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.
MacDonald, G.; Mackenzie, J. A.; Nolan, M.; Insall, R. H.
2016-03-01
In this paper, we devise a moving mesh finite element method for the approximate solution of coupled bulk-surface reaction-diffusion equations on an evolving two dimensional domain. Fundamental to the success of the method is the robust generation of bulk and surface meshes. For this purpose, we use a novel moving mesh partial differential equation (MMPDE) approach. The developed method is applied to model problems with known analytical solutions; these experiments indicate second-order spatial and temporal accuracy. Coupled bulk-surface problems occur frequently in many areas; in particular, in the modelling of eukaryotic cell migration and chemotaxis. We apply the method to a model of the two-way interaction of a migrating cell in a chemotactic field, where the bulk region corresponds to the extracellular region and the surface to the cell membrane.
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.
Effective Medium Equations for Chemotaxis in Porous Media
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.
Global Solutions to the Coupled Chemotaxis-Fluid Equations
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.
Travelling Waves in Hyperbolic Chemotaxis Equations
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.
Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion
Li, Yan; Lankeit, Johannes
2016-05-01
This article deals with an initial-boundary value problem for the coupled chemotaxis-haptotaxis system with nonlinear diffusion under homogeneous Neumann boundary conditions in a bounded smooth domain Ω \\subset {{{R}}n} , n = 2, 3, 4, where χ,ξ and μ are given nonnegative parameters. The diffusivity D(u) is assumed to satisfy D(u)≥slant δ {{u}m-1} for all u > 0 with some δ >0 . It is proved that for sufficiently regular initial data global bounded solutions exist whenever m>2-\\frac{2}{n} . For the case of non-degenerate diffusion (i.e. D(0) > 0) the solutions are classical; for the case of possibly degenerate diffusion (D(0)≥slant 0 ), the existence of bounded weak solutions is shown.
Speed ot travelling waves in reaction-diffusion equations
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)
Speed ot travelling waves in reaction-diffusion equations
International Nuclear Information System (INIS)
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)
Speed ot travelling waves in reaction-diffusion equations
Energy Technology Data Exchange (ETDEWEB)
Benguria, R.D.; Depassier, M.C. [Facultad de Fisica, Pontificia Universidad Catolica de Chile, Avda. Vicuna Mackenna 4860, Santiago (Chile); Mendez, V. [Facultat de Ciencies de la Salut, Universidad Internacional de Catalunya, Gomera s/n 08190 Sant Cugat del Valles, Barcelona (Spain)
2002-07-01
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)
Individual-based models for bacterial chemotaxis in the diffusion asymptotics
Rousset, Mathias
2011-01-01
We discuss velocity-jump models for chemotaxis of bacteria with an internal state that allows the velocity jump rate to depend on the memory of the chemoattractant concentration along their path of motion. Using probabilistic techniques, we provide a pathwise result that shows that the considered process converges to an advection-diffusion process in the (long-time) diffusion limit. We also (re-)prove using the same approach that the same limiting equation arises for a related, simpler process with direct sensing of the chemoattractant gradient. Additionally, we propose a time discretization technique that retains these diffusion limits exactly, i.e., without error that depends on the time discretization. In the companion paper \\cite{variance}, these results are used to construct a coupling technique that allows numerical simulation of the process with internal state with asymptotic variance reduction, in the sense that the variance vanishes in the diffusion limit.
Numerical study of plume patterns in the chemotaxis-diffusion-convection coupling system
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...
Multinomial diffusion equation
Balter, Ariel; Tartakovsky, Alexandre M.
2011-06-01
We describe a new, microscopic model for diffusion that captures diffusion induced fluctuations at scales where the concept of concentration gives way to discrete particles. We show that in the limit as the number of particles N→∞, our model is equivalent to the classical stochastic diffusion equation (SDE). We test our new model and the SDE against Langevin dynamics in numerical simulations, and show that our model successfully reproduces the correct ensemble statistics, while the classical model fails.
Ibrahim, Moustafa; Saad, Mazen
2015-01-01
This paper is devoted to the mathematical analysis of a degenerate nonlinear parabolic equation. This kind of equations stems either from the modeling of a compressible two phase flow in porous media or from the modeling of a chemotaxis-fluid process. In the degenerate equation, the strong nonlinearities are technically difficult to be controlled by the degenerate dissipative term because the equation itself presents degenerate terms of order 0 and of order 1. In the case of the sub-quadratic...
Planar Gradient Diffusion System to Investigate Chemotaxis in a 3D Collagen Matrix.
Stout, David A; Toyjanova, Jennet; Franck, Christian
2015-01-01
The importance of cell migration can be seen through the development of human life. When cells migrate, they generate forces and transfer these forces to their surrounding area, leading to cell movement and migration. In order to understand the mechanisms that can alter and/or affect cell migration, one can study these forces. In theory, understanding the fundamental mechanisms and forces underlying cell migration holds the promise of effective approaches for treating diseases and promoting cellular transplantation. Unfortunately, modern chemotaxis chambers that have been developed are usually restricted to two dimensions (2D) and have complex diffusion gradients that make the experiment difficult to interpret. To this end, we have developed, and describe in this paper, a direct-viewing chamber for chemotaxis studies, which allows one to overcome modern chemotaxis chamber obstacles able to measure cell forces and specific concentration within the chamber in a 3D environment to study cell 3D migration. More compelling, this approach allows one to successfully model diffusion through 3D collagen matrices and calculate the coefficient of diffusion of a chemoattractant through multiple different concentrations of collagen, while keeping the system simple and user friendly for traction force microscopy (TFM) and digital volume correlation (DVC) analysis. PMID:26131645
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.
