [EN] The neutron diffusion equation is an approximation of the neutron transport equation that describes the neutron population in a nuclear reactor core. In particular, we will consider here VVER-type reactors which use the neutron diffusion equation discretized on hexagonal meshes. Most of the simulation codes of a nuclear power reactor use the multigroup neutron diffusion equation to describe the neutron distribution inside the reactor core.To study the stationary state of a reactor, the r...

  17. Resolution of the time dependent P{sub n} equations by a Godunov type scheme having the diffusion limit; Resolution des equations P{sub n} instationnaires par un schema de type Godunov, ayant la limite diffusion

    Energy Technology Data Exchange (ETDEWEB)

    Cargo, P.; Samba, G

    2007-07-01

    We consider the P{sub n} model to approximate the transport equation in one dimension of space. In a diffusive regime, the solution of this system is solution of a diffusion equation. We are looking for a numerical scheme having the diffusion limit property: in a diffusive regime, it gives the solution of the limiting diffusion equation on a mesh at the diffusion scale. The numerical scheme proposed is an extension of the Godunov type scheme proposed by L. Gosse to solve the P{sub 1} model without absorption term. Moreover, it has the well-balanced property: it preserves the steady solutions of the system. (authors)

  18. New Nonlocal Symmetries of Diffusion-Convection Equations and Their Connection with Generalized Hodograph Transformation

    Directory of Open Access Journals (Sweden)

    Valentyn Tychynin

    2015-09-01

    Full Text Available Additional nonlocal symmetries of diffusion-convection equations and the Burgers equation are obtained. It is shown that these equations are connected via a generalized hodograph transformation and appropriate nonlocal symmetries arise from additional Lie symmetries of intermediate equations. Two entirely different techniques are used to search nonlocal symmetry of a given equation: the first is based on usage of the characteristic equations generated by additional operators, another technique assumes the reconstruction of a parametrical Lie group transformation from such operator. Some of them are based on the nonlocal transformations that contain new independent variable determined by an auxiliary differential equation and allow the interpretation as a nonlocal transformation with additional variables. The formulae derived for construction of exact solutions are used.

  19. The fundamental solution of Cauchy problem for a single equation of the diffusion equation with inertia

    Directory of Open Access Journals (Sweden)

    H. P. Malytska

    2014-12-01

    Full Text Available The paper found the explicit form of the fundamental solution of  Cauchy problem for the equation of Kolmogorov type that has a finite number  groups of spatial variables which are degenerate parabolic.

  20. A nonlinear equation for ionic diffusion in a strong binary electrolyte

    CERN Document Server

    Ghosal, Sandip; 10.1098/rspa.2010.0028

    2012-01-01

    The problem of the one dimensional electro-diffusion of ions in a strong binary electrolyte is considered. In such a system the solute dissociates completely into two species of ions with unlike charges. The mathematical description consists of a diffusion equation for each species augmented by transport due to a self consistent electrostatic field determined by the Poisson equation. This mathematical framework also describes other important problems in physics such as electron and hole diffusion across semi-conductor junctions and the diffusion of ions in plasmas. If concentrations do not vary appreciably over distances of the order of the Debye length, the Poisson equation can be replaced by the condition of local charge neutrality first introduced by Planck. It can then be shown that both species diffuse at the same rate with a common diffusivity that is intermediate between that of the slow and fast species (ambipolar diffusion). Here we derive a more general theory by exploiting the ratio of Debye length...

  1. On the sharp front-type solution of the Nagumo equation with nonlinear diffusion and convection

    Indian Academy of Sciences (India)

    M B A Mansour

    2013-03-01

    This paper is concerned with the Nagumo equation with nonlinear degenerate diffusion and convection which arises in several problems of population dynamics, chemical reactions and others. A sharp front-type solution with a minimum speed to this model equation is analysed using different methods. One of the methods is to solve the travelling wave equations and compute an exact solution which describes the sharp travelling wavefront. The second method is to solve numerically an initial-moving boundary-value problem for the partial differential equation and obtain an approximation for this sharp front-type solution.

  2. Non-Fickian delay reaction-diffusion equations: theoretical and numerical study

    OpenAIRE

    Ferreira, J. A.; Branco, J. R.; Silva, P. da

    2007-01-01

    The Fisher’s equation is established combining the Fick’s law for the flux and the mass conservation law. Assuming that the reaction term depends on the solution at some past time, a delay parameter is introduced and the delay Fisher’s equation is obtained. Modifying the Fick’s law for the flux considering a temporal memory term, integro-differential equations of Volterra type were introduced in the literature. In these paper we study reaction-diffusion equations obtained co...

  3. An Efficient Implicit FEM Scheme for Fractional-in-Space Reaction-Diffusion Equations

    KAUST Repository

    Burrage, Kevin

    2012-01-01

    Fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues that impose a number of computational constraints. In this paper we develop efficient, scalable techniques for solving fractional-in-space reaction diffusion equations using the finite element method on both structured and unstructured grids via robust techniques for computing the fractional power of a matrix times a vector. Our approach is show-cased by solving the fractional Fisher and fractional Allen-Cahn reaction-diffusion equations in two and three spatial dimensions, and analyzing the speed of the traveling wave and size of the interface in terms of the fractional power of the underlying Laplacian operator. © 2012 Society for Industrial and Applied Mathematics.

  4. Diffusive approximation of a time-fractional Burgers equation in nonlinear acoustics

    CERN Document Server

    Lombard, Bruno

    2016-01-01

    A fractional time derivative is introduced into the Burgers equation to model losses of nonlinear waves. This term amounts to a time convolution product, which greatly penalizes the numerical modeling. A diffusive representation of the fractional derivative is adopted here, replacing this nonlocal operator by a continuum of memory variables that satisfy local-in-time ordinary differential equations. Then a quadrature formula yields a system of local partial differential equations, well-suited to numerical integration. The determination of the quadrature coefficients is crucial to ensure both the well-posedness of the system and the computational efficiency of the diffusive approximation. For this purpose, optimization with constraint is shown to be a very efficient strategy. Strang splitting is used to solve successively the hyperbolic part by a shock-capturing scheme, and the diffusive part exactly. Numerical experiments are proposed to assess the efficiency of the numerical modeling, and to illustrate the e...

  5. Generalized Fractional Master Equation for Self-Similar Stochastic Processes Modelling Anomalous Diffusion

    Directory of Open Access Journals (Sweden)

    Gianni Pagnini

    2012-01-01

    inhomogeneity and nonstationarity properties of the medium. For instance, when this superposition is applied to the time-fractional diffusion process, the resulting Master Equation emerges to be the governing equation of the Erdélyi-Kober fractional diffusion, that describes the evolution of the marginal distribution of the so-called generalized grey Brownian motion. This motion is a parametric class of stochastic processes that provides models for both fast and slow anomalous diffusion: it is made up of self-similar processes with stationary increments and depends on two real parameters. The class includes the fractional Brownian motion, the time-fractional diffusion stochastic processes, and the standard Brownian motion. In this framework, the M-Wright function (known also as Mainardi function emerges as a natural generalization of the Gaussian distribution, recovering the same key role of the Gaussian density for the standard and the fractional Brownian motion.

  6. Long-term behavior of reaction-diffusion equations with nonlocal boundary conditions on rough domains

    Science.gov (United States)

    Gal, Ciprian G.; Warma, Mahamadi

    2016-08-01

    We investigate the long term behavior in terms of finite dimensional global and exponential attractors, as time goes to infinity, of solutions to a semilinear reaction-diffusion equation on non-smooth domains subject to nonlocal Robin boundary conditions, characterized by the presence of fractional diffusion on the boundary. Our results are of general character and apply to a large class of irregular domains, including domains whose boundary is Hölder continuous and domains which have fractal-like geometry. In addition to recovering most of the existing results on existence, regularity, uniqueness, stability, attractor existence, and dimension, for the well-known reaction-diffusion equation in smooth domains, the framework we develop also makes possible a number of new results for all diffusion models in other non-smooth settings.

  7. Asymptotic Properties of Solutions of Parabolic Equations Arising from Transient Diffusions

    Institute of Scientific and Technical Information of China (English)

    A.M. Il'in; R.Z. Khasminskii; G. Yin

    2002-01-01

    This work is concerned with asymptotic properties of a class of parabolic systems arising from singularly perturbed diffusions. The underlying system has a fast varying component and a slowly changing component. One of the distinct features is that the fast varying diffusion is transient. Under such a setup, this paper presents an asymptotic analysis of the solutions of such parabolic equations. Asymptotic expansions of functional satisfying the parabolic system are obtained. Error bounds are derived.

  8. A Fully Discrete Galerkin Method for a Nonlinear Space-Fractional Diffusion Equation

    Directory of Open Access Journals (Sweden)

    Yunying Zheng

    2011-01-01

    Full Text Available The spatial transport process in fractal media is generally anomalous. The space-fractional advection-diffusion equation can be used to characterize such a process. In this paper, a fully discrete scheme is given for a type of nonlinear space-fractional anomalous advection-diffusion equation. In the spatial direction, we use the finite element method, and in the temporal direction, we use the modified Crank-Nicolson approximation. Here the fractional derivative indicates the Caputo derivative. The error estimate for the fully discrete scheme is derived. And the numerical examples are also included which are in line with the theoretical analysis.

  9. Incremental Unknowns Method for Solving Three-Dimensional Convection-Diffusion Equations

    Institute of Scientific and Technical Information of China (English)

    Lunji Song; Yujiang Wu

    2007-01-01

    We use the incremental unknowns method in conjunction with the iterative methods to approximate the solution of the nonsymmetric and positive-definite linear systems generated from a multilevel discretization of three-dimensional convection-diffusion equations. The condition numbers of incremental unknowns matrices associated with the convection-diffusion equations and the number of iterations needed to attain an acceptable accuracy are estimated. Numerical results are presented with two-level approximations,which demonstrate that the incremental unknowns method when combined with some iterative methods is very efficient.

  10. Analytical versus discretized solutions of four-group diffusion equations to thermal reactors

    International Nuclear Information System (INIS)

    This paper presents the application of four-group Diffusion theory to thermal reactor criticality calculation. The four-group diffusion equations are applied to the spherical nucleus and reflector of an example reactor. The neutrons fluxes depend upon the radial coordinate. The simultaneous linear ordinary differential equations are solved given the solutions for the fluxes. The neutron fluxes for the nucleus are functions of the eight functions linearly independent consisting of sin, cos, sinh, cosh, sin sinh, sin cosh, cos sinh, and cos cosh. The analytical and discretized calculations of keff value give excellent agreement, an error around 0,03%. (author)

  11. A piecewise linear finite element discretization of the diffusion equation for arbitrary polyhedral grids

    International Nuclear Information System (INIS)

    We develop a piecewise linear (PWL) Galerkin finite element spatial discretization for the multi-dimensional radiation diffusion equation. It uses piecewise linear weight and basis functions in the finite element approximation, and it can be applied on arbitrary polygonal (2-dimensional) or polyhedral (3-dimensional) grids. We show that this new PWL method gives solutions comparable to those from Palmer's finite-volume method. However, since the PWL method produces a symmetric positive definite coefficient matrix, it should be substantially more computationally efficient than Palmer's method, which produces an asymmetric matrix. We conclude that the Galerkin PWL method is an attractive option for solving diffusion equations on unstructured grids. (authors)

  12. A Piecewise Linear Finite Element Discretization of the Diffusion Equation for Arbitrary Polyhedral Grids

    International Nuclear Information System (INIS)

    We develop a piecewise linear (PWL) Galerkin finite element spatial discretization for the multi-dimensional radiation diffusion equation. It uses piecewise linear weight and basis functions in the finite element approximation, and it can be applied on arbitrary polygonal (2D) or polyhedral (3D) grids. We show that this new PWL method gives solutions comparable to those from Palmer's finite-volume method. However, since the PWL method produces a symmetric positive definite coefficient matrix, it should be substantially more computationally efficient than Palmer's method, which produces an asymmetric matrix. We conclude that the Galerkin PWL method is an attractive option for solving diffusion equations on unstructured grids

  13. An analytical solution for the two-group kinetic neutron diffusion equation in cylindrical geometry

    Energy Technology Data Exchange (ETDEWEB)

    Fernandes, Julio Cesar L.; Vilhena, Marco Tullio, E-mail: julio.lombaldo@ufrgs.br, E-mail: vilhena@pq.cnpq.br [Programa de Pos Graduacao em Matematica Aplicada (DMPA/UFRGS), Universidade Federal do Rio Grande do Sul Porto Alegre, RS (Brazil); Bodmann, Bardo Ernst, E-mail: bardo.bodmann@ufrgs.br [Programa de Pos-Graduacao em Engenharia Mecanica (PROMEC/UFRGS), Universidade Federal do Rio Grande do Sul, Porto Alegre, RS (Brazil)

    2011-07-01

    Recently the two-group Kinetic Neutron Diffusion Equation with six groups of delay neutron precursor in a rectangle was solved by the Laplace Transform Technique. In this work, we report on an analytical solution for this sort of problem but in cylindrical geometry, assuming a homogeneous and infinite height cylinder. The solution is obtained applying the Hankel Transform to the Kinetic Diffusion equation and solving the transformed problem by the same procedure used in the rectangle. We also present numerical simulations and comparisons against results available in literature. (author)

  14. Numerical solution of the one-dimensional fractional convection diffusion equations based on Chebyshev operational matrix.

    Science.gov (United States)

    Xie, Jiaquan; Huang, Qingxue; Yang, Xia

    2016-01-01

    In this paper, we are concerned with nonlinear one-dimensional fractional convection diffusion equations. An effective approach based on Chebyshev operational matrix is constructed to obtain the numerical solution of fractional convection diffusion equations with variable coefficients. The principal characteristic of the approach is the new orthogonal functions based on Chebyshev polynomials to the fractional calculus. The corresponding fractional differential operational matrix is derived. Then the matrix with the Tau method is utilized to transform the solution of this problem into the solution of a system of linear algebraic equations. By solving the linear algebraic equations, the numerical solution is obtained. The approach is tested via examples. It is shown that the proposed algorithm yields better results. Finally, error analysis shows that the algorithm is convergent. PMID:27504247

  15. Numerical Solution of Advection-Diffusion Equation Using a Sixth-Order Compact Finite Difference Method

    Directory of Open Access Journals (Sweden)

    Gurhan Gurarslan

    2013-01-01

    Full Text Available This study aims to produce numerical solutions of one-dimensional advection-diffusion equation using a sixth-order compact difference scheme in space and a fourth-order Runge-Kutta scheme in time. The suggested scheme here has been seen to be very accurate and a relatively flexible solution approach in solving the contaminant transport equation for Pe≤5. For the solution of the present equation, the combined technique has been used instead of conventional solution techniques. The accuracy and validity of the numerical model are verified through the presented results and the literature. The computed results showed that the use of the current method in the simulation is very applicable for the solution of the advection-diffusion equation. The present technique is seen to be a very reliable alternative to existing techniques for these kinds of applications.

  16. assessment of concentration of air pollutants using analytical and numerical solution of the atmospheric diffusion equation

    International Nuclear Information System (INIS)

    The mathematical formulation of numerous physical problems a results in differential equations actually partial or ordinary differential equations.In our study we are interested in solutions of partial differential equations.The aim of this work is to calculate the concentrations of the pollution, by solving the atmospheric diffusion equation(ADE) using different mathematical methods of solution. It is difficult to solve the general form of ADE analytically, so we use some assumptions to get its solution.The solutions of it depend on the eddy diffusivity profiles(k) and the wind speed u. We use some physical assumptions to simplify its formula and solve it. In the present work, we solve the ADE analytically in three dimensions using Green's function method, Laplace transform method, normal mode method and these separation of variables method. Also, we use ADM as a numerical method. Finally, comparisons are made with the results predicted by the previous methods and the observed data.

  17. On the numerical solution of the one-dimensional convection-diffusion equation

    Directory of Open Access Journals (Sweden)

    Dehghan Mehdi

    2005-01-01

    Full Text Available The numerical solution of convection-diffusion transport problems arises in many important applications in science and engineering. These problems occur in many applications such as in the transport of air and ground water pollutants, oil reservoir flow, in the modeling of semiconductors, and so forth. This paper describes several finite difference schemes for solving the one-dimensional convection-diffusion equation with constant coefficients. In this research the use of modified equivalent partial differential equation (MEPDE as a means of estimating the order of accuracy of a given finite difference technique is emphasized. This approach can unify the deduction of arbitrary techniques for the numerical solution of convection-diffusion equation. It is also used to develop new methods of high accuracy. This approach allows simple comparison of the errors associated with the partial differential equation. Various difference approximations are derived for the one-dimensional constant coefficient convection-diffusion equation. The results of a numerical experiment are provided, to verify the efficiency of the designed new algorithms. The paper ends with a concluding remark.

  18. VARIATIONAL DISCRETIZATION FOR OPTIMAL CONTROL GOVERNED BY CONVECTION DOMINATED DIFFUSION EQUATIONS

    Institute of Scientific and Technical Information of China (English)

    Michael Hinze; Ningning Yan; Zhaojie Zhou

    2009-01-01

    In this paper, we study variational discretization for the constrained optimal control problem governed by convection dominated diffusion equations, where the state equation is approximated by the edge stabilization Galerkin method. A priori error estimates are derived for the state, the adjoint state and the control. Moreover, residual type a posteriori error estimates in the L2-norm are obtained. Finally, two numerical experiments are presented to illustrate the theoretical results.

  19. Incompressible limit of the compressible non-isentropic magnetohydrodynamic equations with zero magnetic diffusivity

    OpenAIRE

    Jiang, Song; Ju, Qiangchang; Li, Fucai

    2011-01-01

    We study the incompressible limit of the compressible non-isentropic magnetohydrodynamic equations with zero magnetic diffusivity and general initial data in the whole space $\\mathbb{R}^d$ $(d=2,3)$. We first establish the existence of classic solutions on a time interval independent of the Mach number. Then, by deriving uniform a priori estimates, we obtain the convergence of the solution to that of the incompressible magnetohydrodynamic equations as the Mach number tends to zero.

  20. Wright functions governed by fractional directional derivatives and fractional advection diffusion equations

    CERN Document Server

    D'Ovidio, Mirko

    2012-01-01

    We consider fractional directional derivatives and establish some connection with stable densities. Solutions to advection equations involving fractional directional derivatives are presented and some properties investigated. In particular we obtain solutions written in terms of Wright functions by exploiting operational rules involving the shift operator. We also consider fractional advection diffusion equations involving fractional powers of the negative Laplace operator and directional derivatives of fractional order and discuss the probabilistic interpretations of solutions.

  1. Finite Difference and Sinc-Collocation Approximations to a Class of Fractional Diffusion-Wave Equations

    OpenAIRE

    Zhi Mao; Aiguo Xiao; Zuguo Yu; Long Shi

    2014-01-01

    We propose an efficient numerical method for a class of fractional diffusion-wave equations with the Caputo fractional derivative of order $\\alpha $ . This approach is based on the finite difference in time and the global sinc collocation in space. By utilizing the collocation technique and some properties of the sinc functions, the problem is reduced to the solution of a system of linear algebraic equations at each time step. Stability and convergence of the proposed method are rigorously an...

  2. L1 Error Estimates for Difference Approximations Of Degenerate Convection-Diffusion Equations

    CERN Document Server

    Karlsen, Kenneth H; Storrøsten, Erlend B

    2012-01-01

    We analyze a class of semi-discrete monotone difference schemes for degenerate convection-diffusion equations in one spatial dimension. These nonlinear equations are well-posed within a class of (discontinuous) entropy solutions. We prove that the L1 difference between the approximate solutions and the unique entropy solution is bounded above by a constant times the cube root of the spatial discretization parameter.

  3. Mixed time discontinuous space-time finite element method for convection diffusion equations

    Institute of Scientific and Technical Information of China (English)

    2008-01-01

    A mixed time discontinuous space-time finite element scheme for second-order convection diffusion problems is constructed and analyzed. Order of the equation is lowered by the mixed finite element method. The low order equation is discretized with a space-time finite element method, continuous in space but discontinuous in time. Stability, existence, uniqueness and convergence of the approximate solutions are proved. Numerical results are presented to illustrate efficiency of the proposed method.

  4. Strang-type preconditioners for solving fractional diffusion equations by boundary value methods

    OpenAIRE

    Gu, Xian-Ming; Huang, Ting-Zhu; Zhao, Xi-Le; Li, Hou-Biao; Li, Liang

    2013-01-01

    The finite difference scheme with the shifted Gr\\"{u}nwarld formula is employed to semi-discrete the fractional diffusion equations. This spatial discretization can reduce to the large system of ordinary differential equations (ODEs) with initial values. Recently, boundary value method (BVM) was developed as a popular algorithm for solving large systems of ODEs. This method requires the solutions of one or more nonsymmetric, large and sparse linear systems. In this paper, the GMRES method wit...

  5. Identifying space-dependent coefficients and the order of fractionality in fractional advection diffusion equation

    CERN Document Server

    Maryshev, Boris; Latrille, Christelle; Néel, Marie-Christine

    2016-01-01

    Tracer tests in natural porous media sometimes show abnormalities that suggest considering a fractional variant of the Advection Diffusion Equation supplemented by a time derivative of non-integer order. We are describing an inverse method for this equation: it finds the order of the fractional derivative and the coefficients that achieve minimum discrepancy between solution and tracer data. Using an adjoint equation divides the computational effort by an amount proportional to the number of freedom degrees, which becomes large when some coefficients depend on space. Method accuracy is checked on synthetical data, and applicability to actual tracer test is demonstrated.

  6. An Exact Formal Solution to Reaction-Diffusion Equations from Biomathematics

    Institute of Scientific and Technical Information of China (English)

    2002-01-01

    We study the exact formal solution to the simplified Keller-Segel system modelling chemotaxis. The method we use is series expanding. The main result is to attain the formal solution to the simplified Keller-Segel system.

  7. LONG-TIME BEHAVIOR OF A CLASS OF REACTION DIFFUSION EQUATIONS WITH TIME DELAYS

    Institute of Scientific and Technical Information of China (English)

    2006-01-01

    The present paper devotes to the long-time behavior of a class of reaction diffusion equations with delays under Dirichlet boundary conditions. The stability and global attractability for the zero solution are provided, and the existence, stability and attractability for the positive stationary solution are also obtained.

  8. Global stability of travelling wave fronts for non-local diffusion equations with delay

    International Nuclear Information System (INIS)

    This paper is concerned with the global stability of travelling wave fronts for non-local diffusion equations with delay. We prove that the non-critical travelling wave fronts are globally exponentially stable under perturbations in some exponentially weighted L∞-spaces. Moreover, we obtain the decay rates of supx∈R|u(x,t)−φ(x+ct)| using weighted energy estimates

  9. SINGULARLY PERTURBED SOLUTION TO SEMILINEAR HIGHER ORDER REACTION DIFFUSION EQUATIONS WITH TWO PARAMETERS

    Institute of Scientific and Technical Information of China (English)

    Yao Jingsun; Mo Jiaqi

    2009-01-01

    A boundary value problem for semilinear higher order reaction diffusion equa-tions with two parameters is considered. Under suitable conditions, by the theory of differential inequalities, the existence and asymptotic behavior of solutions to initial boundary value problem are studied.

  10. On the exact solutions of nonlinear diffusion-reaction equations with quadratic and cubic nonlinearities

    Indian Academy of Sciences (India)

    R S Kaushal; Ranjit Kumar; Awadhesh Prasad

    2006-08-01

    Attempts have been made to look for the soliton content in the solutions of the recently studied nonlinear diffusion-reaction equations [R S Kaushal, J. Phys. 38, 3897 (2005)] involving quadratic or cubic nonlinearities in addition to the convective flux term which renders the system nonconservative and the corresponding Hamiltonian non-Hermitian.

  11. Fokker-Planck Type Equations with Sobolev Diffusion Coefficients and BV Drift Coefficients

    Institute of Scientific and Technical Information of China (English)

    De Jun LUO

    2013-01-01

    Combining Le Bris and Lions' arguments with Ambrosio's commutator estimate for BV vector fields,we prove in this paper the existence and uniqueness of solutions to the Fokker-Planck type equations with Sobolev diffusion coefficients and BV drift coefficients.

  12. Existence and Asymptotic Behavior of Traveling Wave Fronts for a Time-Delayed Degenerate Diffusion Equation

    OpenAIRE

    Weifang Yan; Rui Liu

    2013-01-01

    This paper is concerned with traveling wave fronts for a degenerate diffusion equation with time delay. We first establish the necessary and sufficient conditions to the existence of monotone increasing and decreasing traveling wave fronts, respectively. Moreover, special attention is paid to the asymptotic behavior of traveling wave fronts connecting two uniform steady states. Some previous results are extended.

  13. A CLASS OF NONLINEAR SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS

    Institute of Scientific and Technical Information of China (English)

    莫嘉琪

    2003-01-01

    A class of nonlinear singularly perturbed problems for reaction diffusion equations are considered.Under suitable conditions,by using the theory of differential inequalities,the asymptotic behavior of solutions for the initial boundary value problems are studied,reduced problems of which possess two intersecting solutions.

  14. Bifurcation Analysis of Gene Propagation Model Governed by Reaction-Diffusion Equations

    Directory of Open Access Journals (Sweden)

    Guichen Lu

    2016-01-01

    Full Text Available We present a theoretical analysis of the attractor bifurcation for gene propagation model governed by reaction-diffusion equations. We investigate the dynamical transition problems of the model under the homogeneous boundary conditions. By using the dynamical transition theory, we give a complete characterization of the bifurcated objects in terms of the biological parameters of the problem.

  15. Convective-diffusive Cahn-Hilliard Equation with Concentration Dependent Mobility

    Institute of Scientific and Technical Information of China (English)

    刘长春; 尹景学

    2003-01-01

    In this paer,we study the global existence of classical solutions for the convective-diffusive Cahn-Hilliard Equation with concentration dependent mobility.Based on the Schauder type estimates,we establish the global existence of classical solutions.

  16. Locally one-dimensional difference scheme for the convective diffusion equation

    Energy Technology Data Exchange (ETDEWEB)

    Bugai, D.A. [Kiev Univ. (Ukraine)

    1994-11-10

    We consider the mixed boundary-value problem for the nonstationary convective diffusion equation in a rectangular region. The summation approximation method is applied to construct a locally homogeneous difference scheme with O({tau}{sup 1/2} + h{sup 3/2}) rate of convergence in the L{sub 2} grid metric.

  17. Multigrid solution of the convection-diffusion equation with high-Reynolds number

    Energy Technology Data Exchange (ETDEWEB)

    Zhang, Jun [George Washington Univ., Washington, DC (United States)

    1996-12-31

    A fourth-order compact finite difference scheme is employed with the multigrid technique to solve the variable coefficient convection-diffusion equation with high-Reynolds number. Scaled inter-grid transfer operators and potential on vectorization and parallelization are discussed. The high-order multigrid method is unconditionally stable and produces solution of 4th-order accuracy. Numerical experiments are included.

  18. A Four Group Reference Code for Solving Neutron Diffusion Equation in a VVER-440 Core

    Energy Technology Data Exchange (ETDEWEB)

    Saarinen, Simo [Fortum Nuclear Services Ltd., P.O. Box 100, 00048 Fortum (Finland)

    2008-07-01

    Nuclear reactor core power calculation is essential in the analysis of the nuclear power plant and especially the core. Currently, the core power distribution in Loviisa VVER-440 core is calculated using nodal code HEXBU-3D and pin-power reconstruction code ELSI-1440 that solve the two group neutron diffusion equation. The computer power available has increased significantly during the last decades allowing us to develop a fine mesh code HEXRE for solving the four group diffusion equation. The diffusion equations are discretized using piecewise linear polynomials. The core is discretized using one node per fuel pin cell. The axial discretization can be chosen freely. The boundary conditions are described using diffusion theory and albedos. Burnup dependence is modelled by tabulating diffusion parameters at certain burnup values and using interpolation for the intermediate values. A two degree polynomial is used for the modelling of the feedback effects. Eigenvalue calculation for both boron concentration and multiplication factor control has been formulated. A possibility to perform fuel loading and shuffling operations is implemented. HEXRE has been thoroughly compared with HEXBU-3D and ELSI-1440. The effect of the different energy and space discretizations used is investigated. Some safety criteria for the core calculated with the HEXRE and HEXBU-3D/ELSI-1440 have been compared. From the calculations (e.g. the safety criteria) we can estimate whether there exists systematic deviations in HEXBU- 3D/ELSI-1440 calculations or not. (author)

  19. The Conforming Virtual Element Method for the convection-diffusion-reaction equation with variable coeffcients.

    Energy Technology Data Exchange (ETDEWEB)

    Manzini, Gianmarco [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Cangiani, Andrea [University of Leicester, Leicester (United Kingdom); Sutton, Oliver [University of Leicester, Leicester (United Kingdom)

    2014-10-02

    This document describes the conforming formulations for virtual element approximation of the convection-reaction-diffusion equation with variable coefficients. Emphasis is given to construction of the projection operators onto polynomial spaces of appropriate order. These projections make it possible the virtual formulation to achieve any order of accuracy. We present the construction of the internal and the external formulation. The difference between the two is in the way the projection operators act on the derivatives (laplacian, gradient) of the partial differential equation. For the diffusive regime we prove the well-posedness of the external formulation and we derive an estimate of the approximation error in the H1-norm. For the convection-dominated case, the streamline diffusion stabilization (aka SUPG) is also discussed.

  20. An asymptotic-preserving scheme for linear kinetic equation with fractional diffusion limit

    Science.gov (United States)

    Wang, Li; Yan, Bokai

    2016-05-01

    We present a new asymptotic-preserving scheme for the linear Boltzmann equation which, under appropriate scaling, leads to a fractional diffusion limit. Our scheme rests on novel micro-macro decomposition to the distribution function, which splits the original kinetic equation following a reshuffled Hilbert expansion. As opposed to classical diffusion limit, a major difficulty comes from the fat tail in the equilibrium which makes the truncation in velocity space depending on the small parameter. Our idea is, while solving the macro-micro part in a truncated velocity domain (truncation only depends on numerical accuracy), to incorporate an integrated tail over the velocity space that is beyond the truncation, and its major component can be precomputed once with any accuracy. Such an addition is essential to drive the solution to the correct asymptotic limit. Numerical experiments validate its efficiency in both kinetic and fractional diffusive regimes.

  1. Efficient numerical solution of the time fractional diffusion equation by mapping from its Brownian counterpart

    CERN Document Server

    Stokes, Peter W; Read, Wayne; White, Ronald D

    2014-01-01

    The solution of a Caputo time fractional diffusion equation of order $0<\\alpha<1$ is found in terms of the solution of a corresponding integer order diffusion equation. We demonstrate a linear time mapping between these solutions that allows for accelerated computation of the solution of the fractional order problem. In the context of an $N$-point finite difference time discretisation, the mapping allows for an improvement in time computational complexity from $O\\left(N^{2}\\right)$ to $O\\left(N^{\\alpha}\\right)$, given a precomputation of $O\\left(N^{1+\\alpha}\\ln N\\right)$. The mapping is applied successfully to the least-squares fitting of a fractional advection diffusion model for the current in a time-of-flight experiment, resulting in a computational speed up in the range of one to three orders of magnitude for realistic problem sizes.

  2. Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings

    Energy Technology Data Exchange (ETDEWEB)

    Jin, Shi, E-mail: jin@math.wisc.edu [Department of Mathematics, Institute of Natural Sciences and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240 (China); Department of Mathematics, University of Wisconsin, Madison, WI 53706 (United States); Xiu, Dongbin, E-mail: dongbin.xiu@utah.edu [Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, UT 84112 (United States); Department of Mathematics, University of Utah, Salt Lake City, UT 84112 (United States); Zhu, Xueyu, E-mail: xzhu@sci.utah.edu [Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, UT 84112 (United States)

    2015-05-15

    In this paper we develop a set of stochastic numerical schemes for hyperbolic and transport equations with diffusive scalings and subject to random inputs. The schemes are asymptotic preserving (AP), in the sense that they preserve the diffusive limits of the equations in discrete setting, without requiring excessive refinement of the discretization. Our stochastic AP schemes are extensions of the well-developed deterministic AP schemes. To handle the random inputs, we employ generalized polynomial chaos (gPC) expansion and combine it with stochastic Galerkin procedure. We apply the gPC Galerkin scheme to a set of representative hyperbolic and transport equations and establish the AP property in the stochastic setting. We then provide several numerical examples to illustrate the accuracy and effectiveness of the stochastic AP schemes.

  3. Finite Element Solutions for the Space Fractional Diffusion Equation with a Nonlinear Source Term

    Directory of Open Access Journals (Sweden)

    Y. J. Choi

    2012-01-01

    Full Text Available We consider finite element Galerkin solutions for the space fractional diffusion equation with a nonlinear source term. Existence, stability, and order of convergence of approximate solutions for the backward Euler fully discrete scheme have been discussed as well as for the semidiscrete scheme. The analytical convergent orders are obtained as O(k+hγ˜, where γ˜ is a constant depending on the order of fractional derivative. Numerical computations are presented, which confirm the theoretical results when the equation has a linear source term. When the equation has a nonlinear source term, numerical results show that the diffusivity depends on the order of fractional derivative as we expect.

  4. Generalized and standard multigroup neutron diffusion equation eigenvalue problem with the finite volume method

    International Nuclear Information System (INIS)

    Most of the neutron diffusion codes use numerical methods giving accurate results in structured meshes. However, the application of these methods in unstructured meshes to deal with complex geometries is not straightforward and it may cause problems of stability and convergence of the solution. By contrast, the Finite Volume Method (FVM) is easily applied to unstructured meshes and is typically used in the transport equations due to the conservation of the transported quantity within the volume. In this paper, the FVM algorithm implemented in the ARB Partial Differential Equations Solver has been used to discretize the multigroup neutron diffusion equation to obtain the matrices of the generalized eigenvalue problem, which has been solved by means of the SLEPc library. Nevertheless, these matrices could be large for fine meshes and the eigenvalue problem resolution could require a high calculation time. Therefore, a transformation of the generalized eigenvalue problem into a standard one is performed in order to reduce the calculation time. (author)

  5. Prediction equations for diffusing capacity (transfer factor of lung for North Indians

    Directory of Open Access Journals (Sweden)

    Sunil Kumar Chhabra

    2016-01-01

    Full Text Available Background: Prediction equations for diffusing capacity of lung for carbon monoxide (DLCO, alveolar volume (VA, and DLCO/VA using the current standardization guidelines are not available for Indian population. The present study was carried out to develop equations for these parameters for North Indian adults and examine the ethnic diversity in predictions. Materials and Methods: DLCO was measured by single-breath technique and VA by single-breath helium dilution using standardized methodology in 357 (258 males, 99 females normal nonsmoker adult North Indians and DLCO/VA was computed. The subjects were randomized into training and test datasets for development of prediction equations by multiple linear regressions and for validation, respectively. Results: For males, the following equations were developed: DLCO, −7.813 + 0.318 × ht −0.624 × age + 0.00552 × age 2 ; VA, −8.152 + 0.087 × ht −0.019 × wt; DLCO/VA, 7.315 − 0.037 × age. For females, the equations were: DLCO, −44.15 + 0.449 × ht −0.099 × age; VA, −6.893 + 0.068 × ht. A statistically acceptable prediction equation was not obtained for DLCO/VA in females. It was therefore computed from predicted DLCO and predicted VA. All equations were internally valid. Predictions of DLCO by Indian equations were lower than most Caucasian predictions in both males and females and greater than the Chinese predictions for males. Conclusion: This study has developed validated prediction equations for DLCO, VA, and DLCO/VA in North Indians. Substantial ethnic diversity exists in predictions for DLCO and VA with Caucasian equations generally yielding higher values than the Indian or Chinese equations. However, DLCO/VA predicted by the Indian equations is slightly higher than that by other equations.

  6. The second boundary value problem for equations of viscoelastic diffusion in polymers

    CERN Document Server

    Vorotnikov, Dmitry A

    2009-01-01

    The classical approach to diffusion processes is based on Fick's law that the flux is proportional to the concentration gradient. Various phenomena occurring during propagation of penetrating liquids in polymers show that this type of diffusion exhibits anomalous behavior and contradicts the just mentioned law. However, they can be explained in the framework of non-Fickian diffusion theories based on viscoelasticity of polymers. Initial-boundary value problems for viscoelastic diffusion equations have been studied by several authors. Most of the studies are devoted to the Dirichlet BVP (the concentration is given on the boundary of the domain). In this chapter we study the second BVP, i.e. when the normal component of the concentration flux is prescribed on the boundary, which is more realistic in many physical situations. We establish existence of weak solutions to this problem. We suggest some conditions on the coefficients and boundary data under which all the solutions tend to the homogeneous state as tim...

  7. Reaction-diffusion equation for quark-hadron transition in heavy-ion collisions

    Science.gov (United States)

    Bagchi, Partha; Das, Arpan; Sengupta, Srikumar; Srivastava, Ajit M.

    2015-09-01

    Reaction-diffusion equations with suitable boundary conditions have special propagating solutions which very closely resemble the moving interfaces in a first-order transition. We show that the dynamics of the chiral order parameter for the chiral symmetry breaking transition in heavy-ion collisions, with dissipative dynamics, is governed by one such equation; specifically, the Newell-Whitehead equation. Furthermore, required boundary conditions are automatically satisfied due to the geometry of the collision. The chiral transition is, therefore, completed by a propagating interface, exactly as for a first-order transition, even though the transition actually is a crossover for relativistic heavy-ion collisions. The same thing also happens when we consider the initial confinement-deconfinement transition with the Polyakov loop order parameter. The resulting equation, again with dissipative dynamics, can then be identified with the reaction-diffusion equation known as the FitzHugh-Nagumo equation which is used in population genetics. Observational constraints imply that the entire phase conversion cannot be achieved by such slow moving fronts, and some alternate faster dynamics needs also to be invoked; for example, involving fluctuations. We discuss the implications of these results for heavy-ion collisions. We also discuss possible extensions for the case of the early universe.

  8. The breakdown of the reaction-diffusion master equation with non-elementary rates

    CERN Document Server

    Smith, Stephen

    2016-01-01

    The chemical master equation (CME) is the exact mathematical formulation of chemical reactions occurring in a dilute and well-mixed volume. The reaction-diffusion master equation (RDME) is a stochastic description of reaction-diffusion processes on a spatial lattice, assuming well-mixing only on the length scale of the lattice. It is clear that, for the sake of consistency, the solution of the RDME of a chemical system should converge to the solution of the CME of the same system in the limit of fast diffusion: indeed, this has been tacitly assumed in most literature concerning the RDME. We show that, in the limit of fast diffusion, the RDME indeed converges to a master equation, but not necessarily the CME. We introduce a class of propensity functions, such that if the RDME has propensities exclusively of this class then the RDME converges to the CME of the same system; while if the RDME has propensities not in this class then convergence is not guaranteed. These are revealed to be elementary and non-element...

  9. Singular solution of the Feller diffusion equation via a spectral decomposition.

    Science.gov (United States)

    Gan, Xinjun; Waxman, David

    2015-01-01

    Feller studied a branching process and found that the distribution for this process approximately obeys a diffusion equation [W. Feller, in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability (University of California Press, Berkeley and Los Angeles, 1951), pp. 227-246]. This diffusion equation and its generalizations play an important role in many scientific problems, including, physics, biology, finance, and probability theory. We work under the assumption that the fundamental solution represents a probability density and should account for all of the probability in the problem. Thus, under the circumstances where the random process can be irreversibly absorbed at the boundary, this should lead to the presence of a Dirac delta function in the fundamental solution at the boundary. However, such a feature is not present in the standard approach (Laplace transformation). Here we require that the total integrated probability is conserved. This yields a fundamental solution which, when appropriate, contains a term proportional to a Dirac delta function at the boundary. We determine the fundamental solution directly from the diffusion equation via spectral decomposition. We obtain exact expressions for the eigenfunctions, and when the fundamental solution contains a Dirac delta function at the boundary, every eigenfunction of the forward diffusion operator contains a delta function. We show how these combine to produce a weight of the delta function at the boundary which ensures the total integrated probability is conserved. The solution we present covers cases where parameters are time dependent, thereby greatly extending its applicability. PMID:25679586

  10. Connecting complexity with spectral entropy using the Laplace transformed solution to the fractional diffusion equation

    Science.gov (United States)

    Liang, Yingjie; Chen, Wen; Magin, Richard L.

    2016-07-01

    Analytical solutions to the fractional diffusion equation are often obtained by using Laplace and Fourier transforms, which conveniently encode the order of the time and the space derivatives (α and β) as non-integer powers of the conjugate transform variables (s, and k) for the spectral and the spatial frequencies, respectively. This study presents a new solution to the fractional diffusion equation obtained using the Laplace transform and expressed as a Fox's H-function. This result clearly illustrates the kinetics of the underlying stochastic process in terms of the Laplace spectral frequency and entropy. The spectral entropy is numerically calculated by using the direct integration method and the adaptive Gauss-Kronrod quadrature algorithm. Here, the properties of spectral entropy are investigated for the cases of sub-diffusion and super-diffusion. We find that the overall spectral entropy decreases with the increasing α and β, and that the normal or Gaussian case with α = 1 and β = 2, has the lowest spectral entropy (i.e., less information is needed to describe the state of a Gaussian process). In addition, as the neighborhood over which the entropy is calculated increases, the spectral entropy decreases, which implies a spatial averaging or coarse graining of the material properties. Consequently, the spectral entropy is shown to provide a new way to characterize the temporal correlation of anomalous diffusion. Future studies should be designed to examine changes of spectral entropy in physical, chemical and biological systems undergoing phase changes, chemical reactions and tissue regeneration.

  11. Approximate Inertial Manifolds for Chemotaxis-Growth System

    Institute of Scientific and Technical Information of China (English)

    Hong LUO; Zhilin PU

    2012-01-01

    The long-time behaviour of solution to chemotaxis-growth system with Neumann condition is considered in this paper.The approximate inertial manifolds of such equations are constructed based on the contraction principle,and the orders of approximations of the manifolds to the global attractor are derived.

  12. Propagation of fronts in the Fisher-Kolmogorov equation with spatially varying diffusion.

    Science.gov (United States)

    Curtis, Christopher W; Bortz, David M

    2012-12-01

    The propagation of fronts in the Fisher-Kolmogorov equation with spatially varying diffusion coefficients is studied. Using coordinate changes, WKB approximations, and multiple scales analysis, we provide an analytic framework that describes propagation of the front up to the minimum of the diffusion coefficient. We also present results showing the behavior of the front after it passes the minimum. In each case, we show that standard traveling coordinate frames do not properly describe front propagation. Last, we provide numerical simulations to support our analysis and to show, that around the minimum, the motion of the front is arrested on asymptotically significant time scales.

  13. Analysis of nonlinear parabolic equations modeling plasma diffusion across a magnetic field

    International Nuclear Information System (INIS)

    We analyse the evolutionary behavior of the solution of a pair of coupled quasilinear parabolic equations modeling the diffusion of heat and mass of a magnetically confined plasma. The solutions's behavior, due to the nonlinear diffusion coefficients, exhibits many new phenomena. In short time, the solution converges into a highly organized symmetric pattern that is almost completely independent of initial data. The asymptotic dynamics then become very simple and take place in a finite dimensional space. These conclusions are backed by extensive numerical experimentation

  14. Numerical solution of uncertain neutron diffusion equation for imprecisely defined homogeneous triangular bare reactor

    Indian Academy of Sciences (India)

    S Nayak; S Chakraverty

    2015-10-01

    In this paper, neutron diffusion equation of a triangular homogeneous bare reactor with uncertain parameters has been investigated. Here the involved parameters viz. geometry of the reactor, diffusion coefficient and absorption coefficient, etc. are uncertain and these are considered as fuzzy. Fuzzy values are handled through limit method which was defined for interval computations. The concept of fuzziness is hybridised with traditional finite element method to propose fuzzy finite element method. The proposed fuzzy finite element method has been used to obtain the uncertain eigenvalues of the said problem. Further these uncertain eigenvalues are compared with the traditional finite element method in special cases.

  15. MAXIMAL ATTRACTORS OF CLASSICAL SOLUTIONS FOR REACTION DIFFUSION EQUATIONS WITH DISPERSION

    Institute of Scientific and Technical Information of China (English)

    Li Yanling; Ma Yicheng

    2005-01-01

    The paper first deals with the existence of the maximal attractor of classical solution for reaction diffusion equation with dispersion, and gives the sup-norm estimate for the attractor. This estimate is optimal for the attractor under Neumann boundary condition. Next, the same problem is discussed for reaction diffusion system with uniformly contracting rectangle, and it reveals that the maximal attractor of classical solution for such system in the whole space is only necessary to be established in some invariant region.Finally, a few examples of application are given.

  16. Quantification of numerical diffusivity due to TVD schemes in the advection equation

    Science.gov (United States)

    Bidadi, Shreyas; Rani, Sarma L.

    2014-03-01

    In this study, the numerical diffusivity νnum inherent to the Roe-MUSCL scheme has been quantified for the scalar advection equation. The Roe-MUSCL scheme employed is a combination of: (1) the standard extension of the original Roe's formulation to the advection equation, and (2) van Leer's Monotone Upwind Scheme for Conservation Laws (MUSCL) technique that applies a linear variable reconstruction in a cell along with a scaled limiter function. An explicit expression is derived for the numerical diffusivity in terms of the limiter function, the distance between the cell centers on either side of a face, and the face-normal velocity. The numerical diffusivity formulation shows that a scaled limiter function is more appropriate for MUSCL in order to consistently recover the central-differenced flux at the maximum value of the limiter. The significance of the scaling factor is revealed when the Roe-MUSCL scheme, originally developed for 1-D scenarios, is applied to 2-D scalar advection problems. It is seen that without the scaling factor, the MUSCL scheme may not necessarily be monotonic in multi-dimensional scenarios. Numerical diffusivities of the minmod, superbee, van Leer and Barth-Jesperson TVD limiters were quantified for four problems: 1-D advection of a step function profile, and 2-D advection of step, sinusoidal, and double-step profiles. For all the cases, it is shown that the superbee scheme provides the lowest numerical diffusivity that is also most confined to the vicinity of the discontinuity. The minmod scheme is the most diffusive, as well as active in regions away from high gradients. As expected, the grid resolution study demonstrates that the magnitude and the spatial extent of the numerical diffusivity decrease with increasing resolution.

  17. A Bloch-Torrey Equation for Diffusion in a Deforming Media

    Energy Technology Data Exchange (ETDEWEB)

    Rohmer, Damien; Gullberg, Grant T.

    2006-12-29

    Diffusion Tensor Magnetic Resonance Imaging (DTMRI)technique enables the measurement of diffusion parameters and therefore,informs on the structure of the biological tissue. This technique isapplied with success to the static organs such as brain. However, thediffusion measurement on the dynamically deformable organs such as thein-vivo heart is a complex problem that has however a great potential inthe measurement of cardiac health. In order to understand the behavior ofthe Magnetic Resonance (MR)signal in a deforming media, the Bloch-Torreyequation that leads the MR behavior is expressed in general curvilinearcoordinates. These coordinates enable to follow the heart geometry anddeformations through time. The equation is finally discretized andpresented in a numerical formulation using implicit methods, in order toget a stable scheme that can be applied to any smooth deformations.Diffusion process enables the link between the macroscopic behavior ofmolecules and themicroscopic structure in which they evolve. Themeasurement of diffusion in biological tissues is therefore of majorimportance in understanding the complex underlying structure that cannotbe studied directly. The Diffusion Tensor Magnetic ResonanceImaging(DTMRI) technique enables the measurement of diffusion parametersand therefore provides information on the structure of the biologicaltissue. This technique has been applied with success to static organssuch as the brain. However, diffusion measurement of dynamicallydeformable organs such as the in-vivo heart remains a complex problem,which holds great potential in determining cardiac health. In order tounderstand the behavior of the magnetic resonance (MR) signal in adeforming media, the Bloch-Torrey equation that defines the MR behavioris expressed in general curvilinear coordinates. These coordinates enableus to follow the heart geometry and deformations through time. Theequation is finally discretized and presented in a numerical formulationusing

  18. Turing-Hopf bifurcation in the reaction-diffusion equations and its applications

    Science.gov (United States)

    Song, Yongli; Zhang, Tonghua; Peng, Yahong

    2016-04-01

    In this paper, we consider the Turing-Hopf bifurcation arising from the reaction-diffusion equations. It is a degenerate case and where the characteristic equation has a pair of simple purely imaginary roots and a simple zero root. First, the normal form theory for partial differential equations (PDEs) with delays developed by Faria is adopted to this degenerate case so that it can be easily applied to Turing-Hopf bifurcation. Then, we present a rigorous procedure for calculating the normal form associated with the Turing-Hopf bifurcation of PDEs. We show that the reduced dynamics associated with Turing-Hopf bifurcation is exactly the dynamics of codimension-two ordinary differential equations (ODE), which implies the ODE techniques can be employed to classify the reduced dynamics by the unfolding parameters. Finally, we apply our theoretical results to an autocatalysis model governed by reaction-diffusion equations; for such model, the dynamics in the neighbourhood of this bifurcation point can be divided into six categories, each of which is exactly demonstrated by the numerical simulations; and then according to this dynamical classification, a stable spatially inhomogeneous periodic solution has been found.

  19. Existence of Young measure solutions of a class of singular diffusion equations

    Institute of Scientific and Technical Information of China (English)

    2002-01-01

    The first initial-boundary value problem of a class of singular diffusion equations with the flux sublinear growth and the potential without convexity is investigated. Such equations may be strongly degenerate, singular and forward-backward. Inspired by the idea in a recent work of Demoulini, we first discuss the regular case by introducing the Young measure solutions and prove the existence of such solutions. Consequently, we approximate the extreme case by the method of regularization. By means of some uniform estimates and some techniques, the existence of Young measure solutions with bounded variation is established.

  20. A diffusive Fisher-KPP equation with free boundaries and time-periodic advections

    OpenAIRE

    Sun, Ningkui; Lou, Bendong; Zhou, Maolin

    2016-01-01

    We consider a reaction-diffusion-advection equation of the form: $u_t=u_{xx}-\\beta(t)u_x+f(t,u)$ for $x\\in (g(t),h(t))$, where $\\beta(t)$ is a $T$-periodic function representing the intensity of the advection, $f(t,u)$ is a Fisher-KPP type of nonlinearity, $T$-periodic in $t$, $g(t)$ and $h(t)$ are two free boundaries satisfying Stefan conditions. This equation can be used to describe the population dynamics in time-periodic environment with advection. Its homogeneous version (that is, both $...

  1. The Transport Equation in Optically Thick Media: Discussion of IMC and its Diffusion Limit

    Energy Technology Data Exchange (ETDEWEB)

    Szoke, A. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Brooks, E. D. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

    2016-07-12

    We discuss the limits of validity of the Implicit Monte Carlo (IMC) method for the transport of thermally emitted radiation. The weakened coupling between the radiation and material energy of the IMC method causes defects in handling problems with strong transients. We introduce an approach to asymptotic analysis for the transport equation that emphasizes the fact that the radiation and material temperatures are always different in time-dependent problems, and we use it to show that IMC does not produce the correct diffusion limit. As this is a defect of IMC in the continuous equations, no improvement to its discretization can remedy it.

  2. The determination of an unknown boundary condition in a fractional diffusion equation

    KAUST Repository

    Rundell, William

    2013-07-01

    In this article we consider an inverse boundary problem, in which the unknown boundary function ∂u/∂v = f(u) is to be determined from overposed data in a time-fractional diffusion equation. Based upon the free space fundamental solution, we derive a representation for the solution f as a nonlinear Volterra integral equation of second kind with a weakly singular kernel. Uniqueness and reconstructibility by iteration is an immediate result of a priori assumption on f and applying the fixed point theorem. Numerical examples are presented to illustrate the validity and effectiveness of the proposed method. © 2013 Copyright Taylor and Francis Group, LLC.

  3. Biomixing by chemotaxis and efficiency of biological reactions: the critical reaction case

    CERN Document Server

    Kiselev, Alexander

    2012-01-01

    Many phenomena in biology involve both reactions and chemotaxis. These processes can clearly influence each other, and chemotaxis can play an important role in sustaining and speeding up the reaction. In continuation of our earlier work, we consider a model with a single density function involving diffusion, advection, chemotaxis, and absorbing reaction. The model is motivated, in particular, by the studies of coral broadcast spawning, where experimental observations of the efficiency of fertilization rates significantly exceed the data obtained from numerical models that do not take chemotaxis (attraction of sperm gametes by a chemical secreted by egg gametes) into account. We consider the case of the weakly coupled quadratic reaction term, which is the most natural from the biological point of view and was left open. The result is that similarly to higher power coupling, the chemotaxis plays a crucial role in ensuring efficiency of reaction. However, mathematically, the picture is quite different in the qua...

  4. Entropy methods for reaction-diffusion equations: slowly growing a-priori bounds

    KAUST Repository

    Desvillettes, Laurent

    2008-01-01

    In the continuation of [Desvillettes, L., Fellner, K.: Exponential Decay toward Equilibrium via Entropy Methods for Reaction-Diffusion Equations. J. Math. Anal. Appl. 319 (2006), no. 1, 157-176], we study reversible reaction-diffusion equations via entropy methods (based on the free energy functional) for a 1D system of four species. We improve the existing theory by getting 1) almost exponential convergence in L1 to the steady state via a precise entropy-entropy dissipation estimate, 2) an explicit global L∞ bound via interpolation of a polynomially growing H1 bound with the almost exponential L1 convergence, and 3), finally, explicit exponential convergence to the steady state in all Sobolev norms.

  5. A piecewise linear finite element discretization of the diffusion equation for arbitrary polyhedral grids

    International Nuclear Information System (INIS)

    We develop a piecewise linear (PWL) Galerkin finite element spatial discretization for the multi-dimensional radiation diffusion equation. It uses recently introduced piecewise linear weight and basis functions in the finite element approximation and it can be applied on arbitrary polygonal (2D) or polyhedral (3D) grids. We first demonstrate some analytical properties of the PWL method and perform a simple mode analysis to compare the PWL method with Palmer's vertex-centered finite-volume method and with a bilinear continuous finite element method. We then show that this new PWL method gives solutions comparable to those from Palmer's. However, since the PWL method produces a symmetric positive-definite coefficient matrix, it should be substantially more computationally efficient than Palmer's method, which produces an asymmetric matrix. We conclude that the Galerkin PWL method is an attractive option for solving diffusion equations on unstructured grids

  6. A piecewise linear finite element discretization of the diffusion equation for arbitrary polyhedral grids

    Energy Technology Data Exchange (ETDEWEB)

    Bailey, T.S.; Adams, M.L. [Texas A M Univ., Dept. of Nuclear Engineering, College Station, TX (United States); Yang, B.; Zika, M.R. [Lawrence Livermore National Lab., Livermore, CA (United States)

    2005-07-01

    We develop a piecewise linear (PWL) Galerkin finite element spatial discretization for the multi-dimensional radiation diffusion equation. It uses piecewise linear weight and basis functions in the finite element approximation, and it can be applied on arbitrary polygonal (2-dimensional) or polyhedral (3-dimensional) grids. We show that this new PWL method gives solutions comparable to those from Palmer's finite-volume method. However, since the PWL method produces a symmetric positive definite coefficient matrix, it should be substantially more computationally efficient than Palmer's method, which produces an asymmetric matrix. We conclude that the Galerkin PWL method is an attractive option for solving diffusion equations on unstructured grids. (authors)

  7. A Piecewise Linear Finite Element Discretization of the Diffusion Equation for Arbitrary Polyhedral Grids

    Energy Technology Data Exchange (ETDEWEB)

    Bailey, T S; Adams, M L; Yang, B; Zika, M R

    2005-07-15

    We develop a piecewise linear (PWL) Galerkin finite element spatial discretization for the multi-dimensional radiation diffusion equation. It uses piecewise linear weight and basis functions in the finite element approximation, and it can be applied on arbitrary polygonal (2D) or polyhedral (3D) grids. We show that this new PWL method gives solutions comparable to those from Palmer's finite-volume method. However, since the PWL method produces a symmetric positive definite coefficient matrix, it should be substantially more computationally efficient than Palmer's method, which produces an asymmetric matrix. We conclude that the Galerkin PWL method is an attractive option for solving diffusion equations on unstructured grids.

  8. An Implicit Numerical Method for Semilinear Space-Time Fractional Diffusion Equation

    Directory of Open Access Journals (Sweden)

    Gunvant Achutrao BIRAJDAR

    2015-11-01

    Full Text Available The aim of the study is to obtain the solution of semilinear space-time fractional diffusion equation for the first initial boundary value problem (IBVP, by applying an implicit method. The main idea of the method is to convert the problem into an algebraic system which simplifies the computations. We discuss the stability, convergence and error analysis of the implicit finite difference scheme with suitable example using MATLAB.

  9. Stability of travelling wavefronts in discrete reaction–diffusion equations with nonlocal delay effects

    International Nuclear Information System (INIS)

    This paper deals with travelling wavefronts for temporally delayed, spatially discrete reaction–diffusion equations. Using a combination of the weighted energy method and the Green function technique, we prove that all noncritical wavefronts are globally exponentially stable, and critical wavefronts are globally algebraically stable when the initial perturbations around the wavefront decay to zero exponentially near minus infinity regardless of the magnitude of time delay. (paper)

  10. Blowup Analysis for a Nonlocal Diffusion Equation with Reaction and Absorption

    Directory of Open Access Journals (Sweden)

    Yulan Wang

    2012-01-01

    Full Text Available We investigate a nonlocal reaction diffusion equation with absorption under Neumann boundary. We obtain optimal conditions on the exponents of the reaction and absorption terms for the existence of solutions blowing up in finite time, or for the global existence and boundedness of all solutions. For the blowup solutions, we also study the blowup rate estimates and the localization of blowup set. Moreover, we show some numerical experiments which illustrate our results.

  11. Doubly nonlocal reaction-diffusion equation and the emergence of species

    OpenAIRE

    Banerjee, M.; Vougalter, V.; Volpert, V.

    2016-01-01

    The paper is devoted to a reaction-diffusion equation with doubly nonlocal nonlinearity arising in various applications in population dynamics. One of the integral terms corresponds to the nonlocal consumption of resources while another one describes reproduction with different phenotypes. Linear stability analysis of the homogeneous in space stationary solution is carried out. Existence of travelling waves is proved in the case of narrow kernels of the integrals. Periodic travelling waves ar...

  12. 3D λ-modes of the neutron-diffusion equation

    International Nuclear Information System (INIS)

    The paper deals with the calculation of the 3D harmonic lambda modes in a B.W.R. reactor. An algorithm to calculate the harmonic lambda modes corresponding to the steady state two-group 3D neutron-diffusion equation is presented. The algorithm uses subspace iteration method techniques combined with convergence acceleration based on variational principles. The methodology has been tested on two benchmark problems, and applied to obtain the 3D modes of the Cofrentes Nuclear Power Plant. (Author)

  13. Travelling-Wave Solutions and Interfaces for Non-Newtonian Diffusion Equations with Strong Absorption

    Institute of Scientific and Technical Information of China (English)

    Zhongping LI; Wanjuan DU; Chunlai MU

    2013-01-01

    In this paper,we first find finite travelling-wave solutions,and then investigate the short time development of interfaces for non-Newtonian diffusion equations with strong absorption.We show that the initial behavior of the interface depends on the concentration of the mass of u(x,0) near x =0.More precisely,we find a critical value of the concentration,which separates the heating front of interfaces from the cooling front of them.

  14. A posteriori error estimates of constrained optimal control problem governed by convection diffusion equations

    Institute of Scientific and Technical Information of China (English)

    Ningning YAN; Zhaojie ZHOU

    2008-01-01

    In this paper,we study a posteriori error estimates of the edge stabilization Galerkin method for the constrained optimal control problem governed by convection-dominated diffusion equations.The residual-type a posteriori error estimators yield both upper and lower bounds for control u measured in L2-norm and for state y and costate p measured in energy norm.Two numerical examples are presented to illustrate the effectiveness of the error estimators provided in this paper.

  15. An analytic algorithm for the space-time fractional reaction-diffusion equation

    Directory of Open Access Journals (Sweden)

    M. G. Brikaa

    2015-11-01

    Full Text Available In this paper, we solve the space-time fractional reaction-diffusion equation by the fractional homotopy analysis method. Solutions of different examples of the reaction term will be computed and investigated. The approximation solutions of the studied models will be put in the form of convergent series to be easily computed and simulated. Comparison with the approximation solution of the classical case of the studied modeled with their approximation errors will also be studied.

  16. Solution of the Atmospheric Diffusion Equation with Longitudinal Wind Speed Depending on Source Distance

    OpenAIRE

    Davidson Martins Moreira; Taciana Toledo de Almeida Albuquerque

    2016-01-01

    Abstract An integral semi-analytical solution of the atmospheric diffusion equation considering wind speed as a function of both downwind distance from a pollution source and vertical height is presented. The model accounts for transformation and removal mechanisms via both chemical reaction and dry deposition processes. A hypothetical dispersion of contaminants emitted from an urban pollution source in the presence of mesoscale winds in an unstable atmospheric boundary layer is showed. The r...

  17. ASYMPTOTIC SOLUTION OF ACTIVATOR INHIBITOR SYSTEMS FOR NONLINEAR REACTION DIFFUSION EQUATIONS

    Institute of Scientific and Technical Information of China (English)

    Jiaqi MO; Wantao LIN

    2008-01-01

    A nonlinear reaction diffusion equations for activator inhibitor systems is considered. Under suitable conditions, firstly, the outer solution of the original problem is obtained, secondly, using the variables of multiple scales and the expanding theory of power series the formal asymptotic expansions of the solution are constructed, and finally, using the theory of differential inequalities the uniform validity and asymptotic behavior of the solution are studied.

  18. THE LARGE TIME BEHAVIOR OF SPECTRAL APPROXIMATION FOR A CLASS OF PSEUDOPARABOLIC VISCOUS DIFFUSION EQUATION

    Institute of Scientific and Technical Information of China (English)

    2007-01-01

    The asymptotic behavior of the solutions to a class of pseudoparabolic viscous diffusion equation with periodic initial condition is studied by using the spectral method.The semidiscrete Fourier approximate solution of the problem is constructed and the error estimation between spectral approximate solution and exact solution on large time is also obtained. The existence of the approximate attractor AN and the upper semicontinuity d(AN, A) → 0 are proved.

  19. Equation of Diffusion of a Composite Mixture into a Composite Medium

    Science.gov (United States)

    Kravchuk, A. S.; Kravchuk, A. I.; Popova, T. S.