Marangoni-driven chemotaxis, chemotactic collapse, and the Keller-Segel equation
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.
Diffusion equations and turbulent transport
International Nuclear Information System (INIS)
One scrutinized transport equations differing essentially in form from the classical diffusion one. Description of diffusion under strong nonequilibrium and turbulence involved application of equations that took account of transport nonlocality and memory effects. One analyzed ways to derive the mentioned equations starting from quasi-linear approximation and up to equations with fractional derivatives. One points out the generality of the applied theoretical concepts in spite of the essential difference of the exact physical problems. One demonstrated the way of application of the theoretical and probabilistic ideas
Diffusion equations and turbulent transport
International Nuclear Information System (INIS)
Diffusion equations are considered that differ substantially in structure from classical ones. A description of diffusion under strongly nonequilibrium conditions in a highly turbulent plasma requires the use of equations that take into account memory effects and the nonlocal nature of transport. Different methods are developed for constructing such equations, ranging from those in the quasilinear approximation to those with fractional derivatives. It is emphasized that the theoretical concepts underlying the equations proposed are common for a very wide variety of specific physical problems. The ways of applying theoretical probabilistic ideas are demonstrated
The generalized Airy diffusion equation
Directory of Open Access Journals (Sweden)
Frank M. Cholewinski
2003-08-01
Full Text Available Solutions of a generalized Airy diffusion equation and an associated nonlinear partial differential equation are obtained. Trigonometric type functions are derived for a third order generalized radial Euler type operator. An associated complex variable theory and generalized Cauchy-Euler equations are obtained. Further, it is shown that the Airy expansions can be mapped onto the Bessel Calculus of Bochner, Cholewinski and Haimo.
A coupled chemotaxis-fluid model: Global existence
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.
Parabolic equations in biology growth, reaction, movement and diffusion
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.
Perturbative linearization of reaction-diffusion equations
International Nuclear Information System (INIS)
We develop perturbative expansions to obtain solutions for the initial-value problems of two important reaction-diffusion systems, namely the Fisher equation and the time-dependent Ginzburg-Landau equation. The starting point of our expansion is the corresponding singular-perturbation solution. This approach transforms the solution of nonlinear reaction-diffusion equations into the solution of a hierarchy of linear equations. Our numerical results demonstrate that this hierarchy rapidly converges to the exact solution
A System of Non-linear Partial Differential Equations Modeling Chemotaxis with Sensitivity Functions
Post, Katharina
1999-01-01
Wir betrachten ein System nichtlinearer parabolischer partieller Differentialgleichungen zur Modellierung des biologischen Phänomens Chemotaxis, das unter anderem in Aggregationsprozessen in Lebenszyklen bestimmter Einzeller eine wichtige Rolle spielt. Unser Chemotaxismodell benutzt Sensitivitäts funktionen, die die vorkommenden biologischen Prozesse genauer spezifizieren. Trotz der durch die Sensitivitätsfunktionen eingebrachten, zusätzlichen Nichtlinearitäten in den Gleichungen erhalten w...
Fractional diffusion equations coupled by reaction terms
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.
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.
Fractional Diffusion Limit for Collisional Kinetic Equations
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.
Diffusive limits for linear transport equations
International Nuclear Information System (INIS)
The authors show that the Hibert and Chapman-Enskog asymptotic treatments that reduce the nonlinear Boltzmann equation to the Euler and Navier-Stokes fluid equations have analogs in linear transport theory. In this linear setting, these fluid limits are described by diffusion equations, involving familiar and less familiar diffusion coefficients. Because of the linearity extant, one can carry out explicitly the initial and boundary layer analyses required to obtain asymptotically consistent initial and boundary conditions for the diffusion equations. In particular, the effects of boundary curvature and boundary condition variation along the surface can be included in the boundary layer analysis. A brief review of heuristic (nonasymptotic) diffusion description derivations is also included in our discussion
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.
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.
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.
Diffusion phenomenon for linear dissipative wave equations
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.
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.
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.
Relativistic diffusion equation from stochastic quantization
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.
Logarithmic diffusion and porous media equations: a unified description
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...
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.
Existence of global solutions for a chemotaxis-fluid system with nonlinear diffusion
Chung, Yun-Sung; Kang, Kyungkeun
2016-04-01
We consider a coupled system consisting of the Navier-Stokes equations and a porous medium type of Keller-Segel system that model the motion of swimming bacteria living in fluid and consuming oxygen. We establish the global-in-time existence of weak solutions for the Cauchy problem of the system in dimension three. In addition, if the Stokes system, instead Navier-Stokes system, is considered for the fluid equation, we prove that bounded weak solutions exist globally in time.
A Convergent Reaction-Diffusion Master Equation
Isaacson, Samuel A
2012-01-01
The reaction-diffusion master equation (RDME) is a lattice stochastic reaction-diffusion model that has been used to study spatially distributed cellular processes. The RDME has been shown to have the drawback of losing bimolecular reactions in the continuum limit that the lattice spacing approaches zero (in two or more dimensions). In this work we derive a new convergent RDME (CRDME) that eliminates this problem. The CRDME is obtained by finite volume discretization of a spatially-continuous stochastic reaction-diffusion model. We demonstrate the empirical numerical convergence of reaction time statistics associated with the CRDME. Although the reaction time statistics of the RDME diverge as the lattice spacing approaches zero, we show they approach those of the CRDME for sufficiently large lattice spacings or slow bimolecular reaction rates. As such, the RDME may be interpreted as an approximation to the CRDME in several asymptotic limits.
Entropy methods for diffusive partial differential equations
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.
Self-similar dynamics of bacterial chemotaxis
Ngamsaad, Waipot
2012-01-01
We investigate the pattern formation of colony generated by chemotactic bacteria through a continuum model. In a simplified case, the dynamics of system is governed by a density-dependent convection-reaction-diffusion equation, $u_t = (u^{m})_{xx} - 2\\kappa(u^m)_{x}+ u - u^{m}$. This equation admits the analytical solutions that show the self-similarity of the bacterial colony's morphogenesis. In addition, we found that the colony evolves long time as the sharp traveling wave. The roles of chemotaxis on the regulation of pattern formation in these results are also discussed.
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.
Turbulence and diffusion scaling versus equations
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.
Differential constraints and exact solutions of nonlinear diffusion equations
International Nuclear Information System (INIS)
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
Differential constraints and exact solutions of nonlinear diffusion equations
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.
Nikitin, A. G.
2005-01-01
Group classification of systems of two coupled nonlinear reaction-diffusion equation with a general diffusion matrix started in papers math-ph/0411027,, math-ph/0411028 is completed in present paper where all non-equivalent equations with triangular diffusion matrix are classified. In addition, symmetries of diffusion systems with nilpotent diffusion matrix and additional first order derivative terms are described.
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)
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
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)
On an explicit finite difference method for fractional diffusion equations
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...
Voter Model Perturbations and Reaction Diffusion Equations
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 ...
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.
The Non-Classical Boltzmann Equation, and Diffusion-Based Approximations to the Boltzmann Equation
Frank, Martin; Larsen, Edward W; Vasques, Richard
2014-01-01
We show that several diffusion-based approximations (classical diffusion or SP1, SP2, SP3) to the linear Boltzmann equation can (for an infinite, homogeneous medium) be represented exactly by a non-classical transport equation. As a consequence, we indicate a method to solve diffusion-based approximations to the Boltzmann equation via Monte Carlo, with only statistical errors - no truncation errors.
Partial differential equations and non-diffusive structures
International Nuclear Information System (INIS)
In this paper we give a short introduction to open problems and recent studies of classes of partial differential equations, which—in contrast to reaction–diffusion systems—describe phenomena with local interactions. Partial differential equations coupled with ordinary differential equations, models of transport type and hyperbolic systems are discussed with respect to their pattern forming behaviour. (open problem)
Solution of diffusive wave equation using fem for flood forecasting
International Nuclear Information System (INIS)
Flood forecasting can be predicted numerically by the solution of 1D (One-Dimensional) unsteady diffusive wave equation. This research paper presents the development of a fem (Finite Element Model) for flood routing using diffusive wave with and without lateral inflow/outflow. FEM is based on two-step semi implicit Taylor Galerkin technique. The accuracy of the model has been verified by comparing computed results with available solution of the diffusive wave problems available in open literature. Numerical results demonstrate that the technique is an efficient and accurate tool to simulate diffusive wave equation for outflow hydrograph for temporal variation of lateral inflow and outflow. (author)
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)
International Nuclear Information System (INIS)
Two lines in improved neutron diffusion theory extending the efficiency of finite-difference diffusion codes to the field of optically small systems, are here reviewed. The firs involves the nodal solution for tensorial diffusion equation in slab geometry and tensorial formulation in parallelepiped and cylindrical gemometry; the dependence of critical eigenvalue from small slab thicknesses is also analitically investigated and finally a regularized tensorial diffusion equation is derived for slab. The other line refer to diffusion models formally unchanged with respect to the classical one, but where new size-dependent RTGB definitions for diffusion parameters are adopted, requiring that they allow to reproduce, in diffusion approach, the terms of neutron transport global balance; the trascendental equation for the buckling, arising in slab, sphere and parallelepiped geometry from the above requirement, are reported and the sizedependence of the new diffusion coefficient and extrapolated end point is investigated
LAGRANGE STABILITY IN MEAN SQUARE OF STOCHASTIC REACTION DIFFUSION EQUATIONS
Institute of Scientific and Technical Information of China (English)
无
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.
Biomixing by chemotaxis and enhancement of biological reactions
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...
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.
Measure solutions for the Smoluchowski coagulation-diffusion equation
Norris, James
2014-01-01
A notion of measure solution is formulated for a coagulation-diffusion equation, which is the natural counterpart of Smoluchowski's coagulation equation in a spatially inhomogeneous setting. Some general properties of such solutions are established. Sufficient conditions are identified on the diffusivity, coagulation rates and initial data for existence, uniqueness and mass conservation of solutions. These conditions impose no form of monotonicity on the coagulation kernel, which may depend o...
Hyperbolic reaction-diffusion equations for a forest fire model
Méndez López, Vicenç; Llebot, Josep Enric,
1997-01-01
Forest fire models have been widely studied from the context of self-organized criticality and from the ecological properties of the forest and combustion. On the other hand, reaction-diffusion equations have interesting applications in biology and physics. We propose here a model for fire propagation in a forest by using hyperbolic reaction-diffusion equations. The dynamical and thermodynamical aspects of the model are analyzed in detail.