    2016-07-01

    The equation of diffusion of a composite mixture into a composite medium has been obtained for the first time. The assumption used is that the macropoint of the medium, i.e., an elementary macrovolume, in which the statistical parameters of distribution of inhomogeneities coincide with the corresponding values assigned for the medium as a whole, is small compared to the geometric dimensions of the volume considered. The ″Reuss-Voigt fork″ has been obtained for determining the limits of the change in the diffusion coefficient. Thereafter the fork is narrowed to the ″Kravchuk-Tarasyuk fork.″ Effective diffusion coefficients are obtained as an arithmetic mean value of the Kravchuk-Tarasyuk fork. The found averaged physical parameters can be used in solving specific physical problems for inhomogeneous media.

  20. Accelerated molecular dynamics and equation-free methods for simulating diffusion in solids.

    Energy Technology Data Exchange (ETDEWEB)

    Deng, Jie; Zimmerman, Jonathan A.; Thompson, Aidan Patrick; Brown, William Michael (Oak Ridge National Laboratories, Oak Ridge, TN); Plimpton, Steven James; Zhou, Xiao Wang; Wagner, Gregory John; Erickson, Lindsay Crowl

    2011-09-01

    Many of the most important and hardest-to-solve problems related to the synthesis, performance, and aging of materials involve diffusion through the material or along surfaces and interfaces. These diffusion processes are driven by motions at the atomic scale, but traditional atomistic simulation methods such as molecular dynamics are limited to very short timescales on the order of the atomic vibration period (less than a picosecond), while macroscale diffusion takes place over timescales many orders of magnitude larger. We have completed an LDRD project with the goal of developing and implementing new simulation tools to overcome this timescale problem. In particular, we have focused on two main classes of methods: accelerated molecular dynamics methods that seek to extend the timescale attainable in atomistic simulations, and so-called 'equation-free' methods that combine a fine scale atomistic description of a system with a slower, coarse scale description in order to project the system forward over long times.

  1. Travelling Waves for a Density Dependent Diffusion Nagumo Equation over the Real Line

    Institute of Scientific and Technical Information of China (English)

    Robert A. Van Gorder

    2012-01-01

    We consider the density dependent diffusion Nagumo equation, where the diffusion coefficient is a simple power function. This equation is used in modelling electrical pulse propagation in nerve axons and in population genetics (amongst other areas). In the present paper, the δ-expansion method is applied to a travelling wave reduction of the problem, so that we may obtain globally valid perturbation solutions (in the sense that the perturbation solutions are valid over the entire infinite domain, not just locally; hence the results are a generalization of the local solutions considered recently in the literature). The resulting boundary value problem is solved on the real line subject to conditions at z →±∞. Whenever a perturbative method is applied, it is important to discuss the accuracy and convergence properties of the resulting perturbation expansions. We compare our results with those of two different numerical methods (designed for initial and boundary value problems, respectively) and deduce that the perturbation expansions agree with the numerical results after a reasonable number of iterations. Finally, we are able to discuss the influence of the wave speed c and the asymptotic concentration value α on the obtained solutions. Upon recasting the density dependent diffusion Nagumo equation as a two-dimensional dynamical system, we are also able to discuss the influence of the nonlinear density dependence (which is governed by a power-law parameter m) on oscillations of the travelling wave solutions.

  2. Direct method of solving finite difference nonlinear equations for multicomponent diffusion in a gas centrifuge

    International Nuclear Information System (INIS)

    This paper describes the the next evolution step in development of the direct method for solving systems of Nonlinear Algebraic Equations (SNAE). These equations arise from the finite difference approximation of original nonlinear partial differential equations (PDE). This method has been extended on the SNAE with three variables. The solving SNAE bases on Reiterating General Singular Value Decomposition of rectangular matrix pencils (RGSVD-algorithm). In contrast to the computer algebra algorithm in integer arithmetic based on the reduction to the Groebner's basis that algorithm is working in floating point arithmetic and realizes the reduction to the Kronecker's form. The possibilities of the method are illustrated on the example of solving the one-dimensional diffusion equation for 3-component model isotope mixture in a ga centrifuge. The implicit scheme for the finite difference equations without simplifying the nonlinear properties of the original equations is realized. The technique offered provides convergence to the solution for the single run. The Toolbox SNAE is developed in the framework of the high performance numeric computation and visualization software MATLAB. It includes more than 30 modules in MATLAB language for solving SNAE with two and three variables. (author)

  3. On the master equation approach to diffusive grain-surface chemistry: the H, O, CO system

    CERN Document Server

    Stantcheva, T; Herbst, E

    2002-01-01

    We have used the master equation approach to study a moderately complex network of diffusive reactions occurring on the surfaces of interstellar dust particles. This network is meant to apply to dense clouds in which a large portion of the gas-phase carbon has already been converted to carbon monoxide. Hydrogen atoms, oxygen atoms, and CO molecules are allowed to accrete onto dust particles and their chemistry is followed. The stable molecules produced are oxygen, hydrogen, water, carbon dioxide (CO2), formaldehyde (H2CO), and methanol (CH3OH). The surface abundances calculated via the master equation approach are in good agreement with those obtained via a Monte Carlo method but can differ considerably from those obtained with standard rate equations.

  4. Long-time behavior of a finite volume discretization for a fourth order diffusion equation

    Science.gov (United States)

    Maas, Jan; Matthes, Daniel

    2016-07-01

    We consider a non-standard finite-volume discretization of a strongly non-linear fourth order diffusion equation on the d-dimensional cube, for arbitrary d≥slant 1 . The scheme preserves two important structural properties of the equation: the first is the interpretation as a gradient flow in a mass transportation metric, and the second is an intimate relation to a linear Fokker–Planck equation. Thanks to these structural properties, the scheme possesses two discrete Lyapunov functionals. These functionals approximate the entropy and the Fisher information, respectively, and their dissipation rates converge to the optimal ones in the discrete-to-continuous limit. Using the dissipation, we derive estimates on the long-time asymptotics of the discrete solutions. Finally, we present results from numerical experiments which indicate that our discretization is able to capture significant features of the complex original dynamics, even with a rather coarse spatial resolution.

  5. Local error estimates for adaptive simulation of the reaction-diffusion master equation via operator splitting

    Science.gov (United States)

    Hellander, Andreas; Lawson, Michael J.; Drawert, Brian; Petzold, Linda

    2014-06-01

    The efficiency of exact simulation methods for the reaction-diffusion master equation (RDME) is severely limited by the large number of diffusion events if the mesh is fine or if diffusion constants are large. Furthermore, inherent properties of exact kinetic-Monte Carlo simulation methods limit the efficiency of parallel implementations. Several approximate and hybrid methods have appeared that enable more efficient simulation of the RDME. A common feature to most of them is that they rely on splitting the system into its reaction and diffusion parts and updating them sequentially over a discrete timestep. This use of operator splitting enables more efficient simulation but it comes at the price of a temporal discretization error that depends on the size of the timestep. So far, existing methods have not attempted to estimate or control this error in a systematic manner. This makes the solvers hard to use for practitioners since they must guess an appropriate timestep. It also makes the solvers potentially less efficient than if the timesteps were adapted to control the error. Here, we derive estimates of the local error and propose a strategy to adaptively select the timestep when the RDME is simulated via a first order operator splitting. While the strategy is general and applicable to a wide range of approximate and hybrid methods, we exemplify it here by extending a previously published approximate method, the diffusive finite-state projection (DFSP) method, to incorporate temporal adaptivity.

  6. Local error estimates for adaptive simulation of the Reaction–Diffusion Master Equation via operator splitting

    Science.gov (United States)

    Hellander, Andreas; Lawson, Michael J; Drawert, Brian; Petzold, Linda

    2015-01-01

    The efficiency of exact simulation methods for the reaction-diffusion master equation (RDME) is severely limited by the large number of diffusion events if the mesh is fine or if diffusion constants are large. Furthermore, inherent properties of exact kinetic-Monte Carlo simulation methods limit the efficiency of parallel implementations. Several approximate and hybrid methods have appeared that enable more efficient simulation of the RDME. A common feature to most of them is that they rely on splitting the system into its reaction and diffusion parts and updating them sequentially over a discrete timestep. This use of operator splitting enables more efficient simulation but it comes at the price of a temporal discretization error that depends on the size of the timestep. So far, existing methods have not attempted to estimate or control this error in a systematic manner. This makes the solvers hard to use for practitioners since they must guess an appropriate timestep. It also makes the solvers potentially less efficient than if the timesteps are adapted to control the error. Here, we derive estimates of the local error and propose a strategy to adaptively select the timestep when the RDME is simulated via a first order operator splitting. While the strategy is general and applicable to a wide range of approximate and hybrid methods, we exemplify it here by extending a previously published approximate method, the Diffusive Finite-State Projection (DFSP) method, to incorporate temporal adaptivity. PMID:26865735

  7. Development of Multigrid Methods for diffusion, Advection, and the incompressible Navier-Stokes Equations

    Energy Technology Data Exchange (ETDEWEB)

    Gjesdal, Thor

    1997-12-31

    This thesis discusses the development and application of efficient numerical methods for the simulation of fluid flows, in particular the flow of incompressible fluids. The emphasis is on practical aspects of algorithm development and on application of the methods either to linear scalar model equations or to the non-linear incompressible Navier-Stokes equations. The first part deals with cell centred multigrid methods and linear correction scheme and presents papers on (1) generalization of the method to arbitrary sized grids for diffusion problems, (2) low order method for advection-diffusion problems, (3) attempt to extend the basic method to advection-diffusion problems, (4) Fourier smoothing analysis of multicolour relaxation schemes, and (5) analysis of high-order discretizations for advection terms. The second part discusses a multigrid based on pressure correction methods, non-linear full approximation scheme, and papers on (1) systematic comparison of the performance of different pressure correction smoothers and some other algorithmic variants, low to moderate Reynolds numbers, and (2) systematic study of implementation strategies for high order advection schemes, high-Re flow. An appendix contains Fortran 90 data structures for multigrid development. 160 refs., 26 figs., 22 tabs.

  8. Thermal diffusion segregation in granular binary mixtures described by the Enskog equation

    Energy Technology Data Exchange (ETDEWEB)

    Garzo, Vicente, E-mail: vicenteg@unex.es [Departamento de Fisica, Universidad de Extremadura, E-06071 Badajoz (Spain)

    2011-05-15

    The diffusion induced by a thermal gradient in a granular binary mixture is analyzed here in the context of the (inelastic) Enskog equation. Although the Enskog equation neglects velocity correlations among particles that are about to collide, it retains the spatial correlations arising from volume exclusion effects and thus is expected to be applicable for moderate densities. In the steady state with gradients only along a given direction, a segregation criterion is obtained from the thermal diffusion factor {Lambda} by measuring the amount of segregation parallel to the thermal gradient. As expected, the sign of the factor {Lambda} provides a criterion for the transition between the Brazil-nut effect (BNE) and the reverse Brazil-nut effect (RBNE) by varying the parameters of the mixture (the masses and sizes of particles, concentration, solid volume fraction and coefficients of restitution). The form of the phase diagrams for the BNE/RBNE transition is illustrated in detail for several systems, with special emphasis on the significant role played by the inelasticity of collisions. In particular, an effect already found in dilute gases (segregation in a binary mixture of identical masses and sizes but different coefficients of restitution) is extended to dense systems. A comparison with recent computer simulation results reveals good qualitative agreement at the level of the thermal diffusion factor. The present analysis generalizes to arbitrary concentration previous theoretical results derived in the tracer limit case.

  9. Hybrid simplified spherical harmonics with diffusion equation for light propagation in tissues

    International Nuclear Information System (INIS)

    Aiming at the limitations of the simplified spherical harmonics approximation (SPN) and diffusion equation (DE) in describing the light propagation in tissues, a hybrid simplified spherical harmonics with diffusion equation (HSDE) based diffuse light transport model is proposed. In the HSDE model, the living body is first segmented into several major organs, and then the organs are divided into high scattering tissues and other tissues. DE and SPN are employed to describe the light propagation in these two kinds of tissues respectively, which are finally coupled using the established boundary coupling condition. The HSDE model makes full use of the advantages of SPN and DE, and abandons their disadvantages, so that it can provide a perfect balance between accuracy and computation time. Using the finite element method, the HSDE is solved for light flux density map on body surface. The accuracy and efficiency of the HSDE are validated with both regular geometries and digital mouse model based simulations. Corresponding results reveal that a comparable accuracy and much less computation time are achieved compared with the SPN model as well as a much better accuracy compared with the DE one. (paper)

  10. Space-Time Fractional Diffusion-Advection Equation with Caputo Derivative

    Directory of Open Access Journals (Sweden)

    José Francisco Gómez Aguilar

    2014-01-01

    Full Text Available An alternative construction for the space-time fractional diffusion-advection equation for the sedimentation phenomena is presented. The order of the derivative is considered as 0<β, γ≤1 for the space and time domain, respectively. The fractional derivative of Caputo type is considered. In the spatial case we obtain the fractional solution for the underdamped, undamped, and overdamped case. In the temporal case we show that the concentration has amplitude which exhibits an algebraic decay at asymptotically large times and also shows numerical simulations where both derivatives are taken in simultaneous form. In order that the equation preserves the physical units of the system two auxiliary parameters σx and σt are introduced characterizing the existence of fractional space and time components, respectively. A physical relation between these parameters is reported and the solutions in space-time are given in terms of the Mittag-Leffler function depending on the parameters β and γ. The generalization of the fractional diffusion-advection equation in space-time exhibits anomalous behavior.

  11. Mixed-hybrid finite element method for the transport equation and diffusion approximation of transport problems

    International Nuclear Information System (INIS)

    This thesis focuses on mathematical analysis, numerical resolution and modelling of the transport equations. First of all, we deal with numerical approximation of the solution of the transport equations by using a mixed-hybrid scheme. We derive and study a mixed formulation of the transport equation, then we analyse the related variational problem and present the discretization and the main properties of the scheme. We particularly pay attention to the behavior of the scheme and we show its efficiency in the diffusion limit (when the mean free path is small in comparison with the characteristic length of the physical domain). We present academical benchmarks in order to compare our scheme with other methods in many physical configurations and validate our method on analytical test cases. Unstructured and very distorted meshes are used to validate our scheme. The second part of this thesis deals with two transport problems. The first one is devoted to the study of diffusion due to boundary conditions in a transport problem between two plane plates. The second one consists in modelling and simulating radiative transfer phenomenon in case of the industrial context of inertial confinement fusion. (author)

  12. A CLASS OF NONLINEAR SINGULARLY PERTURBED INITIAL BOUNDARY VALUE PROBLEMS FOR REACTION DIFFUSION EQUATIONS WITH BOUNDARY PERTURBATION

    Institute of Scientific and Technical Information of China (English)

    Mo Jiaqi

    2007-01-01

    A class of nonlinear initial boundary value problems for reaction diffusion equations with boundary perturbation is considered. Under suitable conditions and using the theory of differential inequalities the asymptotic solution of the initial boundary value problems is studied.

  13. Dictyostelium Chemotaxis studied with fluorescence fluctuation spectroscopy

    NARCIS (Netherlands)

    Ruchira, A.

    2005-01-01

    The movement of cells in the direction of a chemical gradient, also known as chemotaxis, is a vital biological process. During chemotaxis, minute extracellular signals are translated into complex cellular responses such as change in morphology and motility. To understand the chemotaxis mechanism at

  14. Regularity and mass conservation for discrete coagulation–fragmentation equations with diffusion

    KAUST Repository

    Cañizo, J.A.

    2010-03-01

    We present a new a priori estimate for discrete coagulation-fragmentation systems with size-dependent diffusion within a bounded, regular domain confined by homogeneous Neumann boundary conditions. Following from a duality argument, this a priori estimate provides a global L2 bound on the mass density and was previously used, for instance, in the context of reaction-diffusion equations. In this paper we demonstrate two lines of applications for such an estimate: On the one hand, it enables to simplify parts of the known existence theory and allows to show existence of solutions for generalised models involving collision-induced, quadratic fragmentation terms for which the previous existence theory seems difficult to apply. On the other hand and most prominently, it proves mass conservation (and thus the absence of gelation) for almost all the coagulation coefficients for which mass conservation is known to hold true in the space homogeneous case. © 2009 Elsevier Masson SAS. All rights reserved.

  15. Analytical Solutions of a Fractional Diffusion-advection Equation for Solar Cosmic-Ray Transport

    Science.gov (United States)

    Litvinenko, Yuri E.; Effenberger, Frederic

    2014-12-01

    Motivated by recent applications of superdiffusive transport models to shock-accelerated particle distributions in the heliosphere, we analytically solve a one-dimensional fractional diffusion-advection equation for the particle density. We derive an exact Fourier transform solution, simplify it in a weak diffusion approximation, and compare the new solution with previously available analytical results and with a semi-numerical solution based on a Fourier series expansion. We apply the results to the problem of describing the transport of energetic particles, accelerated at a traveling heliospheric shock. Our analysis shows that significant errors may result from assuming an infinite initial distance between the shock and the observer. We argue that the shock travel time should be a parameter of a realistic superdiffusive transport model.

  16. Analytical solutions of a fractional diffusion-advection equation for solar cosmic-ray transport

    CERN Document Server

    Litvinenko, Yuri E

    2014-01-01

    Motivated by recent applications of superdiffusive transport models to shock-accelerated particle distributions in the heliosphere, we solve analytically a one-dimensional fractional diffusion-advection equation for the particle density. We derive an exact Fourier transform solution, simplify it in a weak diffusion approximation, and compare the new solution with previously available analytical results and with a semi-numerical solution based on a Fourier series expansion. We apply the results to the problem of describing the transport of energetic particles, accelerated at a traveling heliospheric shock. Our analysis shows that significant errors may result from assuming an infinite initial distance between the shock and the observer. We argue that the shock travel time should be a parameter of a realistic superdiffusive transport model.

  17. Retinal Image Enhancement Using Robust Inverse Diffusion Equation and Self-Similarity Filtering

    Science.gov (United States)

    Fu, Shujun; Xu, Lingzhong; Zhao, Kun; Zhang, Caiming

    2016-01-01

    As a common ocular complication for diabetic patients, diabetic retinopathy has become an important public health problem in the world. Early diagnosis and early treatment with the help of fundus imaging technology is an effective control method. In this paper, a robust inverse diffusion equation combining a self-similarity filtering is presented to detect and evaluate diabetic retinopathy using retinal image enhancement. A flux corrected transport technique is used to control diffusion flux adaptively, which eliminates overshoots inherent in the Laplacian operation. Feature preserving denoising by the self-similarity filtering ensures a robust enhancement of noisy and blurry retinal images. Experimental results demonstrate that this algorithm can enhance important details of retinal image data effectively, affording an opportunity for better medical interpretation and subsequent processing. PMID:27388503

  18. An alternative solution of the neutron diffusion equation in cylindrical symmetry

    Energy Technology Data Exchange (ETDEWEB)

    Dababneh, Saed, E-mail: dababneh@bau.edu.jo [Prince Abdullah Bin Ghazi Faculty of Science and IT, Al-Balqa Applied University, Salt 19117 (Jordan); Jordan Nuclear Regulatory Commission, P.O. Box 2587, Amman 11941 (Jordan); Khasawneh, Kafa, E-mail: k.khasawneh@bau.edu.jo [Prince Abdullah Bin Ghazi Faculty of Science and IT, Al-Balqa Applied University, Salt 19117 (Jordan); Odibat, Zaid [Prince Abdullah Bin Ghazi Faculty of Science and IT, Al-Balqa Applied University, Salt 19117 (Jordan)

    2011-05-15

    Research highlights: > The homotopy perturbation method is applied in cylindrical symmetry. > Analytical solutions are found for neutron diffusion in infinite and finite cylinders. > Different boundary conditions are applied. > The method reproduces critical size and flux as calculated using canonical methods. - Abstract: Alternative analytical solutions of the neutron diffusion equation for both infinite and finite cylinders of fissile material are formulated using the homotopy perturbation method. Zero flux boundary conditions are investigated on boundary as well as on extrapolated boundary. Numerical results are provided for one-speed fast neutrons in {sup 235}U. The results reveal that the homotopy perturbation method provides an accurate alternative to the Bessel function based solutions for these geometries.

  19. Periodic and solitary wave solutions of cubic–quintic nonlinear reaction-diffusion equation with variable convection coefficients

    Indian Academy of Sciences (India)

    BHARDWAJ S B; SINGH RAM MEHAR; SHARMA KUSHAL; MISHRA S C

    2016-06-01

    Attempts have been made to explore the exact periodic and solitary wave solutions of nonlinear reaction diffusion (RD) equation involving cubic–quintic nonlinearity along with timedependent convection coefficients. Effect of varying model coefficients on the physical parameters of solitary wave solutions is demonstrated. Depending upon the parametric condition, the periodic,double-kink, bell and antikink-type solutions for cubic–quintic nonlinear reaction-diffusion equation are extracted. Such solutions can be used to explain various biological and physical phenomena.

  20. Perturbation theory for the one-dimensional reaction-diffusion equation with a quadratic nonlinearity of coalescence/annihilation type

    Energy Technology Data Exchange (ETDEWEB)

    Abad, E [Departamento de Fisica, Universidad de Extremadura, E-06071 Badajoz (Spain); Frisch, H L, E-mail: eabad@unex.e [Department of Chemistry, State University of New York at Albany, Albany, NY 12222 (United States)

    2010-05-01

    We take a reaction-diffusion equation describing two-particle coalescence (or, alternatively, annihilation) in an infinite domain as a starting point to develop a perturbation scheme based in an expansion in terms of the effective reaction constant valid in the slow reaction limit. We compute the first-order correction to the diffusion equation and show that the underlying approximation breaks down at late times, suggesting that higher-order terms become increasingly important in this regime.

  1. Smoothed particle hydrodynamics model for Landau-Lifshitz-Navier-Stokes and advection-diffusion equations

    Science.gov (United States)

    Kordilla, Jannes; Pan, Wenxiao; Tartakovsky, Alexandre

    2014-12-01

    We propose a novel smoothed particle hydrodynamics (SPH) discretization of the fully coupled Landau-Lifshitz-Navier-Stokes (LLNS) and stochastic advection-diffusion equations. The accuracy of the SPH solution of the LLNS equations is demonstrated by comparing the scaling of velocity variance and the self-diffusion coefficient with kinetic temperature and particle mass obtained from the SPH simulations and analytical solutions. The spatial covariance of pressure and velocity fluctuations is found to be in a good agreement with theoretical models. To validate the accuracy of the SPH method for coupled LLNS and advection-diffusion equations, we simulate the interface between two miscible fluids. We study formation of the so-called "giant fluctuations" of the front between light and heavy fluids with and without gravity, where the light fluid lies on the top of the heavy fluid. We find that the power spectra of the simulated concentration field are in good agreement with the experiments and analytical solutions. In the absence of gravity, the power spectra decay as the power -4 of the wavenumber—except for small wavenumbers that diverge from this power law behavior due to the effect of finite domain size. Gravity suppresses the fluctuations, resulting in much weaker dependence of the power spectra on the wavenumber. Finally, the model is used to study the effect of thermal fluctuation on the Rayleigh-Taylor instability, an unstable dynamics of the front between a heavy fluid overlaying a light fluid. The front dynamics is shown to agree well with the analytical solutions.

  2. The stochastic dance of circling sperm cells: sperm chemotaxis in the plane

    Energy Technology Data Exchange (ETDEWEB)

    Friedrich, B M; Juelicher, F [Max Planck Institute for the Physics of Complex Systems, Noethnitzer Strasse 38, 01187 Dresden (Germany)], E-mail: ben@pks.mpg.de, E-mail: julicher@pks.mpg.de

    2008-12-15

    Biological systems such as single cells must function in the presence of fluctuations. It has been shown in a two-dimensional experimental setup that sea urchin sperm cells move toward a source of chemoattractant along planar trochoidal swimming paths, i.e. drifting circles. In these experiments, a pronounced variability of the swimming paths is observed. We present a theoretical description of sperm chemotaxis in two dimensions which takes fluctuations into account. We derive a coarse-grained theory of stochastic sperm swimming paths in a concentration field of chemoattractant. Fluctuations enter as multiplicative noise in the equations for the sperm swimming path. We discuss the stochastic properties of sperm swimming and predict a concentration-dependence of the effective diffusion constant of sperm swimming which could be tested in experiments.

  3. The stochastic dance of circling sperm cells: sperm chemotaxis in the plane

    International Nuclear Information System (INIS)

    Biological systems such as single cells must function in the presence of fluctuations. It has been shown in a two-dimensional experimental setup that sea urchin sperm cells move toward a source of chemoattractant along planar trochoidal swimming paths, i.e. drifting circles. In these experiments, a pronounced variability of the swimming paths is observed. We present a theoretical description of sperm chemotaxis in two dimensions which takes fluctuations into account. We derive a coarse-grained theory of stochastic sperm swimming paths in a concentration field of chemoattractant. Fluctuations enter as multiplicative noise in the equations for the sperm swimming path. We discuss the stochastic properties of sperm swimming and predict a concentration-dependence of the effective diffusion constant of sperm swimming which could be tested in experiments.

  4. Free boundary value problems for a class of generalized diffusion equation

    Institute of Scientific and Technical Information of China (English)

    2002-01-01

    The transport behavior of free boundary value problems for a class of generalized diffusion equations was studied. Suitable similarity transformations were used to convert the problems into a class of singular nonlinear two-point boundary value problems and similarity solutions were numerical presented for different representations of heat conduction function, convection function, heat flux function, and power law parameters by utilizing the shooting technique. The results revealed the flux transfer mechanism and the character as well as the effects of parameters on the solutions.

  5. Relation between the complex Ginzburg-Landau equation and reaction-diffusion System

    Institute of Scientific and Technical Information of China (English)

    Shao Xin; Ren Yi; Ouyang Qi

    2006-01-01

    The complex Ginzburg-Landau equation(CGLE)has been used to describe the travelling wave behaviour in reaction-diffusion (RD) systems. We argue that this description is valid only when the RD system is close to the Hopf bifurcation,and is not valid when a RD system is away from the onset.To test this,we study spirals and anti-spirals in the chlorite-iodide-malonic acid (CIMA) reaction and the corresponding CGLE.Numerical simulations confirm that the CGLE can only be applied to the CIMA reaction when it is very near the Hopf onset.

  6. Singular perturbation for the weakly nonlinear reaction diffusion equation with boundary perturbation

    Institute of Scientific and Technical Information of China (English)

    MO Jia-qi

    2008-01-01

    In this paper, a class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with boundary perturbation are considered under suitable conditions. Firstly, by dint of the regular perturbation method, the outer solution of the original problem is obtained. Secondly, by using the stretched variable and the expansion theory of power series the initial layer of the solution is constructed. And then, by using the theory of differential inequalities, the asymptotic behavior of the solution for the initial boundary value problems is studied. Finally, using some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed.

  7. THE NONLINEAR NONLOCAL SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS WITH A BOUNDARY PERTURBATION

    Institute of Scientific and Technical Information of China (English)

    Jingsun Yao; Jiaqi Mo

    2005-01-01

    The nonlinear nonlocal singularly perturbed initial boundary value problems for reaction diffusion equations with a boundary perturbation is considered. Under suitable conditions, the outer solution of the original problem is obtained. Using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer is constructed. And then using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems is studied. Finally the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed.

  8. A CLASS OF NONLINEAR NONLOCAL SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS WITH BOUNDARY PERTURBATION

    Institute of Scientific and Technical Information of China (English)

    MO Jia-qi; WANG Hui; LIN Wan-tao

    2005-01-01

    A class of nonlinear nonlocal for singularly perturbed Robin initial boundary value problems for reaction diffusion equations with boundary perturbation is considered. Under suitable conditions, first, the outer solution of the original problem was obtained. Secondly, using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer was constructed. Finally, using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems was studied, and educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation were discussed.

  9. Asymptotic Analysis to a Diffusion Equation with a Weighted Nonlo cal Source

    Institute of Scientific and Technical Information of China (English)

    JIANG Liang-jun

    2015-01-01

    In this paper, we deal with the blow-up property of the solution to the diffusion equation ut=∆u+a(x)f(u) RΩh(u)dx, x∈Ω, t>0 subject to the null Dirichlet boundary condition. We will show that under certain conditions, the solution blows up in finite time and prove that the set of all blow-up points is the whole region. Especially, in case of f(s)=sp, h(s)=sq, 0≤p≤1, p+q>1, we obtain the asymptotic behavior of the blow up solution.

  10. Stability and error analysis on modified nodal expansion method for transient convection-diffusion equation

    International Nuclear Information System (INIS)

    To further investigate the features of modified nodal expansion method (MNEM) for solving the convection-diffusion equation, the stability and error analysis were carried out. Based on sign preservation principle, the stability analysis reveals that the MNEM has inherent stability. The error analysis was implemented through a series of numerical experiments, and the results show that the MNEM is 3rd order scheme for one dimensional problem, while as 2nd order scheme for multidimensional problem because of using simple transverse leakage approximation. (authors)

  11. Solution of thermal neutron diffusion equation for the two-component system by perturbation calculation

    International Nuclear Information System (INIS)

    A method of solving the diffusion equation for the th ermal neutron flux in a heterogeneous medium is presented. Perturbation calculation is successfully applied for the cylindrical concentric system after testing this method for the spherical concentric geometry analytically solved by Czubek (1981). The method permits to calculate the t hermal neutron decay constant and the space distribution of the thermal neutron flux in a heterogeneous geom etry. The condition of the constant value of the neutron flux in the inner part of the system has to be m et. This method has an application in the measurement of the thermal neutron absorption cross section, presented by Czubek (1981). (author)

  12. The Galerkin finite element method for a multi-term time-fractional diffusion equation

    KAUST Repository

    Jin, Bangti

    2015-01-01

    © 2014 The Authors. We consider the initial/boundary value problem for a diffusion equation involving multiple time-fractional derivatives on a bounded convex polyhedral domain. We analyze a space semidiscrete scheme based on the standard Galerkin finite element method using continuous piecewise linear functions. Nearly optimal error estimates for both cases of initial data and inhomogeneous term are derived, which cover both smooth and nonsmooth data. Further we develop a fully discrete scheme based on a finite difference discretization of the time-fractional derivatives, and discuss its stability and error estimate. Extensive numerical experiments for one- and two-dimensional problems confirm the theoretical convergence rates.

  13. Preconditioned iterative methods for space-time fractional advection-diffusion equations

    Science.gov (United States)

    Zhao, Zhi; Jin, Xiao-Qing; Lin, Matthew M.

    2016-08-01

    In this paper, we propose practical numerical methods for solving a class of initial-boundary value problems of space-time fractional advection-diffusion equations. First, we propose an implicit method based on two-sided Grünwald formulae and discuss its stability and consistency. Then, we develop the preconditioned generalized minimal residual (preconditioned GMRES) method and preconditioned conjugate gradient normal residual (preconditioned CGNR) method with easily constructed preconditioners. Importantly, because resulting systems are Toeplitz-like, fast Fourier transform can be applied to significantly reduce the computational cost. We perform numerical experiments to demonstrate the efficiency of our preconditioners, even in cases with variable coefficients.