The diffusion equation and the steady state. Chapter 2
International Nuclear Information System (INIS)
We shall now study the equations that govern the neutron field in a reactor. These equations are based on the concept of local neutron balance, which takes into account the reaction rates in an element of volume and the net leakage rates out of the volume. The reaction rates are written in terms of the local cross sections, assumed known from a preprocessed database (e.g., ENDF/B-VI). The starting equation is the Maxwell-Boltzmann transport equation, in its integro-differential form. The various approximations required to go from the transport equation to the neutron diffusion equation will be presented first, because all finite-reactor calculations are based on the diffusion approximation. We shall then discuss the multi-group formalism of the diffusion equations and study the mathematical properties of this equation in steady state. This preliminary step will allow us to derive in a more accurate way, in the next chapter, the reactor point-kinetics equations. In the diffusion approximation, neutrons diffuse from regions of high concentration to regions of low concentration, just as heat diffuses from regions of high temperature to those of low temperature, or, rather, as gas molecules diffuse to reduce spatial variations in concentration. While it is sufficiently accurate to treat the transport of gas molecules as a diffusion process, this approach is too limiting for neutron transport. In contrast to a gas, where collisions are very frequent, the cross sections for the interaction of neutrons with nuclei are relatively small, as we saw in chapter 1 (of the order of barns, i.e., 10-24cm2) . This implies that neutrons traverse appreciable distances (of the order of a centimetre) between collisions. This relatively long neutron mean free path, together with the heterogeneity of the physical medium, requires that a more complete treatment be carried out, taking account of variations in the angular distribution of neutron speed in the vicinity of highly absorbing
Notes on Stefan-Maxwell Equation versus Grahan's Diffusion Law
Institute of Scientific and Technical Information of China (English)
无
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.
Some Remarks on Some Strongly Coupled Reaction-Diffusion Equations
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.
A Fluctuating Lattice Boltzmann Method for the Diffusion Equation
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.
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...
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.
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.
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.
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.
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.
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.
Exact solutions for logistic reaction-diffusion equations in biology
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.
Linear fractional diffusion-wave equation for scientists and engineers
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 ...
Lattice Boltzmann method for the fractional advection-diffusion equation
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.
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.
Stochastic differential equations and diffusion processes
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
Nonlinear delay reaction-diffusion equations: exact traveling wave solutions
International Nuclear Information System (INIS)
We consider nonlinear delay reaction-diffusion equations. We present a number of exact traveling wave solutions, which can be represented in terms of elementary functions. We consider equations with quadratic, power, exponential, and logarithmic nonlinearities, as well as more complex equations with the kinetic function depending on one or two arbitrary functions of a single argument. All of the solutions obtained involve several free parameters and can be used to test approximate analytical and numerical methods for solving similar and more complex nonlinear differential-difference equations
Ho, C. -L.; Lee, C.-C.
2015-01-01
We consider solvability of the generalized reaction-diffusion equation with both space- and time-dependent diffusion and reaction terms by means of the similarity method. By introducing the similarity variable, the reaction-diffusion equation is reduced to an ordinary differential equation. Matching the resulting ordinary differential equation with known exactly solvable equations, one can obtain corresponding exactly solvable reaction-diffusion systems. Several representative examples of exa...
Unconditionally stable diffusion-acceleration of the transport equation
International Nuclear Information System (INIS)
The standard iterative procedure for solving fixed-source discrete-ordinates problems converges very slowly for problems in optically large regions with scattering ratios c near unity. The diffusion-synthetic acceleration method has been proposed to make use of the fact that for this class of problems the diffusion equation is often an accurate approximation to the transport equation. However, stability difficulties have historically hampered the implementation of this method for general transport differencing schemes. In this article we discuss a recently developed procedure for obtaining unconditionally stable diffusion-synthetic acceleration methods for various transport differencing schemes. We motivate the analysis by first discussing the exact transport equation; then we illustrate the procedure by deriving a new stable acceleration method for the linear discontinuous transport differencing scheme. We also provide some numerical results
Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation
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.
Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation
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.
Reaction-diffusion equations with spatially distributed hysteresis
Gurevich, Pavel; Shamin, Roman; Tikhomirov, Sergey
2012-01-01
The paper deals with reaction-diffusion equations involving a hysteretic discontinuity in the source term, which is defined at each spatial point. In particular, such problems describe chemical reactions and biological processes in which diffusive and nondiffusive substances interact according to hysteresis law. We find sufficient conditions that guarantee the existence and uniqueness of solutions as well as their continuous dependence on initial data.
Traveling waves for a boundary reaction-diffusion equation
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).
Phasefield theory for fractional diffusion-reaction equations and applications
Imbert, Cyril
2009-01-01
This paper is concerned with diffusion-reaction equations where the classical diffusion term, such as the Laplacian operator, is replaced with a singular integral term, such as the fractional Laplacian operator. As far as the reaction term is concerned, we consider bistable non-linearities. After properly rescaling (in time and space) these integro-differential evolution equations, we show that the limits of their solutions as the scaling parameter goes to zero exhibit interfaces moving by anisotropic mean curvature. The singularity and the unbounded support of the potential at stake are both the novelty and the challenging difficulty of this work.
Continuity, the Bloch-Torrey equation, and Diffusion MRI
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...
The Nonclassical Diffusion Approximation to the Nonclassical Linear Boltzmann Equation
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.
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.
COUPLED CHEMOTAXIS FLUID MODEL
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.
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)
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
THE INTERIOR LAYER SOLUTION TO NONLOCAL REACTION DIFFUSION EQUATIONS
Institute of Scientific and Technical Information of China (English)
无
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.