  14. Boundary Element Method with Non—overlapping Domain Decomposition for Diffusion Equation

    Institute of Scientific and Technical Information of China (English)

    ZHUJialin; ZHANGTaiping

    2002-01-01

    A boundary element method based on non-overlapping domain decomposition method to solve the time-dependent diffusion equations is presented.The time-dependent fundamental solution is used in the formulation of boundary integrals and the time integratioin process always restarts from the initial time condition.The process of replacing the interface values,which needs a summation of boundary integrals related to the boundary values at previous time steps can be treated in parallel parallel iterative procedure,Numerical experiments demonstrate that the implementation of the present alogrithm is efficient.

  15. Diffusion coefficients of Fokker-Planck equation for rotating dust grains in a fusion plasma

    International Nuclear Information System (INIS)

    In the fusion devices, ions, H atoms, and H2 molecules collide with dust grains and exert stochastic torques which lead to small variations in angular momentum of the grain. By considering adsorption of the colliding particles, thermal desorption of H atoms and normal H2 molecules, and desorption of the recombined H2 molecules from the surface of an oblate spheroidal grain, we obtain diffusion coefficients of the Fokker-Planck equation for the distribution function of fluctuating angular momentum. Torque coefficients corresponding to the recombination mechanism show that the nonspherical dust grains may rotate with a suprathermal angular velocity

  16. Solution of the Atmospheric Diffusion Equation with Longitudinal Wind Speed Depending on Source Distance

    Directory of Open Access Journals (Sweden)

    Davidson Martins Moreira

    2016-06-01

    Full Text Available Abstract An integral semi-analytical solution of the atmospheric diffusion equation considering wind speed as a function of both downwind distance from a pollution source and vertical height is presented. The model accounts for transformation and removal mechanisms via both chemical reaction and dry deposition processes. A hypothetical dispersion of contaminants emitted from an urban pollution source in the presence of mesoscale winds in an unstable atmospheric boundary layer is showed. The results demonstrate that the mesoscale winds generated by urban heat islands advect contaminants upward, which increases the intensity of air pollution in urban areas.

  17. Stochastic Lotka-Volterra equations: A model of lagged diffusion of technology in an interconnected world

    Science.gov (United States)

    Chakrabarti, Anindya S.

    2016-01-01

    We present a model of technological evolution due to interaction between multiple countries and the resultant effects on the corresponding macro variables. The world consists of a set of economies where some countries are leaders and some are followers in the technology ladder. All of them potentially gain from technological breakthroughs. Applying Lotka-Volterra (LV) equations to model evolution of the technology frontier, we show that the way technology diffuses creates repercussions in the partner economies. This process captures the spill-over effects on major macro variables seen in the current highly globalized world due to trickle-down effects of technology.

  18. Approximate analytical solution of diffusion equation with fractional time derivative using optimal homotopy analysis method

    Directory of Open Access Journals (Sweden)

    S. Das

    2013-12-01

    Full Text Available In this article, optimal homotopy-analysis method is used to obtain approximate analytic solution of the time-fractional diffusion equation with a given initial condition. The fractional derivatives are considered in the Caputo sense. Unlike usual Homotopy analysis method, this method contains at the most three convergence control parameters which describe the faster convergence of the solution. Effects of parameters on the convergence of the approximate series solution by minimizing the averaged residual error with the proper choices of parameters are calculated numerically and presented through graphs and tables for different particular cases.

  19. Diffusion coefficients of Fokker-Planck equation for rotating dust grains in a fusion plasma

    Science.gov (United States)

    Bakhtiyari-Ramezani, M.; Mahmoodi, J.; Alinejad, N.

    2015-11-01

    In the fusion devices, ions, H atoms, and H2 molecules collide with dust grains and exert stochastic torques which lead to small variations in angular momentum of the grain. By considering adsorption of the colliding particles, thermal desorption of H atoms and normal H2 molecules, and desorption of the recombined H2 molecules from the surface of an oblate spheroidal grain, we obtain diffusion coefficients of the Fokker-Planck equation for the distribution function of fluctuating angular momentum. Torque coefficients corresponding to the recombination mechanism show that the nonspherical dust grains may rotate with a suprathermal angular velocity.

  20. Time adaptivity in the diffusive wave approximation to the shallow water equations

    KAUST Repository

    Collier, Nathaniel Oren

    2013-05-01

    We discuss the use of time adaptivity applied to the one dimensional diffusive wave approximation to the shallow water equations. A simple and computationally economical error estimator is discussed which enables time-step size adaptivity. This robust adaptive time discretization corrects the initial time step size to achieve a user specified bound on the discretization error and allows time step size variations of several orders of magnitude. In particular, the one dimensional results presented in this work feature a change of four orders of magnitudes for the time step over the entire simulation. © 2011 Elsevier B.V.

  1. A solution of the neutron diffusion equation in hemispherical symmetry using the homotopy perturbation method

    Energy Technology Data Exchange (ETDEWEB)

    Khasawneh, Kafa [Prince Abdullah Bin Ghazi Faculty of Science and IT, Al-Balqa Applied University, P.O. Box 7051, Salt 19117 (Jordan); Dababneh, Saed, E-mail: dababneh@bau.edu.j [Prince Abdullah Bin Ghazi Faculty of Science and IT, Al-Balqa Applied University, P.O. Box 7051, Salt 19117 (Jordan); Jordan Nuclear Regulatory Commission, P.O. Box 2587, Amman 11941 (Jordan); Odibat, Zaid [Prince Abdullah Bin Ghazi Faculty of Science and IT, Al-Balqa Applied University, P.O. Box 7051, Salt 19117 (Jordan)

    2009-11-15

    The homotopy perturbation method is used to formulate a new analytic solution of the neutron diffusion equation both for a sphere and a hemisphere of fissile material. Different boundary conditions are investigated; including zero flux on boundary, zero flux on extrapolated boundary, and radiation boundary condition. The interaction between two hemispheres with opposite flat faces is also presented. Numerical results are provided for one-speed fast neutrons in {sup 235}U. A comparison with Bessel function based solutions demonstrates that the homotopy perturbation method can exactly reproduce the results. The computational implementation of the analytic solutions was found to improve the numeric results when compared to finite element calculations.

  2. Stability and Bifurcation in a Delayed Reaction-Diffusion Equation with Dirichlet Boundary Condition

    Science.gov (United States)

    Guo, Shangjiang; Ma, Li

    2016-04-01

    In this paper, we study the dynamics of a diffusive equation with time delay subject to Dirichlet boundary condition in a bounded domain. The existence of spatially nonhomogeneous steady-state solution is investigated by applying Lyapunov-Schmidt reduction. The existence of Hopf bifurcation at the spatially nonhomogeneous steady-state solution is derived by analyzing the distribution of the eigenvalues. The direction of Hopf bifurcation and stability of the bifurcating periodic solution are also investigated by means of normal form theory and center manifold reduction. Moreover, we illustrate our general results by applications to the Nicholson's blowflies models with one- dimensional spatial domain.

  3. Application of Runge-Kutta method to solve transient neutron diffusion equation

    International Nuclear Information System (INIS)

    NGFMN-K code was developed to solve transient neutron diffusion equations. Nodal Green's function method based on the second boundary condition was utilized for spatial discreteness. Backward Euler (BE) and a fourth-order accurate diagonally implicit Runge-Kutta (DIRK) method were used for temporal discreteness. Automatic time step control was achieved by embedding a third-order accurate Runge-Kutta solution to estimate the truncation error for DIRK. Numerical evaluations show that results of two methods agree well with reference solution, and DIRK is more accurate and efficient than BE, especially in severe transient. (authors)

  4. Scalable implicit methods for reaction-diffusion equations in two and three space dimensions

    Energy Technology Data Exchange (ETDEWEB)

    Veronese, S.V.; Othmer, H.G. [Univ. of Utah, Salt Lake City, UT (United States)

    1996-12-31

    This paper describes the implementation of a solver for systems of semi-linear parabolic partial differential equations in two and three space dimensions. The solver is based on a parallel implementation of a non-linear Alternating Direction Implicit (ADI) scheme which uses a Cartesian grid in space and an implicit time-stepping algorithm. Various reordering strategies for the linearized equations are used to reduce the stride and improve the overall effectiveness of the parallel implementation. We have successfully used this solver for large-scale reaction-diffusion problems in computational biology and medicine in which the desired solution is a traveling wave that may contain rapid transitions. A number of examples that illustrate the efficiency and accuracy of the method are given here; the theoretical analysis will be presented.

  5. Automatic simplification of systems of reaction-diffusion equations by a posteriori analysis.

    Science.gov (United States)

    Maybank, Philip J; Whiteley, Jonathan P

    2014-02-01

    Many mathematical models in biology and physiology are represented by systems of nonlinear differential equations. In recent years these models have become increasingly complex in order to explain the enormous volume of data now available. A key role of modellers is to determine which components of the model have the greatest effect on a given observed behaviour. An approach for automatically fulfilling this role, based on a posteriori analysis, has recently been developed for nonlinear initial value ordinary differential equations [J.P. Whiteley, Model reduction using a posteriori analysis, Math. Biosci. 225 (2010) 44-52]. In this paper we extend this model reduction technique for application to both steady-state and time-dependent nonlinear reaction-diffusion systems. Exemplar problems drawn from biology are used to demonstrate the applicability of the technique. PMID:24418010

  6. Self-similar singular solution of fast diffusion equation with gradient absorption terms

    Institute of Scientific and Technical Information of China (English)

    SHI Pei-hu; WANG Ming-xin

    2007-01-01

    The self-similar singular solution of the fast diffusion equation with nonlinear gradient absorption terms are studied. By a self-similar transformation, the self-similar solutions satisfy a boundary value problem of nonlinear ordinary differential equation (ODE). Using the shooting arguments, the existence and uniqueness of the solution to the initial data problem of the nonlinear ODE are investigated, and the solutions are classified by the region of the initial data. The necessary and sufficient condition for the existence and uniqueness of self-similar very singular solutions is obtained by investigation of the classification of the solutions. In case of existence, the self-similar singular solution is very singular solution.

  7. THE EXTINCTION BEHAVIOR OF THE SOLUTIONS FOR A CLASS OF REACTION-DIFFUSION EQUATIONS

    Institute of Scientific and Technical Information of China (English)

    陈松林

    2001-01-01

    The methods of Lp estimation are used to discuss the extinction phenomena of the solutions to the following reaction-diffusion equations with initial-boudnary values u/ t = Au-λ |u|γ-1u-βu ((x,t) ∈Ω×(0,+∞)),u(x,t) | Ω×(0.+∞) = 0,u(x,0) = uo(x) ∈ H1 0(Ω) ∩ L1+γ(Ω) (x ∈Ω).Sufficient and necessary conditions about the extinction of the solutions is given Here λ>0, γ> 0, β> 0 are constants, Ω∈ RN is bounded with smooth boundary Ω At last,it is simulated with a higher order equation by using the present methods.

  8. Finite Difference and Sinc-Collocation Approximations to a Class of Fractional Diffusion-Wave Equations

    Directory of Open Access Journals (Sweden)

    Zhi Mao

    2014-01-01

    Full Text Available We propose an efficient numerical method for a class of fractional diffusion-wave equations with the Caputo fractional derivative of order α. This approach is based on the finite difference in time and the global sinc collocation in space. By utilizing the collocation technique and some properties of the sinc functions, the problem is reduced to the solution of a system of linear algebraic equations at each time step. Stability and convergence of the proposed method are rigorously analyzed. The numerical solution is of 3-α order accuracy in time and exponential rate of convergence in space. Numerical experiments demonstrate the validity of the obtained method and support the obtained theoretical results.

  9. Talagrand's T2-Transportation Inequality and Log-Sobolev Inequality for Dissipative SPDEs and Applications to Reaction-Diffusion Equations

    Institute of Scientific and Technical Information of China (English)

    Liming WU; Zhengliang ZHANG

    2006-01-01

    We establish Talagrand's T2-transportation inequalities for infinite dimensional dissipative diffusions with sharp constants, through Galerkin type's approximations and the known results in the finite dimensional case. Furthermore in the additive noise case we prove also logarithmic Sobolev inequalities with sharp constants. Applications to ReactionDiffusion equations are provided.

  10. Parallel computing for homogeneous diffusion and transport equations in neutronics; Calcul parallele pour les equations de diffusion et de transport homogenes en neutronique

    Energy Technology Data Exchange (ETDEWEB)

    Pinchedez, K

    1999-06-01

    Parallel computing meets the ever-increasing requirements for neutronic computer code speed and accuracy. In this work, two different approaches have been considered. We first parallelized the sequential algorithm used by the neutronics code CRONOS developed at the French Atomic Energy Commission. The algorithm computes the dominant eigenvalue associated with PN simplified transport equations by a mixed finite element method. Several parallel algorithms have been developed on distributed memory machines. The performances of the parallel algorithms have been studied experimentally by implementation on a T3D Cray and theoretically by complexity models. A comparison of various parallel algorithms has confirmed the chosen implementations. We next applied a domain sub-division technique to the two-group diffusion Eigen problem. In the modal synthesis-based method, the global spectrum is determined from the partial spectra associated with sub-domains. Then the Eigen problem is expanded on a family composed, on the one hand, from eigenfunctions associated with the sub-domains and, on the other hand, from functions corresponding to the contribution from the interface between the sub-domains. For a 2-D homogeneous core, this modal method has been validated and its accuracy has been measured. (author)

  11. Local and Global Well-posedness of Strong Solutions to the 3D Primitive Equations with Vertical Eddy Diffusivity

    CERN Document Server

    Cao, Chongsheng; Titi, Edriss S

    2013-01-01

    In this paper, we consider the initial-boundary value problem of the viscous 3D primitive equations for oceanic and atmospheric dynamics with only vertical diffusion in the temperature equation. Local and global well-posedness of strong solutions are established for this system with $H^2$ initial data.

  12. Global Well-posedness of Strong Solutions to the 3D Primitive Equations with Horizontal Eddy Diffusivity

    CERN Document Server

    Cao, Chongsheng; Titi, Edriss S

    2014-01-01

    In this paper, we consider the initial-boundary value problem of the 3D primitive equations for oceanic and atmospheric dynamics with only horizontal diffusion in the temperature equation. Global well-posedness of strong solutions are established with $H^2$ initial data.

  13. Variational Multiscale Element Free Galerkin Method Coupled with Low-Pass Filter for Burgers’ Equation with Small Diffusion

    Directory of Open Access Journals (Sweden)

    Ping Zhang

    2016-01-01

    Full Text Available Variational multiscale element free Galerkin (VMEFG method is applied to Burgers’ equation. It can be found that, for the very small diffusivity coefficients, VMEFG method still suffers from instability in the presence of boundary or interior layers. In order to overcome this problem, the high order low-pass filter is used to smooth the solution. Three test examples with very small diffusion are presented and the solutions obtained are compared with exact solutions and some other numerical methods. The numerical results are found in which the VMEFG coupled with low-pass filter works very well for Burgers’ equation with very small diffusivity coefficients.

  14. Solution of the diffusion equations for several groups by the finite elements method

    International Nuclear Information System (INIS)

    The code DELFIN has been implemented for the solution of the neutrons diffusion equations in two dimensions obtained by applying the approximation of several groups of energy. The code works with any number of groups and regions, and can be applied to thermal reactors as well as fast reactor. Providing it with the diffusion coefficients, the effective sections and the fission spectrum we obtain the results for the systems multiplying constant and the flows of each groups. The code was established using the method of finite elements, which is a form of resolution of the variational formulation of the equations applying the Ritz-Galerkin method with continuous polynomial functions by parts, in one case of the Lagrange type with rectangular geometry and up to the third grade. The obtained results and the comparison with the results in the literature, permit to reach the conclusion that it is convenient, to use the rectangular elements in all the cases where the geometry permits it, and demonstrate also that the finite elements method is better than the finite differences method. (author)

  15. Mixed dual finite element methods for the numerical treatment of the diffusion equation in hexagonal geometry

    International Nuclear Information System (INIS)

    The nodal method Minos has been developed to offer a powerful method for the calculation of nuclear reactor cores in rectangular geometry. This method solves the mixed dual form of the diffusion equation and, also of the simplified PN approximation. The discretization is based on Raviart-Thomas' mixed dual finite elements and the iterative algorithm is an alternating direction method, which uses the current as unknown. The subject of this work is to adapt this method to hexagonal geometry. The guiding idea is to construct and test different methods based on the division of a hexagon into trapeze or rhombi with appropriate mapping of these quadrilaterals onto squares in order to take into advantage what is already available in the Minos solver. The document begins with a review of the neutron diffusion equation. Then we discuss its mixed dual variational formulation from a functional as well as from a numerical point of view. We study conformal and bilinear mappings for the two possible meshing of the hexagon. Thus, four different methods are proposed and are completely described in this work. Because of theoretical and numerical difficulties, a particular treatment has been necessary for methods based on the conformal mapping. Finally, numerical results are presented for a hexagonal benchmark to validate and compare the four methods with respect to pre-defined criteria. (authors)

  16. Diffuse fluorescence tomography based on the radiative transfer equation for small animal imaging

    Science.gov (United States)

    Wang, Yihan; Zhang, Limin; Zhao, Huijuan; Gao, Feng; Li, Jiao

    2014-02-01

    Diffuse florescence tomography (DFT) as a high-sensitivity optical molecular imaging tool, can be applied to in vivo visualize interior cellular and molecular events for small-animal disease model through quantitatively recovering biodistributions of specific molecular probes. In DFT, the radiative transfer equation (RTE) and its approximation, such as the diffuse equation (DE), have been used as the forward models. The RTE-based DFT methodology is more suitable for biological tissue having void-like regions and the near-source area as in the situations of small animal imaging. We present a RTE-based scheme for the steady state DFT, which combines the discrete solid angle method and the finite difference method to obtain numerical solutions of the 2D steady RTE, with the natural boundary condition and collimating light source model. The approach is validated using the forward data from the Monte Carlo simulation for its better performances in the spatial resolution and reconstruction fidelity compared to the DE-based scheme.

  17. A non-scale-invariant form for coarse-grained diffusion-reaction equations

    Science.gov (United States)

    Ostvar, Sassan; Wood, Brian D.

    2016-09-01

    The process of mixing and reaction is a challenging problem to understand mathematically. Although there have been successes in describing the effective properties of mixing and reaction under a number of regimes, process descriptions for early times have been challenging for cases where the structure of the initial conditions is highly segregated. In this paper, we use the method of volume averaging to develop a rigorous theory for diffusive mixing with reactions from initial to asymptotic times under highly segregated initial conditions in a bounded domain. One key feature that arises in this development is that the functional form of the averaged differential mass balance equations is not, in general, scale invariant. Upon upscaling, an additional source term arises that helps to account for the initial configuration of the reacting chemical species. In this development, we derive the macroscopic parameters (a macroscale source term and an effectiveness factor modifying the reaction rate) defined in the macroscale diffusion-reaction equation and provide example applications for several initial configurations.

  18. Langevin equation with multiplicative white noise: Transformation of diffusion processes into the Wiener process in different prescriptions

    Energy Technology Data Exchange (ETDEWEB)

    Kwok, Sau Fa, E-mail: kwok@dfi.uem.br

    2012-08-15

    A Langevin equation with multiplicative white noise and its corresponding Fokker-Planck equation are considered in this work. From the Fokker-Planck equation a transformation into the Wiener process is provided for different orders of prescription in discretization rule for the stochastic integrals. A few applications are also discussed. - Highlights: Black-Right-Pointing-Pointer Fokker-Planck equation corresponding to the Langevin equation with mul- tiplicative white noise is presented. Black-Right-Pointing-Pointer Transformation of diffusion processes into the Wiener process in different prescriptions is provided. Black-Right-Pointing-Pointer The prescription parameter is associated with the growth rate for a Gompertz-type model.

  19. Turbulent diffusion from a point source in stratified and neutral flows around a three-dimensional hill. I. Diffusion equation analysis

    Energy Technology Data Exchange (ETDEWEB)

    Hunt, J.C.R.; Puttock, J.S.; Snyder, W.H.

    1979-01-01

    Modeling of turbulent diffusion around three-dimensional hills is described in terms of horizontal flow and horizontal diffusion. The advective diffusive equation for flows around such hills is solved to show how source positions on and off the center line affect the trajectories and splitting of impinging plumes and the value and position of the maximum surface pollutant concentration on the hill. Plume analysis in a neutrally stable potential flow around an axisymmetric obstacle, using the diffusion equation, is discussed. Because streamlines approach the surface of a three-dimensional hill much more closely than they approach a two-dimensional hill, the maximum surface concentrations on the hill can become very much greater than in the absence of the hill. (4 diagrams, 5 graphs, 23 references, 1 table)

  20. Analyses of the Instabilities in the Discretized Diffusion Equations via Information Theory

    Directory of Open Access Journals (Sweden)

    Maxence Bigerelle

    2016-04-01

    Full Text Available In a previous investigation (Bigerelle and Iost, 2004, the authors have proposed a physical interpretation of the instability λ = Δt/Δx2 > 1/2 of the parabolic partial differential equations when solved by finite differences. However, our results were obtained using integration techniques based on erf functions meaning that no statistical fluctuation was introduced in the mathematical background. In this paper, we showed that the diffusive system can be divided into sub-systems onto which a Brownian motion is applied. Monte Carlo simulations are carried out to reproduce the macroscopic diffusive system. It is shown that the amount of information characterized by the compression ratio of information of the system is pertinent to quantify the entropy of the system according to some concepts introduced by the authors (Bigerelle and Iost, 2007. Thanks to this mesoscopic discretization, it is proved that information on each sub-cell of the diffusion map decreases with time before the unstable equality λ = 1/2 and increases after this threshold involving an increase in negentropy, i.e., a decrease in entropy contrarily to the second principle of thermodynamics.

  1. Sound energy decay in coupled spaces using a parametric analytical solution of a diffusion equation.

    Science.gov (United States)

    Luizard, Paul; Polack, Jean-Dominique; Katz, Brian F G

    2014-05-01

    Sound field behavior in performance spaces is a complex phenomenon. Issues regarding coupled spaces present additional concerns due to sound energy exchanges. Coupled volume concert halls have been of increasing interest in recent decades because this architectural principle offers the possibility to modify the hall's acoustical environment in a passive way by modifying the coupling area. Under specific conditions, the use of coupled reverberation chambers can provide non-exponential sound energy decay in the main room, resulting in both high clarity and long reverberation which are antagonistic parameters in a single volume room. Previous studies have proposed various sound energy decay models based on statistical acoustics and diffusion theory. Statistical acoustics assumes a perfectly uniform sound field within a given room whereas measurements show an attenuation of energy with increasing source-receiver distance. While previously proposed models based on diffusion theory use numerical solvers, the present study proposes a heuristic model of sound energy behavior based on an analytical solution of the commonly used diffusion equation and physically justified approximations. This model is validated by means of comparisons to scale model measurements and numerical geometrical acoustics simulations, both applied to the same simple concert hall geometry. PMID:24815259

  2. On the Initial-Boundary-Value Problem for the Time-Fractional Diffusion Equation on the Real Positive Semiaxis

    Directory of Open Access Journals (Sweden)

    D. Goos

    2015-01-01

    Full Text Available We consider the time-fractional derivative in the Caputo sense of order α∈(0, 1. Taking into account the asymptotic behavior and the existence of bounds for the Mainardi and the Wright function in R+, two different initial-boundary-value problems for the time-fractional diffusion equation on the real positive semiaxis are solved. Moreover, the limit when α↗1 of the respective solutions is analyzed, recovering the solutions of the classical boundary-value problems when α = 1, and the fractional diffusion equation becomes the heat equation.

  3. A Priori and A Posteriori Error Estimates of Streamline Diffusion Finite Element Method for Optimal Control Problem Governed by Convection Dominated Diffusion Equation

    Institute of Scientific and Technical Information of China (English)

    Ningning Yan; Zhaojie Zhou

    2008-01-01

    In this paper, we investigate a streamline diffusion finite element approximation scheme for the constrained optimal control problem governed by linear convection dominated diffusion equations. We prove the existence and uniqueness of the discretized scheme. Then a priori and a posteriori error estimates are derived for the state, the co-state and the control. Three numerical examples are presented to illustrate our theoretical results.

  4. Flow and Diffusion Equations for Fluid Flow in Porous Rocks for the Multiphase Flow Phenomena

    Directory of Open Access Journals (Sweden)

    Mohammad Miyan

    2015-07-01

    Full Text Available The multiphase flow in porous media is a subject of great complexities with a long rich history in the field of fluid mechanics. This is a subject with important technical applications, most notably in oil recovery from petroleum reservoirs and so on. The single-phase fluid flow through a porous medium is well characterized by Darcy’s law. In the petroleum industry and in other technical applications, transport is modeled by postulating a multiphase generalization of the Darcy’s law. In this connection, distinct pressures are defined for each constituent phase with the difference known as capillary pressure, determined by the interfacial tension, micro pore geometry and surface chemistry of the solid medium. For flow rates, relative permeability is defined that relates the volume flow rate of each fluid to its pressure gradient. In the present paper, there is a derivation and analysis about the diffusion equation for the fluid flow in porous rocks and some important results have been founded. The permeability is a function of rock type that varies with stress, temperature etc., and does not depend on the fluid. The effect of the fluid on the flow rate is accounted for by the term of viscosity. The numerical value of permeability for a given rock depends on the size of the pores in the rock as well as on the degree of interconnectivity of the void space. The pressure pulses obey the diffusion equation not the wave equation. Then they travel at a speed which continually decreases with time rather than travelling at a constant speed. The results shown in this paper are much useful in earth sciences and petroleum industry.

  5. Fast resolution of the neutron diffusion equation through public domain Ode codes

    International Nuclear Information System (INIS)

    The time-dependent neutron diffusion equation is a partial differential equation with source terms. The resolution method usually includes discretizing the spatial domain, obtaining a large system of linear, stiff ordinary differential equations (ODEs), whose resolution is computationally very expensive. Some standard techniques use a fixed time step to solve the ODE system. This can result in errors (if the time step is too large) or in long computing times (if the time step is too little). To speed up the resolution method, two well-known public domain codes have been selected: DASPK and FCVODE that are powerful codes for the resolution of large systems of stiff ODEs. These codes can estimate the error after each time step, and, depending on this estimation can decide which is the new time step and, possibly, which is the integration method to be used in the next step. With these mechanisms, it is possible to keep the overall error below the chosen tolerances, and, when the system behaves smoothly, to take large time steps increasing the execution speed. In this paper we address the use of the public domain codes DASPK and FCVODE for the resolution of the time-dependent neutron diffusion equation. The efficiency of these codes depends largely on the preconditioning of the big systems of linear equations that must be solved. Several pre-conditioners have been programmed and tested; it was found that the multigrid method is the best of the pre-conditioners tested. Also, it has been found that DASPK has performed better than FCVODE, being more robust for our problem.We can conclude that the use of specialized codes for solving large systems of ODEs can reduce drastically the computational work needed for the solution; and combining them with appropriate pre-conditioners, the reduction can be still more important. It has other crucial advantages, since it allows the user to specify the allowed error, which cannot be done in fixed step implementations; this, of course

  6. Fast resolution of the neutron diffusion equation through public domain Ode codes

    Energy Technology Data Exchange (ETDEWEB)

    Garcia, V.M.; Vidal, V.; Garayoa, J. [Universidad Politecnica de Valencia, Departamento de Sistemas Informaticos, Valencia (Spain); Verdu, G. [Universidad Politecnica de Valencia, Departamento de Ingenieria Quimica y Nuclear, Valencia (Spain); Gomez, R. [I.E.S. de Tavernes Blanques, Valencia (Spain)

    2003-07-01

    The time-dependent neutron diffusion equation is a partial differential equation with source terms. The resolution method usually includes discretizing the spatial domain, obtaining a large system of linear, stiff ordinary differential equations (ODEs), whose resolution is computationally very expensive. Some standard techniques use a fixed time step to solve the ODE system. This can result in errors (if the time step is too large) or in long computing times (if the time step is too little). To speed up the resolution method, two well-known public domain codes have been selected: DASPK and FCVODE that are powerful codes for the resolution of large systems of stiff ODEs. These codes can estimate the error after each time step, and, depending on this estimation can decide which is the new time step and, possibly, which is the integration method to be used in the next step. With these mechanisms, it is possible to keep the overall error below the chosen tolerances, and, when the system behaves smoothly, to take large time steps increasing the execution speed. In this paper we address the use of the public domain codes DASPK and FCVODE for the resolution of the time-dependent neutron diffusion equation. The efficiency of these codes depends largely on the preconditioning of the big systems of linear equations that must be solved. Several pre-conditioners have been programmed and tested; it was found that the multigrid method is the best of the pre-conditioners tested. Also, it has been found that DASPK has performed better than FCVODE, being more robust for our problem.We can conclude that the use of specialized codes for solving large systems of ODEs can reduce drastically the computational work needed for the solution; and combining them with appropriate pre-conditioners, the reduction can be still more important. It has other crucial advantages, since it allows the user to specify the allowed error, which cannot be done in fixed step implementations; this, of course

  7. Three-dimensional h-adaptivity for the multigroup neutron diffusion equations

    KAUST Repository

    Wang, Yaqi

    2009-04-01

    Adaptive mesh refinement (AMR) has been shown to allow solving partial differential equations to significantly higher accuracy at reduced numerical cost. This paper presents a state-of-the-art AMR algorithm applied to the multigroup neutron diffusion equation for reactor applications. In order to follow the physics closely, energy group-dependent meshes are employed. We present a novel algorithm for assembling the terms coupling shape functions from different meshes and show how it can be made efficient by deriving all meshes from a common coarse mesh by hierarchic refinement. Our methods are formulated using conforming finite elements of any order, for any number of energy groups. The spatial error distribution is assessed with a generalization of an error estimator originally derived for the Poisson equation. Our implementation of this algorithm is based on the widely used Open Source adaptive finite element library deal.II and is made available as part of this library\\'s extensively documented tutorial. We illustrate our methods with results for 2-D and 3-D reactor simulations using 2 and 7 energy groups, and using conforming finite elements of polynomial degree up to 6. © 2008 Elsevier Ltd. All rights reserved.

  8. Preconditioned time-difference methods for advection-diffusion-reaction equations

    Energy Technology Data Exchange (ETDEWEB)

    Aro, C.; Rodrigue, G. [Lawrence Livermore National Lab., CA (United States); Wolitzer, D. [California State Univ., Hayward, CA (United States)

    1994-12-31

    Explicit time differencing methods for solving differential equations are advantageous in that they are easy to implement on a computer and are intrinsically very parallel. The disadvantage of explicit methods is the severe restrictions placed on stepsize due to stability. Stability bounds for explicit time differencing methods on advection-diffusion-reaction problems are generally quite severe and implicit methods are used instead. The linear systems arising from these implicit methods are large and sparse so that iterative methods must be used to solve them. In this paper the authors develop a methodology for increasing the stability bounds of standard explicit finite differencing methods by combining explicit methods, implicit methods, and iterative methods in a novel way to generate new time-difference schemes, called preconditioned time-difference methods.

  9. TRAVELING WAVE SPEED AND SOLUTION IN REACTION-DIFFUSION EQUATION IN ONE DIMENSION

    Institute of Scientific and Technical Information of China (English)

    周天寿; 张锁春

    2001-01-01

    By Painlevé analysis, traveling wave speed and solution of reaction-diffusion equations for the concentration of one species in one spatial dimension are in detail investigated. When the exponent of the creation term is larger than the one of the annihilation term, two typical cases are studied, one with the exact traveling wave solutions, yielding the values of speeds, the other with the series expansion solution, also yielding the value of speed. Conversely, when the exponent of creation term is smaller than the one of the annihilation term, two typical cases are also studied, but only for one of them, there is a series development solution, yielding the value of speed, and for the other, traveling wave solution cannot exist. Besides, the formula of calculating speeds and solutions of planar wave within the thin boundary layer are given for a class of typical excitable media.