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.
Gradient estimates for a nonlinear diffusion equation on complete manifolds
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-\
Diffusive Limits of the Master Equation in Inhomogeneous Media
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.
Numerical solution of a reaction-diffusion equation
International Nuclear Information System (INIS)
The purpose of the present work to continue the observations and the numerical experiences on a reaction-diffusion model, that is a simplified form of the neutronic flux equation. The model is parabolic, nonlinear, with Dirichlet boundary conditions. The purpose is to approximate non trivial solutions, asymptotically stables for t → ∞, that is solutions that tend to the elliptic problem, in the Lyapunov sense. It belongs to the so-called reaction-diffusion equations of semi linear kind, that is, linear equations in the heat operator and they have a nonlinear reaction function, in this case f (u, a, b) = u (a - b u), being u concentration, a and b parameters. The study of the incidence of these parameters take an interest to the neutronic flux physics. So that we search non trivial, positive and bounded solutions. The used algorithm is based on the concept of monotone and ordered sequences, and on the existence theorem of Amann and Sattinger. (author)
Kinetic equations for diffusion in the presence of entropic barriers.
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
Langevin equation with fluctuating diffusivity: A two-state model.
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
Langevin equation with fluctuating diffusivity: A two-state model
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.
On analytical solutions for the nonlinear diffusion equation
Directory of Open Access Journals (Sweden)
Ulrich Olivier Dangui-Mbani
2014-09-01
Full Text Available The nonlinear diffusion equation arises in many important areas of nonlinear problems of heat and mass transfer, biological systems and processes involving fluid flow and most of the known exact solutions turn out to be approximate solutions in the form of a series which is the exact solution in the closed form. The approximate results obtained by using Homotopy perturbation transform method (HPTM and have been compared with the exact solutions by using software “mathematica” to show the stability of the solutions of nonlinear equation. The comparisons indicate that there is a very good agreement between the HPTM solutions and exact solutions in terms of accuracy
Simple jumping process with memory: Transport equation and diffusion
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.
Reaction diffusion equation with spatio-temporal delay
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.
Algorithm refinement for stochastic partial differential equations I. linear diffusion
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.
Convergence of the iteration to solve the diffusion equation
International Nuclear Information System (INIS)
Linear boundary value problems of diffusion theory in a symmetric region are analyzed with the help of group theory. If the symmetries of the region commute with the operator of the equation to be solved, the symmetries generate a classification of the solution space. This technique is applied to study the numerical behavior of numerical methods. It is shown that approximate polynomials on the boundary and inside the region must be compatible to get stable results. Compatibility conditions are given. To avoid convergence problems, linear functions on the six faces of a regular hexagon would demand as high as sixth order approximation inside the hexagon. The predicted convergence problems have been observed in the neutron physics code VARIANT, which is a routinely used diffusion equation solver at ANL. (Author)
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.
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.
UPWIND SPLITTING SCHEME FOR CONVECTION-DIFFUSION EQUATIONS
Institute of Scientific and Technical Information of China (English)
无
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.
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)
CONVERGENCE OF MARKOV CHAIN APPROXIMATIONS TO STOCHASTIC REACTION DIFFUSION EQUATIONS
Kouritzin, Michael A.; Hongwei Long
2001-01-01
In the context of simulating the transport of a chemical or bacterial contaminant through a moving sheet of water, we extend a well-established method of approximating reaction-diffusion equations with Markov chains by allowing convection, certain Poisson measure driving sources and a larger class of reaction functions. Our alterations also feature dramatically slower Markov chain state change rates often yielding a ten to one-hundred-fold simulation speed increase over the previous version o...
International Nuclear Information System (INIS)
The three-dimensional generalized (self-adjoint) atmospheric diffusion equation can be solved analytically. Where the source strength is located at any where within the region of interest, and with arbitrary power- law function of wind speed and eddy diffusivities. The Greens method approach is utilized, where the Green's function is splitting into two factors, As a result of the hermeticity the solution of the diffusion equation can be expressed exact, as a multiple of source and the two factors of the Green's function, vertical dispersion factor, and cross wind dispersion factor. The two factors of the Green's function are derived analytically in which, the Green's of the diffusion equation is splitting into two pair of two-dimensional differential equations, by using: separation of variables, Bessel equation, and transform the dependent and independent variables, and the cosine transformation. All amenable cases are adopted such as: Neumann (total reflection), Dirichlet (total adsorption), and mixed boundary conditions. Gaussian plume model is extracted in all the previously cases
Reaction rates for a generalized reaction-diffusion master equation
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.
Analysis of a mixed space-time diffusion equation
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.
A mixed finite element method for nonlinear diffusion equations
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.
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.
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.
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.)
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.
Diffusive Wave Approximation to the Shallow Water Equations: Computational Approach
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.
Dynamic hysteresis modeling including skin effect using diffusion equation model
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.
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.
Multigrid solution of diffusion equations on distributed memory multiprocessor systems
International Nuclear Information System (INIS)
The subject is the solution of partial differential equations for simulation of the reactor core on high-performance computers. The parallelization and implementation of nodal multigrid diffusion algorithms on array and ring configurations of the DIRMU multiprocessor system is outlined. The particular iteration scheme employed in the nodal expansion method appears similarly efficient in serial and parallel environments. The combination of modern multi-level techniques with innovative hardware (vector-multiprocessor systems) provides powerful tools needed for real time simulation of physical systems. The parallel efficiencies range from 70 to 90%. The same performance is estimated for large problems on large multiprocessor systems being designed at present. (orig.)
Mimetic discretization of two-dimensional magnetic diffusion equations
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.
New Parallel Three-level Iterative Method for Diffusion Equation
Directory of Open Access Journals (Sweden)
Tinghuai Ma
2010-01-01
Full Text Available To solve the diffusion equation on parallel computers, we first derived an o(τ2+h6 order implicit finite difference method based on a class of alternating group explicit iterative method. Based on this method, we devised a new alternating group explicit iterative method. Moreover, the absolute stability and convergence of the New Alternating Group Explicit Iterative (N-AGEI method was proved. Finally, the numerical experiments were conducted to verify our method. Both the theoretical analysis and simulation results showed that our proposed difference format had satisfied stability error estimate and convergence.
Poincar\\'e's inequality and diffusive evolution equations
Bjorland, Clayton; Schonbek, Maria E.
2007-01-01
This paper addresses the question of change of decay rate from exponential to algebraic for diffusive evolution equations. We show how the behaviour of the spectrum of the Dirichlet Laplacian in the two cases yields the passage from exponential decay in bounded domains to algebraic decay or no decay at all in the case of unbounded domains. It is well known that such rates of decay exist: the purpose of this paper is to explain what makes the change in decay happen. We also discuss what kind o...
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.
Chaotic dynamics and diffusion in a piecewise linear equation
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.
Nonlinear diffusion equations as asymptotic limits of Cahn-Hilliard systems
Colli, Pierluigi; Fukao, Takeshi
2016-05-01
An asymptotic limit of a class of Cahn-Hilliard systems is investigated to obtain a general nonlinear diffusion equation. The target diffusion equation may reproduce a number of well-known model equations: Stefan problem, porous media equation, Hele-Shaw profile, nonlinear diffusion of singular logarithmic type, nonlinear diffusion of Penrose-Fife type, fast diffusion equation and so on. Namely, by setting the suitable potential of the Cahn-Hilliard systems, all these problems can be obtained as limits of the Cahn-Hilliard related problems. Convergence results and error estimates are proved.
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.
On the supercritically diffusive magneto-geostrophic equations
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...
Travelling Waves in Hybrid Chemotaxis Models
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.
An inherently parallel method for solving discretized diffusion equations
International Nuclear Information System (INIS)
A Monte Carlo approach to solving linear systems of equations is being investigated in the context of the solution of discretized diffusion equations. While the technique was originally devised decades ago, changes in computer architectures (namely, massively parallel machines) have driven the authors to revisit this technique. There are a number of potential advantages to this approach: (1) Analog Monte Carlo techniques are inherently parallel; this is not necessarily true to today's more advanced linear equation solvers (multigrid, conjugate gradient, etc.); (2) Some forms of this technique are adaptive in that they allow the user to specify locations in the problem where resolution is of particular importance and to concentrate the work at those locations; and (3) These techniques permit the solution of very large systems of equations in that matrix elements need not be stored. The user could trade calculational speed for storage if elements of the matrix are calculated on the fly. The goal of this study is to compare the parallel performance of Monte Carlo linear solvers to that of a more traditional parallelized linear solver. The authors observe the linear speedup that they expect from the Monte Carlo algorithm, given that there is no domain decomposition to cause significant communication overhead. Overall, PETSc outperforms the Monte Carlo solver for the test problem. The PETSc parallel performance improves with larger numbers of unknowns for a given number of processors. Parallel performance of the Monte Carlo technique is independent of the size of the matrix and the number of processes. They are investigating modifications to the scheme to accommodate matrix problems with positive off-diagonal elements. They are also currently coding an on-the-fly version of the algorithm to investigate the solution of very large linear systems
Diffusive Boltzmann equation, its fluid dynamics, Couette flow and Knudsen layers
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...
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.
Guiding brine shrimp through mazes by solving reaction diffusion equations
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.
Local multiplicative Schwarz algorithms for convection-diffusion equations
Cai, Xiao-Chuan; Sarkis, Marcus
1995-01-01
We develop a new class of overlapping Schwarz type algorithms for solving scalar convection-diffusion equations discretized by finite element or finite difference methods. The preconditioners consist of two components, namely, the usual two-level additive Schwarz preconditioner and the sum of some quadratic terms constructed by using products of ordered neighboring subdomain preconditioners. The ordering of the subdomain preconditioners is determined by considering the direction of the flow. We prove that the algorithms are optimal in the sense that the convergence rates are independent of the mesh size, as well as the number of subdomains. We show by numerical examples that the new algorithms are less sensitive to the direction of the flow than either the classical multiplicative Schwarz algorithms, and converge faster than the additive Schwarz algorithms. Thus, the new algorithms are more suitable for fluid flow applications than the classical additive or multiplicative Schwarz algorithms.
From baking a cake to solving the diffusion equation
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.
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)
Approximate Lie group analysis and solutions of 2D nonlinear diffusion-convection equations
International Nuclear Information System (INIS)
Approximate Lie symmetries of the (2+1)-dimensional nonlinear diffusion equation with a small convection are completely classified. It is known that the invariance principle furnishes a systematic method of solving initial-value problems. The solutions of instantaneous source type of the 2D diffusion-convection equation are obtained for the case of power-law diffusivity, using a symmetry reduction
On the uniqueness of semi-wavefronts for non-local delayed reaction-diffusion equations
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.