  10. Momentum balance equation for nonelectrolytes in models of coupling between chemical reaction and diffusion in membranes.

    Science.gov (United States)

    Gałdzicki, Z; Miekisz, S

    1984-04-01

    The role of viscosity in coupling between chemical reaction (complex formation) and diffusion in membranes has been investigated. The Fick law was replaced by the momentum balance equation with the viscous term. The irreversible thermodynamics admits coupling of the chemical reaction rate with the gradient of velocity. The proposed model has shown the contrary effect of viscosity and confirmed the experimental results. The chemical reaction rate increases only above the limit value of viscosity. The parameter Q (degree of complex formation) was introduced to investigate coupling. Q equals to the ratio of the chemical contribution into the flux of the complex to the total flux of the substance transported. For different values of the parameters of the model the dependence of Q upon position inside the membrane has been numerically calculated. The assumptions of the model limit it to a specific case and they only roughly model the biological situation. PMID:6537360

  11. The dynamics of nonlinear reaction-diffusion equations with small Lévy noise

    CERN Document Server

    Debussche, Arnaud; Imkeller, Peter

    2013-01-01

    This work considers a small random perturbation of alpha-stable jump type nonlinear reaction-diffusion equations with Dirichlet boundary conditions over an interval. It has two stable points whose domains of attraction meet in a separating manifold with several saddle points. Extending a method developed by Imkeller and Pavlyukevich it proves that in contrast to a Gaussian perturbation, the expected exit and transition times between the domains of attraction depend polynomially on the noise intensity in the small intensity limit. Moreover the solution exhibits metastable behavior: there is a polynomial time scale along which the solution dynamics correspond asymptotically to the dynamic behavior of a finite-state Markov chain switching between the stable states.

  12. Reverse Smoothing Effects, Fine Asymptotics, and Harnack Inequalities for Fast Diffusion Equations

    Directory of Open Access Journals (Sweden)

    Bonforte Matteo

    2007-01-01

    Full Text Available We investigate local and global properties of positive solutions to the fast diffusion equation in the good exponent range , corresponding to general nonnegative initial data. For the Cauchy problem posed in the whole Euclidean space , we prove sharp local positivity estimates (weak Harnack inequalities and elliptic Harnack inequalities; also a slight improvement of the intrinsic Harnack inequality is given. We use them to derive sharp global positivity estimates and a global Harnack principle. Consequences of these latter estimates in terms of fine asymptotics are shown. For the mixed initial and boundary value problem posed in a bounded domain of with homogeneous Dirichlet condition, we prove weak, intrinsic, and elliptic Harnack inequalities for intermediate times. We also prove elliptic Harnack inequalities near the extinction time, as a consequence of the study of the fine asymptotic behavior near the finite extinction time.

  13. An inverse time-dependent source problem for a time-fractional diffusion equation

    Science.gov (United States)

    Wei, T.; Li, X. L.; Li, Y. S.

    2016-08-01

    This paper is devoted to identifying a time-dependent source term in a multi-dimensional time-fractional diffusion equation from boundary Cauchy data. The existence and uniqueness of a strong solution for the corresponding direct problem with homogeneous Neumann boundary condition are firstly proved. We provide the uniqueness and a stability estimate for the inverse time-dependent source problem. Then we use the Tikhonov regularization method to solve the inverse source problem and propose a conjugate gradient algorithm to find a good approximation to the minimizer of the Tikhonov regularization functional. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed method. This paper was supported by the NSF of China (11371181) and the Fundamental Research Funds for the Central Universities (lzujbky-2013-k02).

  14. A Radiation Chemistry Code Based on the Greens Functions of the Diffusion Equation

    Science.gov (United States)

    Plante, Ianik; Wu, Honglu

    2014-01-01

    Ionizing radiation produces several radiolytic species such as.OH, e-aq, and H. when interacting with biological matter. Following their creation, radiolytic species diffuse and chemically react with biological molecules such as DNA. Despite years of research, many questions on the DNA damage by ionizing radiation remains, notably on the indirect effect, i.e. the damage resulting from the reactions of the radiolytic species with DNA. To simulate DNA damage by ionizing radiation, we are developing a step-by-step radiation chemistry code that is based on the Green's functions of the diffusion equation (GFDE), which is able to follow the trajectories of all particles and their reactions with time. In the recent years, simulations based on the GFDE have been used extensively in biochemistry, notably to simulate biochemical networks in time and space and are often used as the "gold standard" to validate diffusion-reaction theories. The exact GFDE for partially diffusion-controlled reactions is difficult to use because of its complex form. Therefore, the radial Green's function, which is much simpler, is often used. Hence, much effort has been devoted to the sampling of the radial Green's functions, for which we have developed a sampling algorithm This algorithm only yields the inter-particle distance vector length after a time step; the sampling of the deviation angle of the inter-particle vector is not taken into consideration. In this work, we show that the radial distribution is predicted by the exact radial Green's function. We also use a technique developed by Clifford et al. to generate the inter-particle vector deviation angles, knowing the inter-particle vector length before and after a time step. The results are compared with those predicted by the exact GFDE and by the analytical angular functions for free diffusion. This first step in the creation of the radiation chemistry code should help the understanding of the contribution of the indirect effect in the

  15. Lattice Boltzmann method for convection-diffusion equations with general interfacial conditions

    Science.gov (United States)

    Hu, Zexi; Huang, Juntao; Yong, Wen-An

    2016-04-01

    In this work, we propose an interfacial scheme accompanying the lattice Boltzmann method for convection-diffusion equations with general interfacial conditions, including conjugate conditions with or without jumps in heat and mass transfer, continuity of macroscopic variables and normal fluxes in ion diffusion in porous media with different porosity, and the Kapitza resistance in heat transfer. The construction of this scheme is based on our boundary schemes [Huang and Yong, J. Comput. Phys. 300, 70 (2015), 10.1016/j.jcp.2015.07.045] for Robin boundary conditions on straight or curved boundaries. It gives second-order accuracy for straight interfaces and first-order accuracy for curved ones. In addition, the new scheme inherits the advantage of the boundary schemes in which only the current lattice nodes are involved. Such an interfacial scheme is highly desirable for problems with complex geometries or in porous media. The interfacial scheme is numerically validated with several examples. The results show the utility of the constructed scheme and very well support our theoretical predications.

  16. Competing computational approaches to reaction-diffusion equations in clusters of cells

    Science.gov (United States)

    Stella, Sabrina; Chignola, Roberto; Milotti, Edoardo

    2014-03-01

    We have developed a numerical model that simulates the growth of small avascular solid tumors. At its core lies a set of partial differential equations that describe diffusion processes as well as transport and reaction mechanisms of a selected number of nutrients. Although the model relies on a restricted subset of molecular pathways, it compares well with experiments, and its emergent properties have recently led us to uncover a metabolic scaling law that stresses the common mechanisms that drive tumor growth. Now we plan to expand the biochemical model at the basis of the simulator to extend its reach. However, the introduction of additional molecular pathways requires an extensive revision of the reaction-diffusion part of the C++ code to make it more modular and to boost performance. To this end, we developed a novel computational abstract model where the individual molecular species represent the basic computational building blocks. Using a simple two-dimensional toy model to benchmark the new code, we find that the new implementation produces a more modular code without affecting performance. Preliminary results also show that a factor 2 speedup can be achieved with OpenMP multithreading, and other very preliminary results indicate that at least an order-of-magnitude speedup can be obtained using an NVidia Fermi GPU with CUDA code.

  17. Space-Time Fractional Reaction-Diffusion Equations Associated with a Generalized Riemann–Liouville Fractional Derivative

    Directory of Open Access Journals (Sweden)

    Ram K. Saxena

    2014-08-01

    Full Text Available This paper deals with the investigation of the computational solutions of a unified fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the generalized Riemann–Liouville fractional derivative defined by others and the space derivative of second order by the Riesz–Feller fractional derivative and adding a function ɸ(x, t. The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of Mittag–Leffler functions. The main result obtained in this paper provides an elegant extension of the fundamental solution for the space-time fractional diffusion equation obtained by others and the result very recently given by others. At the end, extensions of the derived results, associated with a finite number of Riesz–Feller space fractional derivatives, are also investigated.

  18. Fourier spectral methods for fractional-in-space reaction-diffusion equations

    KAUST Repository

    Bueno-Orovio, Alfonso

    2014-04-01

    © 2014, Springer Science+Business Media Dordrecht. Fractional differential equations are becoming increasingly used as a powerful modelling approach for understanding the many aspects of nonlocality and spatial heterogeneity. However, the numerical approximation of these models is demanding and imposes a number of computational constraints. In this paper, we introduce Fourier spectral methods as an attractive and easy-to-code alternative for the integration of fractional-in-space reaction-diffusion equations described by the fractional Laplacian in bounded rectangular domains of ℝ. The main advantages of the proposed schemes is that they yield a fully diagonal representation of the fractional operator, with increased accuracy and efficiency when compared to low-order counterparts, and a completely straightforward extension to two and three spatial dimensions. Our approach is illustrated by solving several problems of practical interest, including the fractional Allen–Cahn, FitzHugh–Nagumo and Gray–Scott models, together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator.

  19. On a Delay Reaction-Diffusion Difference Equation of Neutral Type

    Institute of Scientific and Technical Information of China (English)

    Bao SHI

    2002-01-01

    In this paper, we first consider a delay difference equation of neutral type of the form:△(yn + pyn-k) + qnyn-e = 0 for n ∈ Z+(0) (1*)and give a different condition from that of Yu and Wang (Funkcial Ekvac, 1994, 37(2): 241-248) toguarantee that every non-oscillatory solution of (1*) with p = 1 tends to zero as n →∞. Moreover,we consider a delay reaction-diffusion difference equation of neutral type of the form:△1(un,m+pun-k,m) +qn,mun-e,m=a2△22un+1,m-1 for (n,m) ∈Z+(0)×Ω, (2*)study various cases of p in the neutral term and obtain that if p ≥ -1 then every non-oscillatorysolution of (2*) tends uniformly in m ∈Ω to zero as n →∞; if p = -1 then every solution of(2*) oscillates and if p < -1 then every non-oscillatory solution of (2*) goes uniformly in m ∈Ω toinfinity or minus infinity as n →∞ under some hypotheses.

  20. Solution to the Diffusion equation for multi groups in X Y geometry using Linear Perturbation theory

    International Nuclear Information System (INIS)

    Diverse methods exist to solve numerically the neutron diffusion equation for several energy groups in stationary state among those that highlight those of finite elements. In this work the numerical solution of this equation is presented using Raviart-Thomas nodal methods type finite element, the RT0 and RT1, in combination with iterative techniques that allow to obtain the approached solution in a quick form. Nevertheless the above mentioned, the precision of a method is intimately bound to the dimension of the approach space by cell, 5 for the case RT0 and 12 for the RT1, and/or to the mesh refinement, that makes the order of the problem of own value to solve to grow considerably. By this way if it wants to know an acceptable approach to the value of the effective multiplication factor of the system when this it has experimented a small perturbation it was appeal to the Linear perturbation theory with which is possible to determine it starting from the neutron flow and of the effective multiplication factor of the not perturbed case. Results are presented for a reference problem in which a perturbation is introduced in an assemble that simulates changes in the control bar. (Author)

  1. Numerical Calculation and Exergy Equations of Spray Heat Exchanger Attached to a Main Fan Diffuser

    Science.gov (United States)

    Cui, H.; Wang, H.; Chen, S.

    2015-04-01

    In the present study, the energy depreciation rule of spray heat exchanger, which is attached to a main fan diffuser, is analyzed based on the second law of thermodynamics. Firstly, the exergy equations of the exchanger are deduced. The equations are numerically calculated by the fourth-order Runge-Kutta method, and the exergy destruction is quantitatively effected by the exchanger structure parameters, working fluid (polluted air, i.e., PA; sprayed water, i.e., SW) initial state parameters and the ambient reference parameters. The results are showed: (1) heat transfer is given priority to latent transfer at the bottom of the exchanger, and heat transfer of convection and is equivalent to that of condensation in the upper. (2) With the decrease of initial temperature of SW droplet, the decrease of PA velocity or the ambient reference temperature, and with the increase of a SW droplet size or initial PA temperature, exergy destruction both increase. (3) The exergy efficiency of the exchanger is 72.1 %. An approach to analyze the energy potential of the exchanger may be provided for engineering designs.

  2. Neutrophil Chemotaxis Dysfunction in Human Periodontitis

    OpenAIRE

    Van Dyke, T. E.; Horoszewicz, H. U.; Cianciola, L. J.; Genco, R J

    1980-01-01

    Polymorphonuclear leukocyte (PMNL) chemotaxis studies of 32 patients with localized juvenile periodontitis (periodontosis or LJP), 10 adult patients with a history of LJP (post-LJP), 8 patients with generalized juvenile periodontitis (GJP), and 23 adults with moderate to severe periodontitis were performed: (i) to determine the prevalence of a PMNL chemotaxis defect in a large group of LJP patients; (ii) to study PMNL chemotaxis in patients with other forms of severe periodontal disease; and ...

  3. Signaling mechanisms for regulation of chemotaxis

    Institute of Scientific and Technical Information of China (English)

    Dianqing WU

    2005-01-01

    Chemotaxis is a fascinating biological process, through which a cell migrates along a shallow chemoattractant gradient that is less than 5% difference between the anterior and posterior of the cell. Chemotaxis is composed of two independent,but interrelated processes-motility and directionality, both of which are regulated by extracellular stimuli, chemoattractants.In this mini-review, recent progresses in the understanding of the regulation of leukocyte chemotaxis by chemoattractant signaling are reviewed.

  4. Feeding ducks, bacterial chemotaxis, and the Gini index

    CERN Document Server

    Peaudecerf, Francois J

    2015-01-01

    Classic experiments on the distribution of ducks around separated food sources found consistency with the `ideal free' distribution in which the local population is proportional to the local supply rate. Motivated by this experiment and others, we examine the analogous problem in the microbial world: the distribution of chemotactic bacteria around multiple nearby food sources. In contrast to the optimization of uptake rate that may hold at the level of a single cell in a spatially varying nutrient field, nutrient consumption by a population of chemotactic cells will modify the nutrient field, and the uptake rate will generally vary throughout the population. Through a simple model we study the distribution of resource uptake in the presence of chemotaxis, consumption, and diffusion of both bacteria and nutrients. Borrowing from the field of theoretical economics, we explore how the Gini index can be used as a means to quantify the inequalities of uptake. The redistributive effect of chemotaxis can lead to a p...

  5. Element-free Galerkin modeling of neutron diffusion equation in X–Y geometry

    International Nuclear Information System (INIS)

    Highlights: ► The EFG performs very well and is comparable to the FEM. ► It is shown that the performance of the EFG method is quite adequate. ► The method is attractive in some aspects such as node refinement and dealing with curve boundaries. ► In the EFG methods the shape function construction and integration process are more complicated. - Abstract: The numerical treatment of partial differential equations with element-free discretization techniques has been attractive research area in the recent years. In this paper an Element-free Galerkin, EFG, method is applied to solve the neutron diffusion equation in X–Y geometry. The Moving Least Square (MLS), interpolation is used to construct the shape functions for the weak form of the equation. The constructed shape functions through using Gaussian and cubic weight functions, which are commonly used in Element-free Galerkin method, lack the Kronecker delta property; this causes additional numerical effort for satisfying the essential boundary conditions. In this study a new weight function which almost fulfills the essential boundary conditions with high accuracy is presented. Constructed shape functions along the new weight function provide much stable results for varying support domain size and distorted nodal arrangements. The efficiency and accuracy of the method was evaluated through a number of examples. Results are compared with the analytical and reference solutions. It is revealed that the applied EFG method provides highly accurate results and the method is attractive in some aspects such as nodes refinement and dealing with the curve boundaries.

  6. Resolution of the generalized eigenvalue problem in the neutron diffusion equation discretized by the finite volume method

    OpenAIRE

    Álvaro Bernal; Rafael Miró; Damián Ginestar; Gumersindo Verdú

    2014-01-01

    Numerical methods are usually required to solve the neutron diffusion equation applied to nuclear reactors due to its heterogeneous nature. The most popular numerical techniques are the Finite Difference Method (FDM), the Coarse Mesh Finite Difference Method (CFMD), the Nodal Expansion Method (NEM), and the Nodal Collocation Method (NCM), used virtually in all neutronic diffusion codes, which give accurate results in structured meshes. However, the application of these methods in unstructured...

  7. Resolution of the Generalized Eigenvalue Problem in the Neutron Diffusion Equation Discretized by the Finite Volume Method

    OpenAIRE

    2014-01-01

    Numerical methods are usually required to solve the neutron diffusion equation applied to nuclear reactors due to its heterogeneous nature. The most popular numerical techniques are the Finite Difference Method (FDM), the Coarse Mesh Finite Difference Method (CFMD), the Nodal Expansion Method (NEM), and the Nodal Collocation Method (NCM), used virtually in all neutronic diffusion codes, which give accurate results in structured meshes. However, the application of these methods in unstructured...

  8. Anomalous Diffusion of Dissipative Solitons in the Cubic-Quintic Complex Ginzburg-Landau Equation in Two Spatial Dimensions.

    Science.gov (United States)

    Cisternas, Jaime; Descalzi, Orazio; Albers, Tony; Radons, Günter

    2016-05-20

    We demonstrate the occurrence of anomalous diffusion of dissipative solitons in a "simple" and deterministic prototype model: the cubic-quintic complex Ginzburg-Landau equation in two spatial dimensions. The main features of their dynamics, induced by symmetric-asymmetric explosions, can be modeled by a subdiffusive continuous-time random walk, while in the case dominated by only asymmetric explosions, it becomes characterized by normal diffusion. PMID:27258868

  9. Determination of neutron buildup factor using analytical solution of one-dimensional neutron diffusion equation in cylindrical geometry

    Energy Technology Data Exchange (ETDEWEB)

    Fernandes, Julio Cesar L.; Vilhena, Marco Tullio, E-mail: julio.lombaldo@ufrgs.b, E-mail: vilhena@pq.cnpq.b [Universidade Federal do Rio Grande do Sul (DMPA/UFRGS), Porto Alegre, RS (Brazil). Dept. de Matematica Pura e Aplicada. Programa de Pos Graduacao em Matematica Aplicada; Borges, Volnei; Bodmann, Bardo Ernest, E-mail: bardo.bodmann@ufrgs.b, E-mail: borges@ufrgs.b [Universidade Federal do Rio Grande do Sul (PROMEC/UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica

    2011-07-01

    The principal idea of this work, consist on formulate an analytical method to solved problems for diffusion of neutrons with isotropic scattering in one-dimensional cylindrical geometry. In this area were develop many works that study the same problem in different system of coordinates as well as cartesian system, nevertheless using numerical methods to solve the shielding problem. In view of good results in this works, we starting with the idea that we can represent a source in the origin of the cylindrical system by a Delta Dirac distribution, we describe the physical modeling and solved the neutron diffusion equation inside of cylinder of radius R. For the case of transport equation, the formulation of discrete ordinates S{sub N} consists in discretize the angular variables in N directions and in using a quadrature angular set for approximate the sources of scattering, where the Diffusion equation consist on S{sub 2} approximated transport equation in discrete ordinates. We solved the neutron diffusion equation with an analytical form by the finite Hankel transform. Was presented also the build-up factor for the case that we have neutron flux inside the cylinder. (author)

  10. Diffusive KPP Equations with Free Boundaries in Time Almost Periodic Environments: II. Spreading Speeds and Semi-Wave

    OpenAIRE

    Li, Fang; Liang, Xing; Shen, Wenxian

    2015-01-01

    In this series of papers, we investigate the spreading and vanishing dynamics of time almost periodic diffusive KPP equations with free boundaries. Such equations are used to characterize the spreading of a new species in time almost periodic environments with free boundaries representing the spreading fronts. In the first part of the series, we showed that a spreading-vanishing dichotomy occurs for such free boundary problems (see [16]). In this second part of the series, we investigate the ...

  11. The Exact Solution of Milburn Equation Without Diffusion Approximation for the Two-Model Raman Coupled Model

    Institute of Scientific and Technical Information of China (English)

    XU JingBo; ZOU XuBo; GAO XiaoChun; FU Jian

    2002-01-01

    By making use of the dynamical algebraic approach, we study the two-mode Raman coupled model governed by the Milburn equation and find the exact solution of the Milburn equation without diffusion approximation. The exact solution is then used to discuss the influence of intrinsic decoherence on the revivals of atomic inversion, oscillation of the photon number distribution and squeezing of radiation field in the whole ranges of the decoherence parameter γ.

  12. An Operator Splitting Method with FDM and FEM for the Convection—Diffusion Equation Using Boundary—fitted Coordinate System

    Institute of Scientific and Technical Information of China (English)

    Jie-MinZHAN; Yao-SongCHEN

    1996-01-01

    An operator splitting method combining finite difference method and finite element method is proposed in this paper by using boundary-fitted coordinate system.The governing equation is split into advection and diffusion equations and solved by finit difference method using boundary-fitted coordinate system and finite element method respectively.An example for which analytic solution is available is used to verified the proposed methods and the agreement is very good.Numerical results show that it is very efficient.

  13. From the Highly Compressible Navier-Stokes Equations to Fast Diffusion and Porous Media Equations, Existence of Global Weak Solution for the Quasi-Solutions

    Science.gov (United States)

    Haspot, Boris

    2016-06-01

    We consider the compressible Navier-Stokes equations for viscous and barotropic fluids with density dependent viscosity. The aim is to investigate mathematical properties of solutions of the Navier-Stokes equations using solutions of the pressureless Navier-Stokes equations, that we call quasi solutions. This regime corresponds to the limit of highly compressible flows. In this paper we are interested in proving the announced result in Haspot (Proceedings of the 14th international conference on hyperbolic problems held in Padova, pp 667-674, 2014) concerning the existence of global weak solution for the quasi-solutions, we also observe that for some choice of initial data (irrotationnal) the quasi solutions verify the porous media, the heat equation or the fast diffusion equations in function of the structure of the viscosity coefficients. In particular it implies that it exists classical quasi-solutions in the sense that they are {C^{∞}} on {(0,T)× {R}N} for any {T > 0}. Finally we show the convergence of the global weak solution of compressible Navier-Stokes equations to the quasi solutions in the case of a vanishing pressure limit process. In particular for highly compressible equations the speed of propagation of the density is quasi finite when the viscosity corresponds to {μ(ρ)=ρ^{α}} with {α > 1}. Furthermore the density is not far from converging asymptotically in time to the Barrenblatt solution of mass the initial density {ρ0}.

  14. Elaboration of a nodal method to solve the steady state multigroup diffusion equation. Study and use of the multigroup diffusion code DAHRA

    International Nuclear Information System (INIS)

    The subject is divided in two parts: In the first part a nodal method has been worked out to solve the steady state multigroup diffusion equation. This method belongs to the same set of nodal methods currently used to calculate the exact fission powers and neutron fluxes in a very short computing time. It has been tested on a two dimensional idealized reactors. The effective multiplication factor and the fission powers for each fuel element have been calculated. The second part consists in studying and mastering the multigroup diffusion code DAHRA - a reduced version of DIANE - a two dimensional code using finite difference method

  15. A new Differential Equation for Anomalous Diffusion with Potential Applications to Nonlinear Space Plasmas

    Science.gov (United States)

    Watkins, N. W.; Credgington, D.; Sanchez, R.; Chapman, S. C.

    2007-12-01

    Since the 1960s Mandelbrot has advocated the use of fractals for the description of the non-Euclidean geometry of many aspects of nature. In particular he proposed two kinds of model to capture persistence in time (his Joseph effect, common in hydrology and with fractional Brownian motion as the prototpe) and/or prone to heavy tailed jumps (the Noah effect, typical of economic indices, for which he proposed Lévy flights as an exemplar). Both effects are now well demonstrated in space plasmas, notably in indices quantifying Earth's auroral currents and in the turbulent solar wind. Models have, however, typically emphasised one of the Noah and Joseph parameters (the Lévy exponent μ and the temporal exponent β) at the other's expense. I will describe recent work [1] in which we studied a simple self-affine stable model-linear fractional stable motion, LFSM, which unifies both effects. I will discuss how this resolves some contradictions seen in earlier work. Such Noah-Joseph hybrid ("ambivalent" [2]) behaviour is highly topical in physics but is typically studied in the paradigm of the continuous time random walk (CTRW) [2,3] rather than LFSM. I will clarify the physical differences between these two pictures and present a recently-derived diffusion equation for LFSM. This replaces the second order spatial derivative in the equation of fBm [4] with a fractional derivative of order μ, but retains a diffusion coefficient with a power law time dependence rather than a fractional derivative in time (c.f. [2,3]). Intriguingly the self-similarity exponent extracted from the CTRW differs from that seen in LFSM. In the CTRW it is the ratio of μ to a temporal exponent, in LFSM it is an additive function of them. I will also show work in progress using an LFSM model and simple analytic scaling arguments to study the problem of the area between an LFSM curve and a threshold-related to the burst size measure introduced by Takalo and Consolini into solar- terrestrial physics

  16. Numerical solution of fractional sub-diffusion and time-fractional diffusion-wave equations via fractional-order Legendre functions

    Science.gov (United States)

    Hooshmandasl, M. R.; Heydari, M. H.; Cattani, C.

    2016-08-01

    Fractional calculus has been used to model physical and engineering processes that are best described by fractional differential equations. Therefore designing efficient and reliable techniques for the solution of such equations is an important task. In this paper, we propose an efficient and accurate Galerkin method based on the fractional-order Legendre functions (FLFs) for solving the fractional sub-diffusion equation (FSDE) and the time-fractional diffusion-wave equation (FDWE). The time-fractional derivatives for FSDE are described in the Riemann-Liouville sense, while for FDWE are described in the Caputo sense. To this end, we first derive a new operational matrix of fractional integration (OMFI) in the Riemann-Liouville sense for FLFs. Next, we transform the original FSDE into an equivalent problem with fractional derivatives in the Caputo sense. Then the FLFs and their OMFI together with the Galerkin method are used to transform the problems under consideration into the corresponding linear systems of algebraic equations, which can be simply solved to achieve the numerical solutions of the problems. The proposed method is very convenient for solving such kind of problems, since the initial and boundary conditions are taken into account automatically. Furthermore, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

  17. The intrinsic periodic fluctuation of forest: a theoretical model based on diffusion equation

    Science.gov (United States)

    Zhou, J.; Lin, G., Sr.

    2015-12-01

    Most forest dynamic models predict the stable state of size structure as well as the total basal area and biomass in mature forest, the variation of forest stands are mainly driven by environmental factors after the equilibrium has been reached. However, although the predicted power-law size-frequency distribution does exist in analysis of many forest inventory data sets, the estimated distribution exponents are always shifting between -2 and -4, and has a positive correlation with the mean value of DBH. This regular pattern can not be explained by the effects of stochastic disturbances on forest stands. Here, we adopted the partial differential equation (PDE) approach to deduce the systematic behavior of an ideal forest, by solving the diffusion equation under the restricted condition of invariable resource occupation, a periodic solution was gotten to meet the variable performance of forest size structure while the former models with stable performance were just a special case of the periodic solution when the fluctuation frequency equals zero. In our results, the number of individuals in each size class was the function of individual growth rate(G), mortality(M), size(D) and time(T), by borrowing the conclusion of allometric theory on these parameters, the results perfectly reflected the observed "exponent-mean DBH" relationship and also gave a logically complete description to the time varying form of forest size-frequency distribution. Our model implies that the total biomass of a forest can never reach a stable equilibrium state even in the absence of disturbances and climate regime shift, we propose the idea of intrinsic fluctuation property of forest and hope to provide a new perspective on forest dynamics and carbon cycle research.

  18. 反应扩散方程的奇摄动%SINGULAR PERTURBATION FOR REACTION DIFFUSION EQUATIONS

    Institute of Scientific and Technical Information of China (English)

    莫嘉琪; 王辉; 朱江

    2003-01-01

    The singularly perturbed initial boundary value problems for reaction diffusion equations are considered. Under suitable conditions and by using the theory of differential inequality, the asymptotic behavior of solution for initial boundary value problems are studied, where the reduced problems possess two intersecting solutions.

  19. GLOBAL EXISTENCE AND EXPONENTIAL STABILITY OF SOLUTIONS TO THE QUASILINEAR THERMO-DIFFUSION EQUATIONS WITH SECOND SOUND

    Institute of Scientific and Technical Information of China (English)

    秦玉明; 李海燕

    2014-01-01

    This article is devoted to the study of global existence and exponential stability of solutions to an initial-boundary value problem of the quasilinear thermo-diffusion equations with second sound by means of multiplicative techniques and energy method provided that the initial data are close to the equilibrium and the relaxation kernel is strongly positive definite and decays exponentially.

  20. Traveling wave solutions of a biological reaction-convection-diffusion equation model by using $(G'/G$ expansion method

    Directory of Open Access Journals (Sweden)

    Shahnam Javadi

    2013-07-01

    Full Text Available In this paper, the $(G'/G$-expansion method is applied to solve a biological reaction-convection-diffusion model arising in mathematical biology. Exact traveling wave solutions are obtained by this method. This scheme can be applied to a wide class of nonlinear partial differential equations.

  1. Analytical solutions to the fractional advection-diffusion equation with time-dependent pulses on the boundary

    Science.gov (United States)

    Rubbab, Qammar; Mirza, Itrat Abbas; Qureshi, M. Zubair Akbar

    2016-07-01

    The time-fractional advection-diffusion equation with Caputo-Fabrizio fractional derivatives (fractional derivatives without singular kernel) is considered under the time-dependent emissions on the boundary and the first order chemical reaction. The non-dimensional problem is formulated by using suitable dimensionless variables and the fundamental solutions to the Dirichlet problem for the fractional advection-diffusion equation are determined using the integral transforms technique. The fundamental solutions for the ordinary advection-diffusion equation, fractional and ordinary diffusion equation are obtained as limiting cases of the previous model. Using Duhamel's principle, the analytical solutions to the Dirichlet problem with time-dependent boundary pulses have been obtained. The influence of the fractional parameter and of the drift parameter on the solute concentration in various spatial positions was analyzed by numerical calculations. It is found that the variation of the fractional parameter has a significant effect on the solute concentration, namely, the memory effects lead to the retardation of the mass transport.

  2. Improvement of the symbolic Monte-Carlo method for the transport equation: P1 extension and coupling with diffusion

    Energy Technology Data Exchange (ETDEWEB)

    Clouet, J.F.; Samba, G. [CEA Bruyeres-le-Chatel, 91 (France)

    2005-07-01

    We use asymptotic analysis to study the diffusion limit of the Symbolic Implicit Monte-Carlo (SIMC) method for the transport equation. For standard SIMC with piecewise constant basis functions, we demonstrate mathematically that the solution converges to the solution of a wrong diffusion equation. Nevertheless a simple extension to piecewise linear basis functions enables to obtain the correct solution. This improvement allows the calculation in opaque medium on a mesh resolving the diffusion scale much larger than the transport scale. Anyway, the huge number of particles which is necessary to get a correct answer makes this computation time consuming. Thus, we have derived from this asymptotic study an hybrid method coupling deterministic calculation in the opaque medium and Monte-Carlo calculation in the transparent medium. This method gives exactly the same results as the previous one but at a much lower price. We present numerical examples which illustrate the analysis. (authors)

  3. Uniqueness and reconstruction of an unknown semilinear term in a time-fractional reaction–diffusion equation

    KAUST Repository

    Luchko, Yuri

    2013-05-30

    In this paper, we consider a reaction-diffusion problem with an unknown nonlinear source function that has to be determined from overposed data. The underlying model is in the form of a time-fractional reaction-diffusion equation and the work generalizes some known results for the inverse problems posed for PDEs of parabolic type. For the inverse problem under consideration, a uniqueness result is proved and a numerical algorithm with some theoretical qualification is presented in the one-dimensional case. The key both to the uniqueness result and to the numerical algorithm relies on the maximum principle which has recently been shown to hold for the fractional diffusion equation. In order to show the effectiveness of the proposed method, results of numerical simulations are presented. © 2013 IOP Publishing Ltd.

  4. Reverse Smoothing Effects, Fine Asymptotics, and Harnack Inequalities for Fast Diffusion Equations

    Directory of Open Access Journals (Sweden)

    Juan Luis Vazquez

    2006-11-01

    Full Text Available We investigate local and global properties of positive solutions to the fast diffusion equation ut=Δum in the good exponent range (d−2+/d

  5. Radiative transfer equation modeling by streamline diffusion modified continuous Galerkin method.

    Science.gov (United States)

    Long, Feixiao; Li, Fengyan; Intes, Xavier; Kotha, Shiva P

    2016-03-01

    Optical tomography has a wide range of biomedical applications. Accurate prediction of photon transport in media is critical, as it directly affects the accuracy of the reconstructions. The radiative transfer equation (RTE) is the most accurate deterministic forward model, yet it has not been widely employed in practice due to the challenges in robust and efficient numerical implementations in high dimensions. Herein, we propose a method that combines the discrete ordinate method (DOM) with a streamline diffusion modified continuous Galerkin method to numerically solve RTE. Additionally, a phase function normalization technique was employed to dramatically reduce the instability of the DOM with fewer discrete angular points. To illustrate the accuracy and robustness of our method, the computed solutions to RTE were compared with Monte Carlo (MC) simulations when two types of sources (ideal pencil beam and Gaussian beam) and multiple optical properties were tested. Results show that with standard optical properties of human tissue, photon densities obtained using RTE are, on average, around 5% of those predicted by MC simulations in the entire/deeper region. These results suggest that this implementation of the finite element method-RTE is an accurate forward model for optical tomography in human tissues. PMID:26953662

  6. A generalized spin diffusion equation with four electrochemical potentials for channels with spin-orbit coupling

    Science.gov (United States)

    Sayed, Shehrin; Hong, Seokmin; Datta, Supriyo

    We will present a general semiclassical theory for an arbitrary channel with spin-orbit coupling (SOC), that uses four electrochemical potential (U + , D + , U - , and D -) depending on the sign of z-component of the spin (up (U) , down (D)) and the sign of the x-component of the group velocity (+ , -) . This can be considered as an extension of the standard spin diffusion equation that uses two electrochemical potentials for up and down spin states, allowing us to take into account the unique coupling between charge and spin degrees of freedom in channels with SOC. We will describe applications of this model to answer a number of interesting questions in this field such as: (1) whether topological insulators can switch magnets, (2) how the charge to spin conversion is influenced by the channel resistivity, and (3) how device structures can be designed to enhance spin injection. This work was supported by FAME, one of six centers of STARnet, a Semiconductor Research Corporation program sponsored by MARCO and DARPA.

  7. An analytical solution for the monoenergetic diffusion equation in a multilayered slab: determination of albedo boundary condition

    International Nuclear Information System (INIS)

    In this work, we solve analytically, without losing generality, the neutron diffusion equation for monoenergetic neutrons in a multilayered slab. To this end, we initially determine the solution of the neutron diffusion equation for a generic slab using standard results of second order linear ordinary differential equation with constant coefficients. The global solution for the multilayered slab is then determined applying the boundary condition and continuity of the flux and current at interface. Once the neutron flux is known, the albedo boundary condition is straightly obtained for an arbitrary number of layers in the baffle-reflecting regions, by just using the definition of albedo. We also present numerical simulation for the results neutron flux and comparison with the in literature. (author)

  8. Rigorous Derivation of a Nonlinear Diffusion Equation as Fast-Reaction Limit of a Continuous Coagulation-Fragmentation Model with Diffusion

    KAUST Repository

    Carrillo, J. A.

    2009-10-30

    Weak solutions of the spatially inhomogeneous (diffusive) Aizenmann-Bak model of coagulation-breakup within a bounded domain with homogeneous Neumann boundary conditions are shown to converge, in the fast reaction limit, towards local equilibria determined by their mass. Moreover, this mass is the solution of a nonlinear diffusion equation whose nonlinearity depends on the (size-dependent) diffusion coefficient. Initial data are assumed to have integrable zero order moment and square integrable first order moment in size, and finite entropy. In contrast to our previous result [5], we are able to show the convergence without assuming uniform bounds from above and below on the number density of clusters. © Taylor & Francis Group, LLC.

  9. Qualitative behavior of numerical traveling solutions for reaction–diffusion equations with memory

    OpenAIRE

    Ferreira, J. A.; De Oliveira, P.

    2005-01-01

    In this article the qualitative properties of numerical traveling wave solutions for integro- differential equations, which generalize the well known Fisher equation are studied. The integro-differential equation is replaced by an equivalent hyperbolic equation which allows us to characterize the numerical velocity of traveling wave solutions. Numerical results are presented.

  10. On Fick's first law and the equations of quasi-static motion governing the molecular diffusion

    OpenAIRE

    Désoyer, Thierry

    2008-01-01

    The problem of the molecular diffusion in a biphasic fluid mixture is studied here from the two complementary points of view of Continuum Mechanics - in a somewhat different manner from Truesdell in "Mechanical basis of diffusion" (J. Chem. Physics (U.S.), 37 (1962) 2336-2344) - and that of Thermodynamics. It is established that the force involved in the 'diffusive drag', i.e. in the inter-constituent viscous friction, is necessarily linked to the relative diffusion velocity of one constituen...

  11. Operator splitting methods for degenerate convection-diffusion equations II: Numerical examples with emphasis on reservoir simulation and sedimentation

    Energy Technology Data Exchange (ETDEWEB)

    Holden, Helge; Karlsen, Kenneth Hvistendal; Lie, Knut Andreas

    1999-12-01

    We present an accurate numerical method for a large class of scalar, strongly degenerate convection-diffusion equations. Important subclasses are hyperbolic conservation laws, porous medium type equations, two-phase reservoir flow equations, and strongly degenerate equations coming from the recent theory of sedimentation-consolidation processes. The method is based on splitting the convective and the diffusive terms. The nonlinear, convective part is solved using front tracking and dimensional splitting, while the nonlinear diffusion part is solved by an implicit-explicit finite difference scheme. In addition, one version of the implemented operator splitting method has a mechanism built in for detecting and correcting unphysical entropy loss, which may occur when the time step is large. This mechanism helps us gain a large time step ability for practical computations. A detailed convergence analysis of the operator splitting method was given in Part I. Here we present numerical experiments with the method for examples modelling secondary oil recovery and sedimentation-consolidation processes. We demonstrate that the splitting method resolves sharp gradients accurately, may use large time steps, has first order convergence, exhibits small grid orientation effects, has small mass balance errors, and is rather efficient. (author)

  12. Engineering Hybrid Chemotaxis Receptors in Bacteria.

    Science.gov (United States)

    Bi, Shuangyu; Pollard, Abiola M; Yang, Yiling; Jin, Fan; Sourjik, Victor

    2016-09-16

    Most bacteria use transmembrane sensors to detect a wide range of environmental stimuli. A large class of such sensors are the chemotaxis receptors used by motile bacteria to follow environmental chemical gradients. In Escherichia coli, chemotaxis receptors are known to mediate highly sensitive responses to ligands, making them potentially useful for biosensory applications. However, with only four ligand-binding chemotaxis receptors, the natural ligand spectrum of E. coli is limited. The design of novel chemoreceptors to extend the sensing capabilities of E. coli is therefore a critical aspect of chemotaxis-based biosensor development. One path for novel sensor design is to harvest the large natural diversity of chemosensory functions found in bacteria by creating hybrids that have the signaling domain from E. coli chemotaxis receptors and sensory domains from other species. In this work, we demonstrate that the E. coli receptor Tar can be successfully combined with most typical sensory domains found in chemotaxis receptors and in evolutionary-related two-component histidine kinases. We show that such functional hybrids can be generated using several different fusion points. Our work further illustrates how hybrid receptors could be used to quantitatively characterize ligand specificity of chemotaxis receptors and histidine kinases using standardized assays in E. coli.

  13. Vectorized and multitasked solution of the few-group neutron diffusion equations

    International Nuclear Information System (INIS)

    A numerical algorithm with parallelism was used to solve the two-group, multidimensional neutron diffusion equations on computers characterized by shared memory, vector pipeline, and multi-CPU architecture features. Specifically, solutions were obtained on the Cray X/MP-48, the IBM-3090 with vector facilities, and the FPS-164. The material-centered mesh finite difference method approximation and outer-inner iteration method were employed. Parallelism was introduced in the inner iterations using the cyclic line successive overrelaxation iterative method and solving in parallel across lines. The outer iterations were completed using the Chebyshev semi-iterative method that allows parallelism to be introduced in both space and energy groups. For the three-dimensional model, power, soluble boron, and transient fission product feedbacks were included. Concentrating on the pressurized water reactor (PWR), the thermal-hydraulic calculation of moderator density assumed single-phase flow and a closed flow channel, allowing parallelism to be introduced in the solution across the radial plane. Using a pinwise detail, quarter-core model of a typical PWR in cycle 1, for the two-dimensional model without feedback the measured million floating point operations per second (MFLOPS)/vector speedups were 83/11.7. 18/2.2, and 2.4/5.6 on the Cray, IBM, and FPS without multitasking, respectively. Lower performance was observed with a coarser mesh, i.e., shorter vector length, due to vector pipeline start-up. For an 18 x 18 x 30 (x-y-z) three-dimensional model with feedback of the same core, MFLOPS/vector speedups of --61/6.7 and an execution time of 0.8 CPU seconds on the Cray without multitasking were measured. Finally, using two CPUs and the vector pipelines of the Cray, a multitasking efficiency of 81% was noted for the three-dimensional model

  14. Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations

    Science.gov (United States)

    Kennedy, Christopher A.; Carpenter, Mark H.

    2002-01-01

    Additive Runge-Kutta (ARK) methods are investigated for application to the spatially discretized one- dimensional convection-diffusion-reaction (CDR) equations. Accuracy, stability, conservation, and dense-output are first considered for the general case when N different Runge-Kutta methods are grouped into a single composite method. Then, implicit-explicit, (N = 2), additive Runge-Kutta (ARK(sub 2)) methods from third- to fifth-order are presented that allow for integration of stiff terms by an L-stable, stiffly-accurate explicit, singly diagonally implicit Runge-Kutta (ESDIRK) method while the nonstiff terms are integrated with a traditional explicit Runge-Kutta method (ERK). Coupling error terms of the partitioned method are of equal order to those of the elemental methods. Derived ARK(sub 2) methods have vanishing stability functions for very large values of the stiff scaled eigenvalue, z['] yields -infinity, and retain high stability efficiency in the absence of stiffness, z['] yield 0. Extrapolation-type stage- value predictors are provided based on dense-output formulae. Optimized methods minimize both leading order ARK(sub 2) error terms and Butcher coefficient magnitudes as well as maximize conservation properties. Numerical tests of the new schemes on a CDR problem show negligible stiffness leakage and near classical order convergence rates. However, tests on three simple singular-perturbation problems reveal generally predictable order reduction. Error control is best managed with a PID-controller. While results for the fifth-order method are disappointing, both the new third- and fourth-order methods are at least as efficient as existing ARK(sub 2) methods.

  15. A Solution of the Convective-Diffusion Equation for Solute Mass Transfer inside a Capillary Membrane Bioreactor

    OpenAIRE

    B. Godongwana; Solomons, D.; Sheldon, M. S.

    2010-01-01

    This paper presents an analytical model of substrate mass transfer through the lumen of a membrane bioreactor. The model is a solution of the convective-diffusion equation in two dimensions using a regular perturbation technique. The analysis accounts for radial-convective flow as well as axial diffusion of the substrate specie. The model is applicable to the different modes of operation of membrane bioreactor (MBR) systems (e.g., dead-end, open-shell, or closed-shell mode), as well as the ve...

  16. The Construction of Numerical Scheme for Hyperbolic Equation and Exact Consideration of Numerical Diffusion and Dispersion

    Institute of Scientific and Technical Information of China (English)

    QingdongCAI

    1999-01-01

    A new technique in the formulation of numerical scheme for hyperbolic equation is developed.It is different from the classical FD and FE methods.We begin with the algebraic equations with some undefined parameters,and get the difference equation through Taylor-series expansion.When the parameters in the partial difference equation are defined,the equation is what the scheme will simulated.The numerical example of the viscous BUrgers equation shows the validity of the scheme.This method deals with the numerical viscosity and dispersion exactly ,giving a preliminary explanation to some problems that the CFD face now.

  17. Fokker-Planck equation with memory: the crossover from ballistic to diffusive processes in many-particle systems and incompressible media

    OpenAIRE

    Ilyin, V; Procaccia, I.; Zagorodny, A.

    2012-01-01

    The unified description of diffusion processes that cross over from a ballistic behavior at short times to normal or anomalous diffusion (sub- or superdiffusion) at longer times is constructed on the basis of a non-Markovian generalization of the Fokker-Planck equation. The necessary non-Markovian kinetic coefficients are determined by the observable quantities (mean- and mean square displacements). Solutions of the non-Markovian equation describing diffusive processes in the physical space a...

  18. Strenuous physical exercise adversely affects monocyte chemotaxis

    DEFF Research Database (Denmark)

    Czepluch, Frauke S; Barres, Romain; Caidahl, Kenneth;

    2011-01-01

    Physical exercise is important for proper cardiovascular function and disease prevention, but it may influence the immune system. We evaluated the effect of strenuous exercise on monocyte chemotaxis. Monocytes were isolated from blood of 13 young, healthy, sedentary individuals participating...... in a three-week training program which consisted of repeated exercise bouts. Monocyte chemotaxis and serological biomarkers were investigated at baseline, after three weeks training and after four weeks recovery. Chemotaxis towards vascular endothelial growth factor-A (VEGF-A) and transforming growth factor...

  19. Application of Linearized Anisotropic Diffusion Equation to Edge-Preserving Image Smoothing

    Institute of Scientific and Technical Information of China (English)

    YUANZejian; ZHENGNanning; ZHANGYuanlin

    2004-01-01

    A stable anisotropic diffusion model for edge-preservlng image smoothing is presented. By comparison with existing diffusion models~ this model has two significant new features. The first is that the diffusion model has the ability to adjust the diffusion coefficient and to regulate diffusion orientation simultaneously. The second is that the diffusion model being nonlinear in essence is linearized along time coordinates which makes the model more stable and thus the complexity of computation is reduced greatly. We use an Additive operator splitting scheme (AOS) that is an efficient and stable numerical method to implement the edge-preservlng image smoothing. The experimental results show that the model proposed in this paper is not only stable and efficient but can preserve image edge.

  20. A variable timestep generalized Runge-Kutta method for the numerical integration of the space-time diffusion equations

    Energy Technology Data Exchange (ETDEWEB)

    Aviles, B.N.; Sutton, T.M.; Kelly, D.J. III.

    1991-09-01

    A generalized Runge-Kutta method has been employed in the numerical integration of the stiff space-time diffusion equations. The method is fourth-order accurate, using an embedded third-order solution to arrive at an estimate of the truncation error for automatic timestep control. The efficiency of the Runge-Kutta method is enhanced by a block-factorization technique that exploits the sparse structure of the matrix system resulting from the space and energy discretized form of the time-dependent neutron diffusion equations. Preliminary numerical evaluation using a one-dimensional finite difference code shows the sparse matrix implementation of the generalized Runge-Kutta method to be highly accurate and efficient when compared to an optimized iterative theta method. 12 refs., 5 figs., 4 tabs.

  1. A variable timestep generalized Runge-Kutta method for the numerical integration of the space-time diffusion equations

    International Nuclear Information System (INIS)

    A generalized Runge-Kutta method has been employed in the numerical integration of the stiff space-time diffusion equations. The method is fourth-order accurate, using an embedded third-order solution to arrive at an estimate of the truncation error for automatic timestep control. The efficiency of the Runge-Kutta method is enhanced by a block-factorization technique that exploits the sparse structure of the matrix system resulting from the space and energy discretized form of the time-dependent neutron diffusion equations. Preliminary numerical evaluation using a one-dimensional finite difference code shows the sparse matrix implementation of the generalized Runge-Kutta method to be highly accurate and efficient when compared to an optimized iterative theta method. 12 refs., 5 figs., 4 tabs

  2. Potential use of combining the diffusion equation with the free Shrödinger equation to improve the Optical Coherence Tomography image analysis

    Science.gov (United States)

    Cabrera Fernandez, Delia; Salinas, Harry M.; Somfai, Gabor; Puliafito, Carmen A.

    2006-03-01

    Optical coherence tomography (OCT) is a rapidly emerging medical imaging technology. In ophthalmology, OCT is a powerful tool because it enables visualization of the cross sectional structure of the retina and anterior eye with higher resolutions than any other non-invasive imaging modality. Furthermore, OCT image information can be quantitatively analyzed, enabling objective assessment of features such as macular edema and diabetes retinopathy. We present specific improvements in the quantitative analysis of the OCT system, by combining the diffusion equation with the free Shrödinger equation. In such formulation, important features of the image can be extracted by extending the analysis from the real axis to the complex domain. Experimental results indicate that our proposed novel approach has good performance in speckle noise removal, enhancement and segmentation of the various cellular layers of the retina using the OCT system.

  3. Reciprocity relation for the vector radiative transport equation and its application to diffuse optical tomography with polarized light

    CERN Document Server

    Tricoli, Ugo; Da Silva, Anabela; Markel, Vadim A

    2016-01-01

    We derive a reciprocity relation for vector radiative transport equation (vRTE) that describes propagation of polarized light in multiple-scattering media. We then show how this result, together with translational invariance of a plane-parallel sample, can be used to compute efficiently the sensitivity kernel of diffuse optical tomography (DOT) by Monte Carlo simulations. Numerical examples of polarization-selective sensitivity kernels thus computed are given.

  4. Measurement of cellular chemotaxis with ECIS/Taxis.

    Science.gov (United States)

    Pietrosimone, Kathryn M; Yin, Xiuyin; Knecht, David A; Lynes, Michael A

    2012-01-01

    Cellular movement in response to external stimuli is fundamental to many cellular processes including wound healing, inflammation and the response to infection. A common method to measure chemotaxis is the Boyden chamber assay, in which cells and chemoattractant are separated by a porous membrane. As cells migrate through the membrane toward the chemoattractant, they adhere to the underside of the membrane, or fall into the underlying media, and are subsequently stained and visually counted (1). In this method, cells are exposed to a steep and transient chemoattractant gradient, which is thought to be a poor representation of gradients found in tissues (2). Another assay system, the under-agarose chemotaxis assay, (3, 4) measures cell movement across a solid substrate in a thin aqueous film that forms under the agarose layer. The gradient that develops in the agarose is shallow and is thought to be an appropriate representation of naturally occurring gradients. Chemotaxis can be evaluated by microscopic imaging of the distance traveled. Both the Boyden chamber assay and the under-agarose assay are usually configured as endpoint assays. The automated ECIS/Taxis system combines the under-agarose approach with Electric Cell-substrate Impedance Sensing (ECIS) (5, 6). In this assay, target electrodes are located in each of 8 chambers. A large counter-electrode runs through each of the 8 chambers (Figure 2). Each chamber is filled with agarose and two small wells are the cut in the agarose on either side of the target electrode. One well is filled with the test cell population, while the other holds the sources of diffusing chemoattractant (Figure 3). Current passed through the system can be used to determine the change in resistance that occurs as cells pass over the target electrode. Cells on the target electrode increase the resistance of the system (6). In addition, rapid fluctuations in the resistance represent changes in the interactions of cells with the electrode

  5. The nature and role of advection in advection-diffusion equations used for modelling bed load transport

    Science.gov (United States)

    Ancey, Christophe; Bohorquez, Patricio; Heyman, Joris

    2016-04-01

    The advection-diffusion equation arises quite often in the context of sediment transport, e.g., for describing time and space variations in the particle activity (the solid volume of particles in motion per unit streambed area). Stochastic models can also be used to derive this equation, with the significant advantage that they provide information on the statistical properties of particle activity. Stochastic models are quite useful when sediment transport exhibits large fluctuations (typically at low transport rates), making the measurement of mean values difficult. We develop an approach based on birth-death Markov processes, which involves monitoring the evolution of the number of particles moving within an array of cells of finite length. While the topic has been explored in detail for diffusion-reaction systems, the treatment of advection has received little attention. We show that particle advection produces nonlocal effects, which are more or less significant depending on the cell size and particle velocity. Albeit nonlocal, these effects look like (local) diffusion and add to the intrinsic particle diffusion (dispersal due to velocity fluctuations), with the important consequence that local measurements depend on both the intrinsic properties of particle displacement and the dimensions of the measurement system.

  6. A POSTERIORI ENERGY-NORM ERROR ESTIMATES FOR ADVECTION-DIFFUSION EQUATIONS APPROXIMATED BY WEIGHTED INTERIOR PENALTY METHODS

    Institute of Scientific and Technical Information of China (English)

    Alexandre Ern; Annette F.Stephansen

    2008-01-01

    We propose and analyze a posteriori energy-norm error estimates for weighted interior penalty discontinuous Galerkin approximations of advection-diffusion-reaction equations with heterogeneous and anisotropic diffusion.The weights,which play a key role in the analysis.depend on the diffusion tensor and are used to formulate the consistency terms in the discontinuous Galerkin method.The error upper bounds,in which all the constants are specified.consist of three terms:a residual estimator which depends only on the elementwise fluctuation of the discrete solution residual,a diffusive flux estimator where the weights used in the method enter explicitly,and a non-conforming estimator which is nonzero because of the use of discontinuous finite element spaces.The three estimators can be bounded locally by the approximation error.A particular attention is given to the dependency on problem parameters of the constants in the local lower error bounds,For moderate advection.it.is shown that full robustness with respect to diffusion heterogeneities is achieved owing to the specific design of the weights in the discontinuous Galerkin method,while diffusion anisotropies remain purely local and impact the constants through the square root of the condition number of the diffusion tensor.For dominant advection,the local lower error bounds can be written with constants involving a cut-off for the ratio of local mesh size to the reciprocal of the square root of the lowest local eignevalue of the diffusion tensor.

  7. Predicting in vivo glioma growth with the reaction diffusion equation constrained by quantitative magnetic resonance imaging data

    Science.gov (United States)

    Hormuth, David A., II; Weis, Jared A.; Barnes, Stephanie L.; Miga, Michael I.; Rericha, Erin C.; Quaranta, Vito; Yankeelov, Thomas E.

    2015-07-01

    Reaction-diffusion models have been widely used to model glioma growth. However, it has not been shown how accurately this model can predict future tumor status using model parameters (i.e., tumor cell diffusion and proliferation) estimated from quantitative in vivo imaging data. To this end, we used in silico studies to develop the methods needed to accurately estimate tumor specific reaction-diffusion model parameters, and then tested the accuracy with which these parameters can predict future growth. The analogous study was then performed in a murine model of glioma growth. The parameter estimation approach was tested using an in silico tumor ‘grown’ for ten days as dictated by the reaction-diffusion equation. Parameters were estimated from early time points and used to predict subsequent growth. Prediction accuracy was assessed at global (total volume and Dice value) and local (concordance correlation coefficient, CCC) levels. Guided by the in silico study, rats (n = 9) with C6 gliomas, imaged with diffusion weighted magnetic resonance imaging, were used to evaluate the model’s accuracy for predicting in vivo tumor growth. The in silico study resulted in low global (tumor volume error 0.92) and local (CCC values >0.80) level errors for predictions up to six days into the future. The in vivo study showed higher global (tumor volume error >11.7%, Dice silico study shows that model parameters can be accurately estimated and used to accurately predict future tumor growth at both the global and local scale. However, the poor predictive accuracy in the experimental study suggests the reaction-diffusion equation is an incomplete description of in vivo C6 glioma biology and may require further modeling of intra-tumor interactions including segmentation of (for example) proliferative and necrotic regions.

  8. Asymptotic Bayesian Estimation of a First Order Equation with Small Diffusion

    OpenAIRE

    Hijab, Omar

    1984-01-01

    In this paper a finite-dimensional diffusion is observed in the presence of an additive Brownian motion. A large deviations result is obtained for the conditional probability distribution of the diffusion given the observations as the noise variances go to zero.

  9. Strain-specific chemotaxis of Azospirillum spp.

    OpenAIRE

    Reinhold, B; Hurek, T; Fendrik, I

    1985-01-01

    Chemotactic responses of three Azospirillum strains originating from different host plants were compared to examine the possible role of chemotaxis in the adaptation of these bacteria to their respective hosts. The chemotaxis to several sugars, amino acids, and organic acids was determined qualitatively by an agar plate assay and quantitatively by a channeled-chamber technique. High chemotactic ratios, up to 40, were obtained with the latter technique. The chemotactic response did not rely up...

  10. Semi-implicit finite-difference method with predictor-corrector algorithm for solution of diffusion equation with nonlinear terms

    CERN Document Server

    Lipp, V P; Garcia, M E; Ivanov, D S

    2015-01-01

    We present a finite-difference integration algorithm for solution of a system of differential equations containing a diffusion equation with nonlinear terms. The approach is based on Crank-Nicolson method with predictor-corrector algorithm and provides high stability and precision. Using a specific example of short-pulse laser interaction with semiconductors, we give a detailed description of the method and apply it for the solution of the corresponding system of differential equations, one of which is a nonlinear diffusion equation. The calculated dynamics of the energy density and the number density of photoexcited free carriers upon the absorption of laser energy are presented for the irradiated thin silicon film. The energy conservation within 0.2% has been achieved for the time step $10^4$ times larger than that in case of the explicit scheme, for the chosen numerical setup. We also present a few examples of successful application of the method demonstrating its benefits for the theoretical studies of la...

  11. Global Well-posedness of the 3D Primitive Equations With Partial Vertical Turbulence Mixing Heat Diffusion

    CERN Document Server

    Cao, Chongsheng

    2010-01-01

    The three--dimensional incompressible viscous Boussinesq equations, under the assumption of hydrostatic balance, govern the large scale dynamics of atmospheric and oceanic motion, and are commonly called the primitive equations. To overcome the turbulence mixing a partial vertical diffusion is usually added to the temperature advection (or density stratification) equation. In this paper we prove the global regularity of strong solutions to this model in a three-dimensional infinite horizontal channel, subject to periodic boundary conditions in the horizontal directions, and with no-penetration and stress-free boundary conditions on the solid, top and bottom, boundaries. Specifically, we show that short time strong solutions to the above problem exist globally in time, and that they depend continuously on the initial data.

  12. Localized numerical impulse solutions in diffuse neural networks modeled by the complex fractional Ginzburg-Landau equation

    Science.gov (United States)

    Mvogo, Alain; Tambue, Antoine; Ben-Bolie, Germain H.; Kofané, Timoléon C.

    2016-10-01

    We investigate localized wave solutions in a network of Hindmarsh-Rose neural model taking into account the long-range diffusive couplings. We show by a specific analytical technique that the model equations in the infrared limit (wave number k → 0) can be governed by the complex fractional Ginzburg-Landau (CFGL) equation. According to the stiffness of the system, we propose both the semi and the linearly implicit Riesz fractional finite-difference schemes to solve efficiently the CFGL equation. The obtained fractional numerical solutions for the nerve impulse reveal localized short impulse properties. We also show the equivalence between the continuous CFGL and the discrete Hindmarsh-Rose models for relatively large network.

  13. Modified Taylor solution of equation of oxygen diffusion in a spherical cell with Michaelis-Menten uptake kinetics

    Directory of Open Access Journals (Sweden)

    Hector Vazquez-Leal

    2015-03-01

    Full Text Available This work presents the application of a modified Taylor method to obtain a handy and easily computable approximate solution of the nonlinear differential equation to model the oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics. The obtained solution is fully symbolic in terms of the coefficients of the equation, allowing to use the same solution for different values of the maximum reaction rate, the Michaelis constant, and the permeability of the cell membrane. Additionally, the numerical experiments show the high accuracy of the proposed solution, resulting 1.658509453Å~10−15 as the lowest mean square error for a set of coefficients. The straightforward process to obtain the solution shows that the modified Taylor method is a handy alternative to a more sophisticated method because does not involve the solving of differential equations or calculate complicated integrals.

  14. Determination of transport wind speed in the gaussian plume diffusion equation for low-lying point sources

    Science.gov (United States)

    Wang, I. T.

    A general method for determining the effective transport wind speed, overlineu, in the Gaussian plume equation is discussed. Physical arguments are given for using the generalized overlineu instead of the often adopted release-level wind speed with the plume diffusion equation. Simple analytical expressions for overlineu applicable to low-level point releases and a wide range of atmospheric conditions are developed. A non-linear plume kinematic equation is derived using these expressions. Crosswind-integrated SF 6 concentration data from the 1983 PNL tracer experiment are used to evaluate the proposed analytical procedures along with the usual approach of using the release-level wind speed. Results of the evaluation are briefly discussed.

  15. Numerical solutions and error estimations for the space fractional diffusion equation with variable coefficients via Fibonacci collocation method.

    Science.gov (United States)

    Bahşı, Ayşe Kurt; Yalçınbaş, Salih

    2016-01-01

    In this study, the Fibonacci collocation method based on the Fibonacci polynomials are presented to solve for the fractional diffusion equations with variable coefficients. The fractional derivatives are described in the Caputo sense. This method is derived by expanding the approximate solution with Fibonacci polynomials. Using this method of the fractional derivative this equation can be reduced to a set of linear algebraic equations. Also, an error estimation algorithm which is based on the residual functions is presented for this method. The approximate solutions are improved by using this error estimation algorithm. If the exact solution of the problem is not known, the absolute error function of the problems can be approximately computed by using the Fibonacci polynomial solution. By using this error estimation function, we can find improved solutions which are more efficient than direct numerical solutions. Numerical examples, figures, tables are comparisons have been presented to show efficiency and usable of proposed method. PMID:27610294

  16. DIFFUSIVE-DISPERSIVE TRAVELING WAVES AND KINETIC RELATIONS IV.COMPRESSIBLE EULER EQUATIONS

    Institute of Scientific and Technical Information of China (English)

    2003-01-01

    The authors consider the Euler equations for a compressible fluid in one space dimensionwhen the equation of state of the fluid does not fulfill standard convexity assumptions andviscosity and capillarity effects are taken into account. A typical example of nonconvex con-stitutive equation for fluids is Van der Waals' equation. The first order terms of these partialdifferential equations form a nonlinear system of mixed (hyperbolic-elliptic) type. For a class ofnonconvex equations of state, an existence theorem of traveling waves solutions with arbitrarylarge amplitude is established here. The authors distinguish between classical (compressive) andnonclassical (undercompressive) traveling waves. The latter do not fulfill Lax shock inequali-ties, and are characterized by the so-called kinetic relation, whose properties are investigatedin this paper.