Diffusion-equation representations of landform evolution in the simplest circumstances: Appendix C
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.
Luchko, Yuri; Povstenko, Yuriy
2012-01-01
In this paper, the one-dimensional time-fractional diffusion-wave equation with the fractional derivative of order $1 \\le \\alpha \\le 2$ is revisited. This equation interpolates between the diffusion and the wave equations that behave quite differently regarding their response to a localized disturbance: whereas the diffusion equation describes a process, where a disturbance spreads infinitely fast, the propagation speed of the disturbance is a constant for the wave equation. For the time fractional diffusion-wave equation, the propagation speed of a disturbance is infinite, but its fundamental solution possesses a maximum that disperses with a finite speed. In this paper, the fundamental solution of the Cauchy problem for the time-fractional diffusion-wave equation, its maximum location, maximum value, and other important characteristics are investigated in detail. To illustrate analytical formulas, results of numerical calculations and plots are presented. Numerical algorithms and programs used to produce pl...
Travelling waves in hybrid chemotaxis models
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...
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.
Group Analysis of Variable Coefficient Diffusion-Convection Equations. IV. Potential Symmetries
Ivanova, N M; Sophocleous, C
2007-01-01
This paper completes investigation of symmetry properties of nonlinear variable coefficient diffusion-convection equations of the form $f(x)u_t=(g(x)A(u)u_x)_x+h(x)B(u)u_x$. Potential symmetries of equations from the considered class are found and the connection of them with Lie symmetries of diffusion-type equations is shown. Exact solutions of the Fujita--Storm equation $u_t=(u^{-2}u_x)_x$ are constructed.
International Nuclear Information System (INIS)
The stochastic variational method is applied to particle systems and continuum media. As a brief review of this method, we first discuss the application to particle Lagrangians and derive a diffusion-type equation and the Schrödinger equation with the minimum gauge coupling. We further extend the application of the stochastic variational method to Lagrangians of continuum media and show that the Navier–Stokes, Gross–Pitaevskii and generalized diffusion equations are derived. The correction term for the Navier–Stokes equation is also obtained in this method. We discuss the meaning of this correction by comparing it with the diffusion equation. (paper)
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...
The constructive technique and its application in solving a nonlinear reaction diffusion equation
International Nuclear Information System (INIS)
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. (general)
Parallel computing for homogeneous diffusion and transport equations in neutronics
International Nuclear Information System (INIS)
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)
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)
Regularized lattice Boltzmann model for a class of convection-diffusion equations.
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
Diffusion in periodic potential Langevin versus Fokker-Planck equation approach
International Nuclear Information System (INIS)
Phonon activated diffusion of an interstitial impurity in one dimensional cosine potential is discussed with the use both of Langevin (continuous diffusion model) and Fokker-Planck (jump-diffusion model) equations. Jump rate, jump length and diffusion coefficient as a function of temperature at various barrier heights are calculated. There is some difference between results provided by these two models. Therefore the question arises to what extend these two models of diffusion are equivalent. (author)
Macroscopic dynamics of biological cells interacting via chemotaxis and direct contact
Lushnikov, Pavel M; Alber, Mark
2008-01-01
A connection is established between discrete stochastic model describing microscopic motion of fluctuating cells, and macroscopic equations describing dynamics of cellular density. Cells move towards chemical gradient (process called chemotaxis) with their shapes randomly fluctuating. Nonlinear diffusion equation is derived from microscopic dynamics in dimensions one and two using excluded volume approach. Nonlinear diffusion coefficient depends on cellular volume fraction and it is demonstrated to prevent collapse of cellular density. A very good agreement is shown between Monte Carlo simulations of the microscopic Cellular Potts Model and numerical solutions of the macroscopic equations for relatively large cellular volume fractions. Combination of microscopic and macroscopic models were used to simulate growth of structures similar to early vascular networks.
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...
Innovation diffusion equations on correlated scale-free networks
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...
New Solution of a Fractional Diffusion-Advection Equation Using Ultradistributions
Rocca, M C; Plastino, A R; De Paoli, A L
2014-01-01
In this paper we exactly solve the fractional diffusion-advection equation. For this purpose we use the Theoy of Ultradistributions of J. Sebastiao e Silva, to give a general solution for this equation. From this solution, we obtain several approximations as limiting cases of various situations of physical and astrophysical interest. One of them involves cosmic rays' diffusion.
A Fractional Diffusion Equation for an n-Dimensional Correlated Levy Walk
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.
Modeling bacterial chemotaxis inside a cell
Ouannes, Nesrine; Djedi, Noureddine; Luga, Hervé; Duthen, Yves
2014-01-01
This paper describes a bacterial system that reproduces a population of bacteria that behave by simulating the internal reactions of each bacterial cell. The chemotaxis network of a cell is modulated by a hybrid approach that uses an algebraic model for the receptor clusters activity and an ordinary differential equation for the adaptation dynamics. The experiments are defined in order to simulate bacterial growth in an environment where nutrients are regularly added to it. The results show a...
A generalized Fokker–Planck equation for anomalous diffusion in velocity space
International Nuclear Information System (INIS)
A more general quasi-Fokker–Planck equation is derived to describe particle kinetics in situations when the usual Fokker–Planck equation is not applicable. The equation is valid for an arbitrary value of the transferred in a collision act momentum and for the arbitrary mass ratio of the interacting particles. The only assumption is smallness of the typical velocity of the particles, undergoing diffusion. The developed approach is another tool to study anomalous diffusion, avoiding conventional in such problems fractional differentiation. In this Letter anomalous diffusion in velocity space is considered for hard-sphere, Coulomb and dusty plasma collision models. -- Highlights: ► We expand the Master equation into a series. ► We derive a generalized equation for description of anomalous diffusion. ► We consider special cases of anomalous diffusion for hard-sphere interactions, Coulomb systems and dusty plasmas.