  17. On symmetry groups of a 2D nonlinear diffusion equation with source

    Indian Academy of Sciences (India)

    Radica Cimpoiasu

    2015-04-01

    Symmetry analysis of a 2D nonlinear evolutionary equation with mixed spatial derivative and general source term involving the dependent variable and its spatial derivatives is performed. The source terms for which the equation admits nontrivial Lie symmetries are identified for two different forms of the symmetry operator. In one of these cases, the symmetries do not depend on the form of nonlinearities and in the other case, nonlinearities of power, exponential and trigonometric forms are considered. There are no supplementary nonclassical symmetries for the investigated equation. The results reported here generalize the previous results on the 2D heat equation and the 2D Ricci model.

  18. Convergence to a propagating front in a degenerate Fisher-KPP equation with advection

    CERN Document Server

    Alfaro, Matthieu

    2011-01-01

    We consider a Fisher-KPP equation with density-dependent diffusion and advection, arising from a chemotaxis-growth model. We study its behavior as a small parameter, related to the thickness of a diffuse interface, tends to zero. We analyze, for small times, the emergence of transition layers induced by a balance between reaction and drift effects. Then we investigate the propagation of the layers. Convergence to a free-boundary limit problem is proved and a sharp estimate of the thickness of the layers is provided.

  19. ASYMPTOTIC METHOD OF TRAVELLING WAVE SOLUTIONS FOR A CLASS OF NONLINEAR REACTION DIFFUSION EQUATION

    Institute of Scientific and Technical Information of China (English)

    Mo Jiaqi; Zhang Weijiang; He Ming

    2007-01-01

    In this article the travelling wave solution for a class of nonlinear reaction diffusion problems are considered. Using the homotopic method and the theory of travelling wave transform, the approximate solution for the corresponding problem is obtained.

  20. Toward a General Yet Effective Computational Approach for Diffusive Problems: Variable Diffusion Tensor and DVR Solution of the Smoluchowski Equation along a General One-Dimensional Coordinate.

    Science.gov (United States)

    Piserchia, Andrea; Barone, Vincenzo

    2016-08-01

    A generalization to arbitrary large amplitude motions of a recent approach to the evaluation of diffusion tensors [ J. Comput. Chem. , 2009 , 30 , 2 - 13 ] is presented and implemented in a widely available package for electronic structure computations. A fully black-box tool is obtained, which, starting from the generation of geometric structures along different kinds of paths, proceeds toward the evaluation of an effective diffusion tensor and to the solution of one-dimensional Smoluchowski equations by a robust numerical approach rooted in the discrete variable representation (DVR). Application to a number of case studies shows that the results issuing from our approach are identical to those delivered by previous software (in particular DiTe) for rigid scans along a dihedral angle, but can be improved by employing relaxed scans (i.e., constrained geometry optimizations) or even more general large amplitude paths. The theoretical and numerical background is robust and general enough to allow quite straightforward extensions in several directions (e.g., inclusion of reactive paths, solution of Fokker-Planck or stochastic Liouville equations, multidimensional problems, free-energy rather than electronic-energy driven processes). PMID:27403666

  1. On blowup dynamics in the Keller-Segel model of chemotaxis

    CERN Document Server

    Dejak, S I; Lushnikov, P M; Sigal, I M

    2013-01-01

    We investigate the (reduced) Keller-Segel equations modeling chemotaxis of bio-organisms. We present a formal derivation and partial rigorous results of the blowup dynamics of solution of these equations describing the chemotactic aggregation of the organisms. Our results are confirmed by numerical simulations and the formula we derive coincides with the formula of Herrero and Vel\\'{a}zquez for specially constructed solutions.

  2. THE ASYMPTOTIC BEHAVIOR OF SOLUTION FOR THE SINGULARLY PERTURBED INITIAL BOUNDARY VALUE PROBLEMS OF THE REACTION DIFFUSION EQUATIONS IN A PART OF DOMAIN

    Institute of Scientific and Technical Information of China (English)

    刘其林; 莫嘉琪

    2001-01-01

    A class of singularly perturbed initial boundary value problems for the reaction diffusion equations in a part of domain are considered. Using the operator theory the asymptotic behavior of solution for the problems is studied.

  3. A New 2D-Transport, 1D-Diffusion Approximation of the Boltzmann Transport equation

    Energy Technology Data Exchange (ETDEWEB)

    Larsen, Edward

    2013-06-17

    The work performed in this project consisted of the derivation, implementation, and testing of a new, computationally advantageous approximation to the 3D Boltz- mann transport equation. The solution of the Boltzmann equation is the neutron flux in nuclear reactor cores and shields, but solving this equation is difficult and costly. The new “2D/1D” approximation takes advantage of a special geometric feature of typical 3D reactors to approximate the neutron transport physics in a specific (ax- ial) direction, but not in the other two (radial) directions. The resulting equation is much less expensive to solve computationally, and its solutions are expected to be sufficiently accurate for many practical problems. In this project we formulated the new equation, discretized it using standard methods, developed a stable itera- tion scheme for solving the equation, implemented the new numerical scheme in the MPACT code, and tested the method on several realistic problems. All the hoped- for features of this new approximation were seen. For large, difficult problems, the resulting 2D/1D solution is highly accurate, and is calculated about 100 times faster than a 3D discrete ordinates simulation.

  4. Stochastic processes and applications diffusion processes, the Fokker-Planck and Langevin equations

    CERN Document Server

    Pavliotis, Grigorios A

    2014-01-01

    This book presents various results and techniques from the theory of stochastic processes that are useful in the study of stochastic problems in the natural sciences. The main focus is analytical methods, although numerical methods and statistical inference methodologies for studying diffusion processes are also presented. The goal is the development of techniques that are applicable to a wide variety of stochastic models that appear in physics, chemistry and other natural sciences. Applications such as stochastic resonance, Brownian motion in periodic potentials and Brownian motors are studied and the connection between diffusion processes and time-dependent statistical mechanics is elucidated.                 The book contains a large number of illustrations, examples, and exercises. It will be useful for graduate-level courses on stochastic processes for students in applied mathematics, physics and engineering. Many of the topics covered in this book (reversible diffusions, convergence to eq...

  5. A finite elements method to solve the Bloch–Torrey equation applied to diffusion magnetic resonance imaging

    Energy Technology Data Exchange (ETDEWEB)

    Nguyen, Dang Van [INRIA Saclay, Equipe DEFI, CMAP, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex (France); NeuroSpin, Bat145, Point Courrier 156, CEA Saclay Center, 91191 Gif-sur-Yvette Cedex (France); Li, Jing-Rebecca, E-mail: jingrebecca.li@inria.fr [INRIA Saclay, Equipe DEFI, CMAP, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex (France); NeuroSpin, Bat145, Point Courrier 156, CEA Saclay Center, 91191 Gif-sur-Yvette Cedex (France); Grebenkov, Denis [Laboratoire de Physique de la Matiere Condensee, CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex (France); Le Bihan, Denis [NeuroSpin, Bat145, Point Courrier 156, CEA Saclay Center, 91191 Gif-sur-Yvette Cedex (France)

    2014-04-15

    The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses in a heterogeneous medium can be modeled by the multiple compartment Bloch–Torrey partial differential equation (PDE). In addition, steady-state Laplace PDEs can be formulated to produce the homogenized diffusion tensor that describes the diffusion characteristics of the medium in the long time limit. In spatial domains that model biological tissues at the cellular level, these two types of PDEs have to be completed with permeability conditions on the cellular interfaces. To solve these PDEs, we implemented a finite elements method that allows jumps in the solution at the cell interfaces by using double nodes. Using a transformation of the Bloch–Torrey PDE we reduced oscillations in the searched-for solution and simplified the implementation of the boundary conditions. The spatial discretization was then coupled to the adaptive explicit Runge–Kutta–Chebyshev time-stepping method. Our proposed method is second order accurate in space and second order accurate in time. We implemented this method on the FEniCS C++ platform and show time and spatial convergence results. Finally, this method is applied to study some relevant questions in diffusion MRI.

  6. On progress of the solution of the stationary 2-dimensional neutron diffusion equation: a polynomial approximation method with error analysis

    Energy Technology Data Exchange (ETDEWEB)

    Ceolin, C., E-mail: celina.ceolin@gmail.com [Universidade Federal de Santa Maria (UFSM), Frederico Westphalen, RS (Brazil). Centro de Educacao Superior Norte; Schramm, M.; Bodmann, B.E.J.; Vilhena, M.T., E-mail: celina.ceolin@gmail.com [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica

    2015-07-01

    Recently the stationary neutron diffusion equation in heterogeneous rectangular geometry was solved by the expansion of the scalar fluxes in polynomials in terms of the spatial variables (x; y), considering the two-group energy model. The focus of the present discussion consists in the study of an error analysis of the aforementioned solution. More specifically we show how the spatial subdomain segmentation is related to the degree of the polynomial and the Lipschitz constant. This relation allows to solve the 2-D neutron diffusion problem for second degree polynomials in each subdomain. This solution is exact at the knots where the Lipschitz cone is centered. Moreover, the solution has an analytical representation in each subdomain with supremum and infimum functions that shows the convergence of the solution. We illustrate the analysis with a selection of numerical case studies. (author)

  7. Method for estimating closed-form solutions of the light diffusion equation for turbid media of any boundary shape

    Science.gov (United States)

    Alqasemi, Umar; Salehi, Hassan S.; Zhu, Quing

    2016-01-01

    This paper reports a method of estimating an approximate closed-form solution to the light diffusion equation for any type of geometry involving Dirichlet’s boundary condition with known source location. It is based on estimating the optimum locations of multiple imaginary point sources to cancel the fluence at the extrapolated boundary by constrained optimization using a genetic algorithm. The mathematical derivation of the problem to approach the optimum solution for the direct-current type of diffuse optical systems is described in detail. Our method is first applied to slab geometry and compared with a truncated series solution. After that, it is applied to hemispherical geometry and compared with Monte Carlo simulation results. The method provides a fast and sufficiently accurate fluence distribution for optical reconstruction. PMID:26831771

  8. Sur la premi\\`ere loi de Fick et les \\'equations d'\\'equilibre quasi-statique gouvernant la diffusion mol\\'eculaire

    CERN Document Server

    Désoyer, Thierry

    2008-01-01

    The problem of the molecular diffusion in a biphasic fluid mixture is studied here from the two complementary points of view of Continuum Mechanics - in a somewhat different manner from Truesdell in "Mechanical basis of diffusion" (J. Chem. Physics (U.S.), 37 (1962) 2336-2344) - and that of Thermodynamics. It is established that the force involved in the 'diffusive drag', i.e. in the inter-constituent viscous friction, is necessarily linked to the relative diffusion velocity of one constituent with respect to the other. We also end up with Fick's first law, which appears, however, to be simply a particular form of more general equations governing diffusive motions.

  9. Rhodium SPND's Error Reduction using Extended Kalman Filter combined with Time Dependent Neutron Diffusion Equation

    Energy Technology Data Exchange (ETDEWEB)

    Lee, Jeong Hun; Park, Tong Kyu; Jeon, Seong Su [FNC Technology Co., Ltd., Yongin (Korea, Republic of)

    2014-05-15

    The Rhodium SPND is accurate in steady-state conditions but responds slowly to changes in neutron flux. The slow response time of Rhodium SPND precludes its direct use for control and protection purposes specially when nuclear power plant is used for load following. To shorten the response time of Rhodium SPND, there were some acceleration methods but they could not reflect neutron flux distribution in reactor core. On the other hands, some methods for core power distribution monitoring could not consider the slow response time of Rhodium SPND and noise effect. In this paper, time dependent neutron diffusion equation is directly used to estimate reactor power distribution and extended Kalman filter method is used to correct neutron flux with Rhodium SPND's and to shorten the response time of them. Extended Kalman filter is effective tool to reduce measurement error of Rhodium SPND's and even simple FDM to solve time dependent neutron diffusion equation can be an effective measure. This method reduces random errors of detectors and can follow reactor power level without cross-section change. It means monitoring system may not calculate cross-section at every time steps and computing time will be shorten. To minimize delay of Rhodium SPND's conversion function h should be evaluated in next study. Neutron and Rh-103 reaction has several decay chains and half-lives over 40 seconds causing delay of detection. Time dependent neutron diffusion equation will be combined with decay chains. Power level and distribution change corresponding movement of control rod will be tested with more complicated reference code as well as xenon effect. With these efforts, final result is expected to be used as a powerful monitoring tool of nuclear reactor core.

  10. Rhodium SPND's Error Reduction using Extended Kalman Filter combined with Time Dependent Neutron Diffusion Equation

    International Nuclear Information System (INIS)

    The Rhodium SPND is accurate in steady-state conditions but responds slowly to changes in neutron flux. The slow response time of Rhodium SPND precludes its direct use for control and protection purposes specially when nuclear power plant is used for load following. To shorten the response time of Rhodium SPND, there were some acceleration methods but they could not reflect neutron flux distribution in reactor core. On the other hands, some methods for core power distribution monitoring could not consider the slow response time of Rhodium SPND and noise effect. In this paper, time dependent neutron diffusion equation is directly used to estimate reactor power distribution and extended Kalman filter method is used to correct neutron flux with Rhodium SPND's and to shorten the response time of them. Extended Kalman filter is effective tool to reduce measurement error of Rhodium SPND's and even simple FDM to solve time dependent neutron diffusion equation can be an effective measure. This method reduces random errors of detectors and can follow reactor power level without cross-section change. It means monitoring system may not calculate cross-section at every time steps and computing time will be shorten. To minimize delay of Rhodium SPND's conversion function h should be evaluated in next study. Neutron and Rh-103 reaction has several decay chains and half-lives over 40 seconds causing delay of detection. Time dependent neutron diffusion equation will be combined with decay chains. Power level and distribution change corresponding movement of control rod will be tested with more complicated reference code as well as xenon effect. With these efforts, final result is expected to be used as a powerful monitoring tool of nuclear reactor core

  11. A variable timestep generalized Runge-Kutta method for the numerical integration of the space-time diffusion equations

    International Nuclear Information System (INIS)

    A generalized Runge-Kutta method has been employed in the numerical integration of the stiff space-time diffusion equations. The method is fourth-order accurate, using an embedded third-order solution to arrive at an estimate of the truncation error for automatic timestep control. The efficiency of the Runge-Kutta method is enhanced by a block factorization technique that exploits the sparse structure of the matrix system resulting from the space and energy discretized finite difference code shows the sparse matrix implementation of the generalized Runge-Kutta method to be highly accurate and efficient when compared to an optimized iterative theta method

  12. Explicit and implicit ode solvers using Krylov subspace optimization: Application to the diffusion equation and parabolic Maxwell`s system

    Energy Technology Data Exchange (ETDEWEB)

    Druskin, V.; Knizhnerman, L.

    1994-12-31

    The authors solve the Cauchy problem for an ODE system Au + {partial_derivative}u/{partial_derivative}t = 0, u{vert_bar}{sub t=0} = {var_phi}, where A is a square real nonnegative definite symmetric matrix of the order N, {var_phi} is a vector from R{sup N}. The stiffness matrix A is obtained due to semi-discretization of a parabolic equation or system with time-independent coefficients. The authors are particularly interested in large stiff 3-D problems for the scalar diffusion and vectorial Maxwell`s equations. First they consider an explicit method in which the solution on a whole time interval is projected on a Krylov subspace originated by A. Then they suggest another Krylov subspace with better approximating properties using powers of an implicit transition operator. These Krylov subspace methods generate optimal in a spectral sense polynomial approximations for the solution of the ODE, similar to CG for SLE.

  13. Quantifying equation-of-state and opacity errors using integrated supersonic diffusive radiation flow experiments on the National Ignition Facility

    Energy Technology Data Exchange (ETDEWEB)

    Guymer, T. M., E-mail: Thomas.Guymer@awe.co.uk; Moore, A. S.; Morton, J.; Allan, S.; Bazin, N.; Benstead, J.; Bentley, C.; Comley, A. J.; Garbett, W.; Reed, L.; Stevenson, R. M. [AWE Plc., Aldermaston, Reading RG7 4PR (United Kingdom); Kline, J. L.; Cowan, J.; Flippo, K.; Hamilton, C.; Lanier, N. E.; Mussack, K.; Obrey, K.; Schmidt, D. W.; Taccetti, J. M. [Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States); and others

    2015-04-15

    A well diagnosed campaign of supersonic, diffusive radiation flow experiments has been fielded on the National Ignition Facility. These experiments have used the accurate measurements of delivered laser energy and foam density to enable an investigation into SESAME's tabulated equation-of-state values and CASSANDRA's predicted opacity values for the low-density C{sub 8}H{sub 7}Cl foam used throughout the campaign. We report that the results from initial simulations under-predicted the arrival time of the radiation wave through the foam by ≈22%. A simulation study was conducted that artificially scaled the equation-of-state and opacity with the intended aim of quantifying the systematic offsets in both CASSANDRA and SESAME. Two separate hypotheses which describe these errors have been tested using the entire ensemble of data, with one being supported by these data.

  14. Diffusive KPP equations with free boundaries in time almost periodic environments: II. Spreading speeds and semi-wave solutions

    Science.gov (United States)

    Li, Fang; Liang, Xing; Shen, Wenxian

    2016-08-01

    In this series of papers, we investigate the spreading and vanishing dynamics of time almost periodic diffusive KPP equations with free boundaries. Such equations are used to characterize the spreading of a new species in time almost periodic environments with free boundaries representing the spreading fronts. In the first part of the series, we showed that a spreading-vanishing dichotomy occurs for such free boundary problems (see [16]). In this second part of the series, we investigate the spreading speeds of such free boundary problems in the case that the spreading occurs. We first prove the existence of a unique time almost periodic semi-wave solution associated to such a free boundary problem. Using the semi-wave solution, we then prove that the free boundary problem has a unique spreading speed.

  15. ASYMPTOTIC BEHAVIOR FOR NONOSCILLATORY SOLUTIONS OFREACTION-DIFFUSION DIFFERENTIAL EQUATIONS WITH SEVERAL DELAYS IN THE NEUTRAL TERM

    Institute of Scientific and Technical Information of China (English)

    徐美荣; 时宝

    2004-01-01

    In this paper, we first consider differential equations with several delays inthe neutral term of the form d/dt[y(t)+∑j=1L piy(t-ki)-∑j=1Mrjy(t-Pj)]+q(t)y(t-r)=0 for t∈R+,(1*) study various cases of coefficients in the neutral term and obtain the asymptotic behavior for nonoscillatory solution of (1") under some hypotheses. Moreover,we consider reaction-diffusion differential equations with several delays in the neutral term of the form α/αt[u(t,x)+∑i=1L piu(t-ki,x)-∑j=1M rju(t-Pj,x)]+q(t,x)u(t-r,x)=a2△u(t,x)for (t,x)∈R+ ×Ω. We also study various cases of coefficients in the neutral term and obtain the asymptotic behavior for nonoscillatory solutionun of (2*)under some hypotheses.

  16. Dependence of bacterial chemotaxis on gradient shape and adaptation rate.

    Directory of Open Access Journals (Sweden)

    Nikita Vladimirov

    2008-12-01

    Full Text Available Simulation of cellular behavior on multiple scales requires models that are sufficiently detailed to capture central intracellular processes but at the same time enable the simulation of entire cell populations in a computationally cheap way. In this paper we present RapidCell, a hybrid model of chemotactic Escherichia coli that combines the Monod-Wyman-Changeux signal processing by mixed chemoreceptor clusters, the adaptation dynamics described by ordinary differential equations, and a detailed model of cell tumbling. Our model dramatically reduces computational costs and allows the highly efficient simulation of E. coli chemotaxis. We use the model to investigate chemotaxis in different gradients, and suggest a new, constant-activity type of gradient to systematically study chemotactic behavior of virtual bacteria. Using the unique properties of this gradient, we show that optimal chemotaxis is observed in a narrow range of CheA kinase activity, where concentration of the response regulator CheY-P falls into the operating range of flagellar motors. Our simulations also confirm that the CheB phosphorylation feedback improves chemotactic efficiency by shifting the average CheY-P concentration to fit the motor operating range. Our results suggest that in liquid media the variability in adaptation times among cells may be evolutionary favorable to ensure coexistence of subpopulations that will be optimally tactic in different gradients. However, in a porous medium (agar such variability appears to be less important, because agar structure poses mainly negative selection against subpopulations with low levels of adaptation enzymes. RapidCell is available from the authors upon request.

  17. ASYMPTOTIC SOLUTION TO NONLINEAR REACTION DIFFUSION EQUATION WITH TWO SMALL PARAMETERS

    Institute of Scientific and Technical Information of China (English)

    2010-01-01

    In this paper, a class of nonlinear reaction diffusion singularly perturbed problem with two parameters is studied. Using the singular perturbation method, the structure of solution to the problem is discussed related two small parameters. The asymptotic solution to the problem is given.

  18. On the numerical solution of diffusion-reaction equations with singular source terms

    NARCIS (Netherlands)

    M. Ashyraliyev; J.G. Blom; J.G. Verwer

    2008-01-01

    A numerical study is presented of reaction-diffusion problems having singular reaction source terms, singular in the sense that within the spatial domain the source is defined by a Dirac delta function expression on a lower dimensional surface.A consequence is that solutions will be continuous, but

  19. On the numerical solution of diffusion-reaction equations with singular source terms

    NARCIS (Netherlands)

    Ashyraliyev, M.; Blom, J.G.; Verwer, J.G.

    2008-01-01

    A numerical study is presented of reaction–diffusion problems having singular reaction source terms, singular in the sense that within the spatial domain the source is defined by a Dirac delta function expression on a lower dimensional surface. A consequence is that solutions will be continuous, but

  20. Fundamental constraints on the abundances of chemotaxis proteins

    CERN Document Server

    Bitbol, Anne-Florence

    2015-01-01

    Flagellated bacteria, such as Escherichia coli, perform directed motion in gradients of concentration of attractants and repellents in a process called chemotaxis. The E. coli chemotaxis signaling pathway is a model for signal transduction, but it has unique features. We demonstrate that the need for fast signaling necessitates high abundances of the proteins involved in this pathway. We show that further constraints on the abundances of chemotaxis proteins arise from the requirements of self-assembly, both of flagellar motors and of chemoreceptor arrays. All these constraints are specific to chemotaxis, and published data confirm that chemotaxis proteins tend to be more highly expressed than their homologs in other pathways. Employing a chemotaxis pathway model, we show that the gain of the pathway at the level of the response regulator CheY increases with overall chemotaxis protein abundances. This may explain why, at least in one E. coli strain, the abundance of all chemotaxis proteins is higher in media w...

  1. Mixed dual finite element methods for the numerical treatment of the diffusion equation in hexagonal geometry; Elements finis mixtes duaux pour la resolution numerique de l'equation de la diffusion neutronique en geometrie hexagonale

    Energy Technology Data Exchange (ETDEWEB)

    Schneider, D

    2001-07-01

    The nodal method Minos has been developed to offer a powerful method for the calculation of nuclear reactor cores in rectangular geometry. This method solves the mixed dual form of the diffusion equation and, also of the simplified P{sub N} approximation. The discretization is based on Raviart-Thomas' mixed dual finite elements and the iterative algorithm is an alternating direction method, which uses the current as unknown. The subject of this work is to adapt this method to hexagonal geometry. The guiding idea is to construct and test different methods based on the division of a hexagon into trapeze or rhombi with appropriate mapping of these quadrilaterals onto squares in order to take into advantage what is already available in the Minos solver. The document begins with a review of the neutron diffusion equation. Then we discuss its mixed dual variational formulation from a functional as well as from a numerical point of view. We study conformal and bilinear mappings for the two possible meshing of the hexagon. Thus, four different methods are proposed and are completely described in this work. Because of theoretical and numerical difficulties, a particular treatment has been necessary for methods based on the conformal mapping. Finally, numerical results are presented for a hexagonal benchmark to validate and compare the four methods with respect to pre-defined criteria. (authors)

  2. Application of synthetic diffusion method in the numerical solution of the equations of neutron transport in slab geometry

    International Nuclear Information System (INIS)

    One of the main problems in reactor physics is to determine the neutron distribution in reactor core, since knowing that, it is possible to calculate the rapidity of occurrence of different nuclear reaction inside the reactor core. Within different theories existing in nuclear reactor physics, is neutron transport the one in which equation who govern the exact behavior of neutronic distribution are developed even inside the proper neutron transport theory, there exist different methods of solution which are approximations to exact solution; still more, with the purpose to reach a more precise solution, the majority of methods have been approached to the obtention of solutions in numerical form with the aim of take the advantages of modern computers, and for this reason a great deal of effort is dedicated to numerical solution of the equations of neutron transport. In agreement with the above mentioned, in this work has been developed a computer program which uses a relatively new techniques known as 'acceleration of synthetic diffusion' which has been applied to solve the neutron transport equation with 'classical schemes of spatial integration' obtaining results with a smaller quantity of interactions, if they compare to done without using such equation (Author)

  3. The Transport Equation in Optically Thick Media: Discussion of IMC and its Diffusion Limit

    Energy Technology Data Exchange (ETDEWEB)

    Szoke, A. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Brooks, E. D. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

    2016-07-12

    We discuss the limits of validity of the Implicit Monte Carlo (IMC) method for the transport of thermally emitted radiation. The weakened coupling between the radi- ation and material energy of the IMC method causes defects in handling problems with strong transients. We introduce an approach to asymptotic analysis for the transport equation that emphasizes the fact that the radiation and material tem- peratures are always di erent in time dependent problems, and we use it to show that IMC does not produce the correct di usion limit. As this is a defect of IMC in the continuous equations, no improvement to its discretization can remedy it.

  4. Chemotaxis toward phytoplankton drives organic matter partitioning among marine bacteria.

    Science.gov (United States)

    Smriga, Steven; Fernandez, Vicente I; Mitchell, James G; Stocker, Roman

    2016-02-01

    The microenvironment surrounding individual phytoplankton cells is often rich in dissolved organic matter (DOM), which can attract bacteria by chemotaxis. These "phycospheres" may be prominent sources of resource heterogeneity in the ocean, affecting the growth of bacterial populations and the fate of DOM. However, these effects remain poorly quantified due to a lack of quantitative ecological frameworks. Here, we used video microscopy to dissect with unprecedented resolution the chemotactic accumulation of marine bacteria around individual Chaetoceros affinis diatoms undergoing lysis. The observed spatiotemporal distribution of bacteria was used in a resource utilization model to map the conditions under which competition between different bacterial groups favors chemotaxis. The model predicts that chemotactic, copiotrophic populations outcompete nonmotile, oligotrophic populations during diatom blooms and bloom collapse conditions, resulting in an increase in the ratio of motile to nonmotile cells and in the succession of populations. Partitioning of DOM between the two populations is strongly dependent on the overall concentration of bacteria and the diffusivity of different DOM substances, and within each population, the growth benefit from phycospheres is experienced by only a small fraction of cells. By informing a DOM utilization model with highly resolved behavioral data, the hybrid approach used here represents a new path toward the elusive goal of predicting the consequences of microscale interactions in the ocean.

  5. An age-dependent population equation with diffusion and delayed birth process

    Directory of Open Access Journals (Sweden)

    G. Fragnelli

    2005-01-01

    Full Text Available We propose a new age-dependent population equation which takes into account not only a delay in the birth process, but also other events that may take place during the time between conception and birth. Using semigroup theory, we discuss the well posedness and the asymptotic behavior of the solution.

  6. Feynman path integral application on deriving black-scholes diffusion equation for european option pricing

    Science.gov (United States)

    Utama, Briandhika; Purqon, Acep

    2016-08-01

    Path Integral is a method to transform a function from its initial condition to final condition through multiplying its initial condition with the transition probability function, known as propagator. At the early development, several studies focused to apply this method for solving problems only in Quantum Mechanics. Nevertheless, Path Integral could also apply to other subjects with some modifications in the propagator function. In this study, we investigate the application of Path Integral method in financial derivatives, stock options. Black-Scholes Model (Nobel 1997) was a beginning anchor in Option Pricing study. Though this model did not successfully predict option price perfectly, especially because its sensitivity for the major changing on market, Black-Scholes Model still is a legitimate equation in pricing an option. The derivation of Black-Scholes has a high difficulty level because it is a stochastic partial differential equation. Black-Scholes equation has a similar principle with Path Integral, where in Black-Scholes the share's initial price is transformed to its final price. The Black-Scholes propagator function then derived by introducing a modified Lagrange based on Black-Scholes equation. Furthermore, we study the correlation between path integral analytical solution and Monte-Carlo numeric solution to find the similarity between this two methods.

  7. Asymptotic Analysis of Upwind Discontinuous Galerkin Approximation of the Radiative Transport Equation in the Diffusive Limit

    KAUST Repository

    Guermond, Jean-Luc

    2010-01-01

    We revisit some results from M. L. Adams [Nu cl. Sci. Engrg., 137 (2001), pp. 298- 333]. Using functional analytic tools we prove that a necessary and sufficient condition for the standard upwind discontinuous Galerkin approximation to converge to the correct limit solution in the diffusive regime is that the approximation space contains a linear space of continuous functions, and the restrictions of the functions of this space to each mesh cell contain the linear polynomials. Furthermore, the discrete diffusion limit converges in the Sobolev space H1 to the continuous one if the boundary data is isotropic. With anisotropic boundary data, a boundary layer occurs, and convergence holds in the broken Sobolev space H with s < 1/2 only © 2010 Society for Industrial and Applied Mathematics.

  8. Analysis of some identification problems for the reaction-diffusion-convection equation

    Science.gov (United States)

    Alekseev, G. V.; Mashkov, D. V.; Yashenko, E. N.

    2016-04-01

    Identification problems for a linear stationary reaction-diffusion-convection model, considered in the bounded domain under Dirichlet boundary condition, are studied. Using an optimization method these problems are reduced to respective control problems. The reaction coefficient and the volume density of substance source play the role of controls in this control problem. The solvability of the direct and control problems is proved, the optimality system, which describes the necessary optimality conditions, is derived and the numerical algorithm is developed.

  9. Magnetorotational turbulence in stratified shearing boxes with perfect gas equation of state and finite thermal diffusivity

    OpenAIRE

    G., Bodo; F., Cattaneo; A., Mignone; P., Rossi

    2012-01-01

    We present a numerical study of turbulence and dynamo action in stratified shearing boxes with zero mean magnetic flux. We assume that the fluid obeys the perfect gas law and has finite (constant) thermal diffusivity. The calculations begin from an isothermal state spanning three scale heights above and below the mid-plane. After a long transient the layers settle to a stationary state in which thermal losses out of the boundaries are balanced by dissipative heating. We identify two regimes. ...

  10. Fokker-Planck equation with memory: the cross over from ballistic to diffusive processes in many particle systems and incompressible media

    Directory of Open Access Journals (Sweden)

    V. Ilyin

    2013-03-01

    Full Text Available The unified description of diffusion processes that cross over from a ballistic behavior at short times to normal or anomalous diffusion (sub- or superdiffusion at longer times is constructed on the basis of a non-Markovian generalization of the Fokker-Planck equation. The necessary non- Markovian kinetic coefficients are determined by the observable quantities (mean- and mean square displacements. Solutions of the non-Markovian equation describing diffusive processes in the physical space are obtained. For long times these solutions agree with the predictions of continuous random walk theory; they are however much superior at shorter times when the effect of the ballistic behavior is crucial.

  11. On the modeling of the bottom particles segregation with non-linear diffusion equations: application to the marine sand ripples

    Science.gov (United States)

    Tiguercha, Djlalli; Bennis, Anne-claire; Ezersky, Alexander

    2015-04-01

    The elliptical motion in surface waves causes an oscillating motion of the sand grains leading to the formation of ripple patterns on the bottom. Investigation how the grains with different properties are distributed inside the ripples is a difficult task because of the segration of particle. The work of Fernandez et al. (2003) was extended from one-dimensional to two-dimensional case. A new numerical model, based on these non-linear diffusion equations, was developed to simulate the grain distribution inside the marine sand ripples. The one and two-dimensional models are validated on several test cases where segregation appears. Starting from an homogeneous mixture of grains, the two-dimensional simulations demonstrate different segregation patterns: a) formation of zones with high concentration of light and heavy particles, b) formation of «cat's eye» patterns, c) appearance of inverse Brazil nut effect. Comparisons of numerical results with the new set of field data and wave flume experiments show that the two-dimensional non-linear diffusion equations allow us to reproduce qualitatively experimental results on particles segregation.

  12. A unified nonlinear fractional equation of the diffusion-controlled surfactant adsorption: Reappraisal and new solution of the Ward–Tordai problem

    OpenAIRE

    Jordan Hristov

    2016-01-01

    The article addresses a reappraisal of the famous Ward–Tordai equation describing the equilibrium of surfactants at air/liquid interfaces under diffusion control. The new derivation is entirely developed in the light of fractional calculus. The unified approach demonstrates that this equation can be clearly reformulated as a nonlinear ordinary time-fractional equation of order 1/2. The work formulates versions with different isotherms. A simple solution of the case with the Henry’s isotherm a...

  13. Dynamics of One- and Two-dimensional Kinks in Bistable Reaction-Diffusion Equations with Quasi-Discrete Sources of Reaction

    OpenAIRE

    Rotstein, Horacio G.; Zhabotinsky, Anatol M; Epstein, Irving R.

    2000-01-01

    We study the evolution of fronts in a bistable reaction-diffusion system when the nonlinear reaction term is spatially non-homogeneous. This equation has been used to model wave propagation in various biological systems. Extending previous works on homogeneous reaction terms, we derive asymptotically an equation governing the front motion, which is strongly nonlinear and, for the two-dimensional case, generalizes the classical mean curvature flow equation. We study the motion of one- and two-...

  14. Computational Diffusion Magnetic Resonance Imaging Based on Time-Dependent Bloch NMR Flow Equation and Bessel Functions.

    Science.gov (United States)

    Awojoyogbe, Bamidele O; Dada, Michael O; Onwu, Samuel O; Ige, Taofeeq A; Akinwande, Ninuola I

    2016-04-01

    Magnetic resonance imaging (MRI) uses a powerful magnetic field along with radio waves and a computer to produce highly detailed "slice-by-slice" pictures of virtually all internal structures of matter. The results enable physicians to examine parts of the body in minute detail and identify diseases in ways that are not possible with other techniques. For example, MRI is one of the few imaging tools that can see through bones, making it an excellent tool for examining the brain and other soft tissues. Pulsed-field gradient experiments provide a straightforward means of obtaining information on the translational motion of nuclear spins. However, the interpretation of the data is complicated by the effects of restricting geometries as in the case of most cancerous tissues and the mathematical concept required to account for this becomes very difficult. Most diffusion magnetic resonance techniques are based on the Stejskal-Tanner formulation usually derived from the Bloch-Torrey partial differential equation by including additional terms to accommodate the diffusion effect. Despite the early success of this technique, it has been shown that it has important limitations, the most of which occurs when there is orientation heterogeneity of the fibers in the voxel of interest (VOI). Overcoming this difficulty requires the specification of diffusion coefficients as function of spatial coordinate(s) and such a phenomenon is an indication of non-uniform compartmental conditions which can be analyzed accurately by solving the time-dependent Bloch NMR flow equation analytically. In this study, a mathematical formulation of magnetic resonance flow sequence in restricted geometry is developed based on a general second order partial differential equation derived directly from the fundamental Bloch NMR flow equations. The NMR signal is obtained completely in terms of NMR experimental parameters. The process is described based on Bessel functions and properties that can make it

  15. Numeric algorithms for parallel processors computer architectures with applications to the few-groups neutron diffusion equations

    International Nuclear Information System (INIS)

    A numeric algorithm and an associated computer code were developed for the rapid solution of the finite-difference method representation of the few-group neutron-diffusion equations on parallel computers. Applications of the numeric algorithm on both SIMD (vector pipeline) and MIMD/SIMD (multi-CUP/vector pipeline) architectures were explored. The algorithm was successfully implemented in the two-group, 3-D neutron diffusion computer code named DIFPAR3D (DIFfusion PARallel 3-Dimension). Numerical-solution techniques used in the code include the Chebyshev polynomial acceleration technique in conjunction with the power method of outer iteration. For inner iterations, a parallel form of red-black (cyclic) line SOR with automated determination of group dependent relaxation factors and iteration numbers required to achieve specified inner iteration error tolerance is incorporated. The code employs a macroscopic depletion model with trace capability for selected fission products' transients and critical boron. In addition to this, moderator and fuel temperature feedback models are also incorporated into the DIFPAR3D code, for realistic simulation of power reactor cores. The physics models used were proven acceptable in separate benchmarking studies

  16. On the analytical solution of the multigroup neutron diffusion kinetic equation in a multilayered slab

    Energy Technology Data Exchange (ETDEWEB)

    Ceolin, Celina; Vilhena, Marco T.; Bodmann, Bardo E.J., E-mail: vilhena@pq.cnpq.b, E-mail: bardo.bodmann@ufrgs.b [Universidade Federal do Rio Grande do Sul (PROMEC/UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica; Alvim, Antonio Carlos Marques, E-mail: alvim@nuclear.ufrj.b [Universidade Federal do Rio de Janeiro (PEN/COPPE/UFRJ), RJ (Brazil). Coordenacao dos Programas de Pos-Graduacao de Engenharia. Programa de Energia Nuclear

    2011-07-01

    The authors solved analytically the neutron kinetic equations in a homogeneous slab, assuming the multi group energy model and six delayed neutron precursor groups by the Generalized Integral Laplace Transform Technique (GILTT) for a multi-layered slab. To this end, averaged values for the nuclear parameters in the multi-layered slab are used and the solution is constructed following the idea of Adomian's decomposition method upon reducing the heterogeneous problem to a set of recursive problems with constant parameters in the multi-layered slab. More specifically, the corrections that render the initially homogeneous problem into a heterogeneous one are plugged into the equation as successive source terms. To the best of our knowledge this sort of solution is novel and not found in literature. We further present some numerical simulations. (author)

  17. A nodal method for solving the time-depending diffusion equation in the IQS approximation

    International Nuclear Information System (INIS)

    The fast and slow variation of the neutron flux shape needed for the dynamical description of nuclear reactor cores can be described advantageously in the Improved Quasistatic (IQS) model where the flux is factorized by a fast changing space-independent amplitude and a slow changing shape function. The basic equations of a time-dependent nodal approximation using the IQS method is presented.The calculational procedure of the response matrices is also described. (R.P.) 2 refs

  18. Duality-based Asymptotic-Preserving method for highly anisotropic diffusion equations

    OpenAIRE

    Degond, Pierre; Deluzet, Fabrice; Lozinski, Alexei; Narski, Jacek; Negulescu, Claudia

    2010-01-01

    The present paper introduces an efficient and accurate numerical scheme for the solution of a highly anisotropic elliptic equation, the anisotropy direction being given by a variable vector field. This scheme is based on an asymptotic preserving reformulation of the original system, permitting an accurate resolution independently of the anisotropy strength and without the need of a mesh adapted to this anisotropy. The counterpart of this original procedure is the larger system size, enlarged ...

  19. Deep tissue fluorescence resonance energy transfer imaging and the application of the Bethe-Salpeter equation to non-diffusive wave propagation

    Science.gov (United States)

    Gaind, Vaibhav

    Fluorescence resonance energy transfer (FRET) has found many applications in in vitro imaging as an indicator of molecular activity. However, till now, in vivo FRET imaging has been restricted to near-surface multiphoton microscopy. Optical diffusion tomography (ODT) is an emerging tool for deep tissue imaging. In this work, FRET was incorporated in an ODT framework, thereby allowing FRET to be applied in deep tissue imaging. Using simulations and tissue phantom and small animal imaging experiments, the possibility of imaging molecular activity on the nanometer scale using macroscopic measurements is demonstrated. The diffusion equation model is limited to regions of high scatter and low absorption. The Bethe-Salpeter equation has been used extensively to explain various scattering phenomena and is more fundamental than the Boltzmann transport equation. In this work, the Bethe-Salpeter equation has been investigated for modeling photon transport in the non-diffusive regime.

  20. Imprecision of adaptation in Escherichia coli chemotaxis.

    Directory of Open Access Journals (Sweden)

    Silke Neumann

    Full Text Available Adaptability is an essential property of many sensory systems, enabling maintenance of a sensitive response over a range of background stimulus levels. In bacterial chemotaxis, adaptation to the preset level of pathway activity is achieved through an integral feedback mechanism based on activity-dependent methylation of chemoreceptors. It has been argued that this architecture ensures precise and robust adaptation regardless of the ambient ligand concentration, making perfect adaptation a celebrated property of the chemotaxis system. However, possible deviations from such ideal adaptive behavior and its consequences for chemotaxis have not been explored in detail. Here we show that the chemotaxis pathway in Escherichia coli shows increasingly imprecise adaptation to higher concentrations of attractants, with a clear correlation between the time of adaptation to a step-like stimulus and the extent of imprecision. Our analysis suggests that this imprecision results from a gradual saturation of receptor methylation sites at high levels of stimulation, which prevents full recovery of the pathway activity by violating the conditions required for precise adaptation. We further use computer simulations to show that limited imprecision of adaptation has little effect on the rate of chemotactic drift of a bacterial population in gradients, but hinders precise accumulation at the peak of the gradient. Finally, we show that for two major chemoeffectors, serine and cysteine, failure of adaptation at concentrations above 1 mM might prevent bacteria from accumulating at toxic concentrations of these amino acids.

  1. Duality-based Asymptotic-Preserving method for highly anisotropic diffusion equations

    CERN Document Server

    Degond, Pierre; Lozinski, Alexei; Narski, Jacek; Negulescu, Claudia

    2010-01-01

    The present paper introduces an efficient and accurate numerical scheme for the solution of a highly anisotropic elliptic equation, the anisotropy direction being given by a variable vector field. This scheme is based on an asymptotic preserving reformulation of the original system, permitting an accurate resolution independently of the anisotropy strength and without the need of a mesh adapted to this anisotropy. The counterpart of this original procedure is the larger system size, enlarged by adding auxiliary variables and Lagrange multipliers. This Asymptotic-Preserving method generalizes the method investigated in a previous paper [arXiv:0903.4984v2] to the case of an arbitrary anisotropy direction field.

  2. Toward Synthetic Spatial Patterns in Engineered Cell Populations with Chemotaxis.

    Science.gov (United States)

    Duran-Nebreda, Salva; Solé, Ricard V

    2016-07-15

    A major force shaping form and patterns in biology is based in the presence of amplification mechanisms able to generate ordered, large-scale spatial structures out of local interactions and random initial conditions. Turing patterns are one of the best known candidates for such ordering dynamics, and their existence has been proven in both chemical and physical systems. Their relevance in biology, although strongly supported by indirect evidence, is still under discussion. Extensive modeling approaches have stemmed from Turing's pioneering ideas, but further confirmation from experimental biology is required. An alternative possibility is to engineer cells so that self-organized patterns emerge from local communication. Here we propose a potential synthetic design based on the interaction between population density and a diffusing signal, including also directed motion in the form of chemotaxis. The feasibility of engineering such a system and its implications for developmental biology are also assessed. PMID:27009520

  3. A Parallel Algorithm for the Two-Dimensional Time Fractional Diffusion Equation with Implicit Difference Method

    Directory of Open Access Journals (Sweden)

    Chunye Gong

    2014-01-01

    Full Text Available It is very time consuming to solve fractional differential equations. The computational complexity of two-dimensional fractional differential equation (2D-TFDE with iterative implicit finite difference method is O(MxMyN2. In this paper, we present a parallel algorithm for 2D-TFDE and give an in-depth discussion about this algorithm. A task distribution model and data layout with virtual boundary are designed for this parallel algorithm. The experimental results show that the parallel algorithm compares well with the exact solution. The parallel algorithm on single Intel Xeon X5540 CPU runs 3.16–4.17 times faster than the serial algorithm on single CPU core. The parallel efficiency of 81 processes is up to 88.24% compared with 9 processes on a distributed memory cluster system. We do think that the parallel computing technology will become a very basic method for the computational intensive fractional applications in the near future.

  4. A simple Boltzmann transport equation for ballistic to diffusive transient heat transport

    International Nuclear Information System (INIS)

    Developing simplified, but accurate, theoretical approaches to treat heat transport on all length and time scales is needed to further enable scientific insight and technology innovation. Using a simplified form of the Boltzmann transport equation (BTE), originally developed for electron transport, we demonstrate how ballistic phonon effects and finite-velocity propagation are easily and naturally captured. We show how this approach compares well to the phonon BTE, and readily handles a full phonon dispersion and energy-dependent mean-free-path. This study of transient heat transport shows (i) how fundamental temperature jumps at the contacts depend simply on the ballistic thermal resistance, (ii) that phonon transport at early times approach the ballistic limit in samples of any length, and (iii) perceived reductions in heat conduction, when ballistic effects are present, originate from reductions in temperature gradient. Importantly, this framework can be recast exactly as the Cattaneo and hyperbolic heat equations, and we discuss how the key to capturing ballistic heat effects is to use the correct physical boundary conditions

  5. Nonlocal diffusion problems that approximate a parabolic equation with spatial dependence

    Science.gov (United States)

    Molino, Alexis; Rossi, Julio D.

    2016-06-01

    In this paper, we show that smooth solutions to the Dirichlet problem for the parabolic equation v_t(x,t)=sum_{i,j=1}N a_{ij}(x)partial2v(x,t)/partial{xipartial{x}j} + sum_{i =1}N bi(x)partial{v}(x,t)/partial{x_i} qquad x in Ω, with v( x, t) = g( x, t), {x in partial Ω,} can be approximated uniformly by solutions of nonlocal problems of the form ut^{\\varepsilon}(x,t)=int_{mathbb{R}n} K_{\\varepsilon}(x,y)(u^{\\varepsilon}(y,t)-u^{\\varepsilon}(x,t))dy, quad x in Ω, with {u^{\\varepsilon}(x,t)=g(x,t)}, {x notin Ω}, as {\\varepsilon to 0}, for an appropriate rescaled kernel {K_{\\varepsilon}}. In this way, we show that the usual local evolution problems with spatial dependence can be approximated by nonlocal ones. In the case of an equation in divergence form, we can obtain an approximation with symmetric kernels, that is, {K_{\\varepsilon}(x,y) = K_{\\varepsilon}(y,x)}.

  6. Calculating the vertex unknowns of nine point scheme on quadrilateral meshes for diffusion equation

    Institute of Scientific and Technical Information of China (English)

    2008-01-01

    In the construction of nine point scheme,both vertex unknowns and cell-centered unknowns are introduced,and the vertex unknowns are usually eliminated by using the interpolation of neighboring cell-centered unknowns,which often leads to lose accuracy.Instead of using interpolation,here we propose a different method of calculating the vertex unknowns of nine point scheme,which are solved independently on a new generated mesh.This new mesh is a Vorono¨i mesh based on the vertexes of primary mesh and some additional points on the interface.The advantage of this method is that it is particularly suitable for solving diffusion problems with discontinuous coeffcients on highly distorted meshes,and it leads to a symmetric positive definite matrix.We prove that the method has first-order convergence on distorted meshes.Numerical experiments show that the method obtains nearly second-order accuracy on distorted meshes.

  7. Conjugate gradient like methods and their application to eigenvalue problems for neutron diffusion equation

    International Nuclear Information System (INIS)

    This paper presents conjugate gradient (CG) like methods for solving generalized eigenvalue problems for neutron diffusion. The CG method to minimize the Rayleigh quotient has been proposed for one-group eigenvalue problems. In order to accelerate the convergence, we employed preconditioners based on incomplete factorization of the coefficient matrix. In the case of few-group eigenvalue problems, we introduced a residual norm and minimized it by the ORTHOMIN(1) (OR) method. We also proposed preconditioned OR methods. In order to vectorize the preconditioned CG like methods based on incomplete factorization, the hyperplane method was used and implemented on vector computers. Numerical experiments showed that the preconditioned CG like methods required a much smaller number of iterations and computational time compared with the conventional inner-outer iterative scheme for both one-group and few-group eigenvalue problems. (author)

  8. Two-Phase Fluid Simulation Using a Diffuse Interface Model with Peng--Robinson Equation of State

    KAUST Repository

    Qiao, Zhonghua

    2014-01-01

    In this paper, two-phase fluid systems are simulated using a diffusive interface model with the Peng-Robinson equation of state (EOS), a widely used realistic EOS for hydrocarbon fluid in the petroleum industry. We first utilize the gradient theory of thermodynamics and variational calculus to derive a generalized chemical equilibrium equation, which is mathematically a second-order elliptic partial differential equation (PDE) in molar density with a strongly nonlinear source term. To solve this PDE, we convert it to a time-dependent parabolic PDE with the main interest in its final steady state solution. A Lagrange multiplier is used to enforce mass conservation. The parabolic PDE is then solved by mixed finite element methods with a semi-implicit time marching scheme. Convex splitting of the energy functional is proposed to construct this time marching scheme, where the volume exclusion effect of an EOS is treated implicitly while the pairwise attraction effect of EOS is calculated explicitly. This scheme is proved to be unconditionally energy stable. Our proposed algorithm is able to solve successfully the spatially heterogeneous two-phase systems with the Peng-Robinson EOS in multiple spatial dimensions, the first time in the literature. Numerical examples are provided with realistic hydrocarbon components to illustrate the theory. Furthermore, our computational results are compared with laboratory experimental data and verified with the Young-Laplace equation with good agreement. This work sets the stage for a broad extension of efficient convex-splitting semi-implicit schemes for numerical simulation of phase field models with a realistic EOS in complex geometries of multiple spatial dimensions.

  9. Negative chemotaxis does not control quail neural crest cell dispersion.

    Science.gov (United States)

    Erickson, C A; Olivier, K R

    1983-04-01

    Negative chemotaxis has been proposed to direct dispersion of amphibian neural crest cells away from the neural tube (V. C. Twitty, 1949, Growth 13(Suppl. 9), 133-161). We have reexamined this hypothesis using quail neural crest and do not find evidence for it. When pigmented or freshly isolated neural crest cells are covered by glass shards to prevent diffusion of a "putative" chemotactic agent away from the cells and into the medium, we find a decrease in density of cells beneath the coverslip as did Twitty and Niu (1948, J. Exp. Zool. 108, 405-437). Unlike those investigators, however, we find the covered cells move slower than uncovered cells and that the decrease in density can be attributed to cessation of cell division and increased cell death in older cultures, rather than directed migration away from each other. In cell systems where negative chemotaxis has been demonstrated, a "no man's land" forms between two confronted explants (Oldfield, 1963, Exp. Cell Res. 30, 125-138). No such cell-free space forms between confronted neural crest explants, even if the explants are closely covered to prevent diffusion of the negative chemotactic material. If crest cell aggregates are drawn into capillary tubes to allow accumulation of the putative material, the cells disperse farther, the wider the capillary tube bore. This is contrary to what would be expected if dispersion depended on accumulation of this material. Also, no difference in dispersion is noted between cells in the center of the tubes versus cells near the mouth of the tubes where the tube medium is freely exchanging with external fresh medium. Alternative hypotheses for directionality of crest migration in vivo are discussed. PMID:6832483

  10. UNIFORM BLOWUP PROFILES FOR DIFFUSION EQUATIONS WITH NONLOCAL SOURCE AND NONLOCAL BOUNDARY

    Institute of Scientific and Technical Information of China (English)

    林支桂; 刘玉荣

    2004-01-01

    Long time behavior of solutions to semilinear parabolic equations with nonlocal nonlinear source ut - △u = ∫Ω g(u)dx inΩ× (0, T) and with nonlocal boundary condition u(x, t) = ∫Ω f(x, y)u(y, t)dy on(e) Ω× (0, T) is studied. The authors establish local existence, global existence and nonexistence of solutions and discuss the blowup properties of solutions. Moveover, they derive the uniform blowup estimates for g(s) = sp(p > 1) and g(s) = es under the assumption fΩ f(x, y)dy < 1 for x ∈(e)Ω.

  11. Numerical Treatment of Degenerate Diffusion Equations via Feller's Boundary Classification, and Applications

    Science.gov (United States)

    Cacio, Emanuela; Cohn, Stephen E.; Spigler, Renato

    2011-01-01

    A numerical method is devised to solve a class of linear boundary-value problems for one-dimensional parabolic equations degenerate at the boundaries. Feller theory, which classifies the nature of the boundary points, is used to decide whether boundary conditions are needed to ensure uniqueness, and, if so, which ones they are. The algorithm is based on a suitable preconditioned implicit finite-difference scheme, grid, and treatment of the boundary data. Second-order accuracy, unconditional stability, and unconditional convergence of solutions of the finite-difference scheme to a constant as the time-step index tends to infinity are further properties of the method. Several examples, pertaining to financial mathematics, physics, and genetics, are presented for the purpose of illustration.

  12. Bernstein dual-Petrov-Galerkin method: application to 2D time fractional diffusion equation

    CERN Document Server

    Jani, Mostafa; Babolian, Esmail

    2016-01-01

    In this paper, we develop a dual-Petrov-Galerkin method using Bernstein polynomials. The method is then implemented for the numerical simulation of the two-dimensional subdiffusion equation. The method is based on a finite difference discretization in time and a spectral method in space utilizing a suitable compact combinations of dual Bernstein basis as the test functions and the Bernstein polynomials as the trial ones. We derive the exact sparse operational matrix of differentiation for the dual Bernstein basis which provides a matrix-based approach for spatial discretization of the problem. It is also shown that the proposed method leads to banded linear systems. Finally some numerical examples are provided to show the efficiency and accuracy of the method.

  13. Multiscale analysis of collective motion and decision-making in swarms: an advection-diffusion equation with memory approach.

    Science.gov (United States)

    Raghib, M; Levin, S A; Kevrekidis, I G

    2010-06-01

    We propose a (time) multiscale method for the coarse-grained analysis of collective motion and decision-making in self-propelled particle models of swarms comprising a mixture of 'naïve' and 'informed' individuals. The method is based on projecting the particle configuration onto a single 'meta-particle' that consists of the elongation of the flock together with the mean group velocity and position. We find that the collective states can be associated with the transient and asymptotic transport properties of the random walk followed by the meta-particle, which we assume follows a continuous time random walk (CTRW). These properties can be accurately predicted at the macroscopic level by an advection-diffusion equation with memory (ADEM) whose parameters are obtained from a mean group velocity time series obtained from a single simulation run of the individual-based model.

  14. Compactness result for periodic structures and its application to the homogenization of a diffusion-convection equation

    Directory of Open Access Journals (Sweden)

    Anvarbek M. Meirmanov

    2011-09-01

    Full Text Available We prove the strong compactness of the sequence ${c^{varepsilon}(mathbf{x},t}$ in $L_2(Omega_T$, $Omega_T={(mathbf{x},t:mathbf{x}inOmega subset mathbb{R}^3, tin(0,T}$, bounded in $W^{1,0}_2(Omega_T$ with the sequence of time derivative ${partial/partial tig(chi(mathbf{x}/varepsilon c^{varepsilon}ig}$ bounded in the space $L_2ig((0,T; W^{-1}_2(Omegaig$. As an application we consider the homogenization of a diffusion-convection equation with a sequence of divergence-free velocities ${mathbf{v}^{varepsilon}(mathbf{x},t}$ weakly convergent in $L_2(Omega_T$.

  15. Analysis of two-level method for anisotropic diffusion equations on aligned and non-aligned grids

    CERN Document Server

    Yu, Guozhu; Zikatanov, Ludmil

    2011-01-01

    This paper is devoted to the multigrid convergence analysis for the linear systems arising from the conforming linear finite element discretization of the second order elliptic equations with anisotropic diffusion. The multigrid convergence behavior is known to strongly depend on whether the discretization grid is aligned or non-aligned with the anisotropic direction and analyses in the paper will be mainly focused on two-level algorithms. For an aligned grid case, a lower bound is given for point-wise smoother which shows deterioration of convergence rate. In both aligned and non-aligned cases we show that for a specially designed block smoother the convergence is uniform with respect to both anisotropy ratio and mesh size in the energy norm. The analysis is complemented with numerical experiments which confirm the theoretical results

  16. Transient oscillatory patterns in the diffusive non-local blowfly equation with delay under the zero-flux boundary condition

    International Nuclear Information System (INIS)

    In this paper, we study the spatial-temporal patterns of the solutions to the diffusive non-local Nicholson's blowflies equations with time delay (maturation time) subject to the no flux boundary condition. We establish the existence of both spatially homogeneous periodic solutions and various spatially inhomogeneous periodic solutions by investigating the Hopf bifurcations at the spatially homogeneous steady state. We also compute the normal form on the centre manifold, by which the bifurcation direction and stability of the bifurcated periodic solutions can be determined. The results show that the bifurcated homogeneous periodic solutions are stable, while the bifurcated inhomogeneous periodic solutions can only be stable on the corresponding centre manifold, implying that generically the model can only allow transient oscillatory patterns. Finally, we present some numerical simulations to demonstrate the theoretic results. For these transient patterns, we derive approximation formulas which are confirmed by numerical simulations. (paper)

  17. A variational nodal expansion method for the solution of multigroup neutron diffusion equations

    International Nuclear Information System (INIS)

    An accurate neutronics analysis method is needed for light water reactor core monitoring systems to efficiently operate the core with a smaller margin to limiting parameters. It is also required in in-core fuel management systems to optimize the core loading patterns, and the fuel designs with a higher reliability. When mixed oxide fuel or much higher burnup fuel is used, a new higher order nodal method seems necessary to introduce. Based on these considerations, a new nodal diffusion method for the neutronics analysis of light water reactor cores has been developed. The method is based on an approximation of neutron fluxes by expanding them with a set of functions defined within a node. The expansion coefficients are determined in such a way that the solution becomes the most accurate approximation to the exact solution by utilizing the variational principle. The expansion functions are obtained only from single assembly diffusion calculations. The present method includes no homogenization procedure, and the assembly heterogeneity effect on neutron fluxes is taken into account in a consistent way. The intra-nodal pin-power distribution can also be determined in a consistent way with high accuracy. The present method was implemented in a two-dimensional nodal code, and tested for benchmark cases. The results proved that the accuracy of the present method was excellent. The root mean square errors of both nodal powers and nodal maximum pin powers were observed to be less than 1%. The computing time of the code was measured to be about 3% of the reference, fine-mesh calculation. A three-dimensional version is currently being developed, and since the heterogeneity effect is of less importance in axial direction, a more efficient calculation method can be adopted for the axial solution of the neutron flux. The new method can be used as a ''plug-in'' module to existing core simulators to increase the accuracy of the neutronics part of existing core models, including the

  18. The distributed-order fractional diffusion-wave equation of groundwater flow: Theory and application to pumping and slug tests

    Science.gov (United States)

    Su, Ninghu; Nelson, Paul N.; Connor, Sarah

    2015-10-01

    We present a distributed-order fractional diffusion-wave equation (dofDWE) to describe radial groundwater flow to or from a well, and three sets of solutions of the dofDWE for flow from a well for aquifer tests: one for pumping tests, and two for slug tests. The dofDWE is featured by two temporal orders of fractional derivatives, β1 and β2, which characterise small and large pores, respectively. By fitting the approximate solutions of the dofDWE to data from slug tests in the field, we determined the effective saturated hydraulic conductivity, Ke, transmissivity, Tf, and the order of fractional derivatives, β2 in one test and β2 and β1 in the second test. We found that the patterns of groundwater flow from a well during the slug tests at this site belong to the class of sub-diffusion with β2 concept of the critical time to link Ke as a function of β2 and β1. The importance of the orders of fractional derivatives is obvious in the approximate solutions: for short time slug tests only the parameter β2 for flow in large pores is present while for long time slug tests the parameters β2 and β1 are present indicating both large and small pores are functioning.

  19. A Splitting Scheme for Solving Reaction-Diffusion Equations Modeling Dislocation Dynamics in Materials Subjected to Cyclic Loading

    Science.gov (United States)

    Pontes, J.; Walgraef, D.; Christov, C. I.

    2010-11-01

    Strain localization and dislocation pattern formation are typical features of plastic deformation in metals and alloys. Glide and climb dislocation motion along with accompanying production/annihilation processes of dislocations lead to the occurrence of instabilities of initially uniform dislocation distributions. These instabilities result into the development of various types of dislocation micro-structures, such as dislocation cells, slip and kink bands, persistent slip bands, labyrinth structures, etc., depending on the externally applied loading and the intrinsic lattice constraints. The Walgraef-Aifantis (WA) (Walgraef and Aifanits, J. Appl. Phys., 58, 668, 1985) model is an example of a reaction-diffusion model of coupled nonlinear equations which describe 0 formation of forest (immobile) and gliding (mobile) dislocation densities in the presence of cyclic loading. This paper discuss two versions of the WA model and focus on a finite difference, second order in time 1-Nicolson semi-implicit scheme, with internal iterations at each time step and a spatial splitting using the Stabilizing, Correction (Christov and Pontes, Mathematical and Computer Modelling, 35, 87, 2002) for solving the model evolution equations in two dimensions. The results of two simulations are presented. More complete results will appear in a forthcoming paper.

  20. Coupled third-order simplified spherical harmonics and diffusion equation-based fluorescence tomographic imaging of liver cancer

    Science.gov (United States)

    Chen, Xueli; Sun, Fangfang; Yang, Defu; Liang, Jimin

    2015-09-01

    For fluorescence tomographic imaging of small animals, the liver is usually regarded as a low-scattering tissue and is surrounded by adipose, kidneys, and heart, all of which have a high scattering property. This leads to a breakdown of the diffusion equation (DE)-based reconstruction method as well as a heavy computational burden for the simplified spherical harmonics equation (SPN). Coupling the SPN and DE provides a perfect balance between the imaging accuracy and computational burden. The coupled third-order SPN and DE (CSDE)-based reconstruction method is developed for fluorescence tomographic imaging. This is achieved by doubly using the CSDE for the excitation and emission processes of the fluorescence propagation. At the same time, the finite-element method and hybrid multilevel regularization strategy are incorporated in inverse reconstruction. The CSDE-based reconstruction method is first demonstrated with a digital mouse-based liver cancer simulation, which reveals superior performance compared with the SPN and DE-based methods. It is more accurate than the DE-based method and has lesser computational burden than the SPN-based method. The feasibility of the proposed approach in applications of in vivo studies is also illustrated with a liver cancer mouse-based in situ experiment, revealing its potential application in whole-body imaging of small animals.