Coupled Oscillators with Chemotaxis
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.
Air Pollution Steady-State Advection-Diffusion Equation: The General Three-Dimensional Solution
Bardo Bodmann; Tiziano Tirabassi; Marco Túllio Vilhena; Daniela Buske
2012-01-01
Atmospheric air pollution turbulent fluxes can be assumed to be proportional to the mean concentration gradient. This assumption, along with the equation of continuity, leads to the advection-diffusion equation. Many models simulating air pollution dispersion are based upon the solution (numerical or analytical) of the advection-diffusion equation as- suming turbulence parameterization for realistic physical scenarios. We present the general steady three-dimensional solution of the advection-...
Koide, T.; Kodama, T.
2011-01-01
The stochastic variational method is applied to particle systems and continuum mediums. As the brief review of this method, we first discuss the application to particle Lagrangians and derive a diffusion-type equation and the Schr\\"{o}dinger equation with the minimum gauge coupling. We further extend the application of the stochastic variational method to Lagrangians of continuum mediums and show that the Navier-Stokes, Gross-Pitaevskii and generalized diffusion equations are derived. The cor...
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.
Perfect and near perfect adaptation in a model of bacterial chemotaxis
Mello, Bernardo A.; Tu, Yuhai
2002-01-01
The signaling apparatus mediating bacterial chemotaxis can adapt to a wide range of persistent external stimuli. In many cases, the bacterial activity returns to its pre-stimulus level exactly and this "perfect adaptability" is robust against variations in various chemotaxis protein concentrations. We model the bacterial chemotaxis signaling pathway, from ligand binding to CheY phosphorylation. By solving the steady-state equations of the model analytically, we derive a full set of conditions...
International Nuclear Information System (INIS)
In this work we report an analytical solution for the monoenergetic neutron diffusion kinetic equation in cartesian geometry. Bearing in mind that the equation for the delayed neutron precursor concentration is a first order linear differential equation in the time variable, to make possible the application of the GITT approach to the kinetic equation, we introduce a fictitious diffusion term multiplied by a positive small value ε. By this procedure, we are able to solve this set of equations. Indeed, applying the GITT technique to the modified diffusion kinetic equation, we come out with a matrix differential equation which has a well known analytical solution when ε goes to zero. We report numerical simulations as well study of numerical convergence of the results attained. (author)
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.
Darboux transformations for (1+2)-dimensional Fokker-Planck equations with constant diffusion matrix
International Nuclear Information System (INIS)
We construct a Darboux transformation for (1+2)-dimensional Fokker-Planck equations with constant diffusion matrix. Our transformation is based on the two-dimensional supersymmetry formalism for the Schrödinger equation. The transformed Fokker-Planck equation and its solutions are obtained in explicit form.
The second boundary value problem for equations of viscoelastic diffusion in polymers
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...
On the integrability of the generalized Fisher-type nonlinear diffusion equations
International Nuclear Information System (INIS)
In this paper, the geometric integrability and Lax integrability of the generalized Fisher-type nonlinear diffusion equations with modified diffusion in (1+1) and (2+1) dimensions are studied by the pseudo-spherical surface geometry method and prolongation technique. It is shown that the (1+1)-dimensional Fisher-type nonlinear diffusion equation is geometrically integrable in the sense of describing a pseudo-spherical surface of constant curvature -1 only for m = 2, and the generalized Fisher-type nonlinear diffusion equations in (1+1) and (2+1) dimensions are Lax integrable only for m = 2. This paper extends the results in Bindu et al 2001 (J. Phys. A: Math. Gen. 34 L689) and further provides the integrability information of (1+1)- and (2+1)-dimensional Fisher-type nonlinear diffusion equations for m = 2
Finite element approximation of the Fokker-Planck equation for diffuse optical tomography
International Nuclear Information System (INIS)
In diffuse optical tomography, light transport theory is used to describe photon propagation inside turbid medium. A commonly used simplification for the radiative transport equation is the diffusion approximation due to computational feasibility. However, it is known that the diffusion approximation is not valid close to the sources and boundary and in low-scattering regions. Fokker-Planck equation describes light propagation when scattering is forward-peaked. In this article a numerical solution of the Fokker-Planck equation using finite element method is developed. Approach is validated against Monte Carlo simulation and compared with the diffusion approximation. The results show that the Fokker-Planck equation gives equal or better results than the diffusion approximation on the boundary of a homogeneous medium and in turbid medium containing a low-scattering region when scattering is forward-peaked.
Transformed Fourier and Fick equations for the control of heat and mass diffusion
International Nuclear Information System (INIS)
We review recent advances in the control of diffusion processes in thermodynamics and life sciences through geometric transforms in the Fourier and Fick equations, which govern heat and mass diffusion, respectively. We propose to further encompass transport properties in the transformed equations, whereby the temperature is governed by a three-dimensional, time-dependent, anisotropic heterogeneous convection-diffusion equation, which is a parabolic partial differential equation combining the diffusion equation and the advection equation. We perform two dimensional finite element computations for cloaks, concentrators and rotators of a complex shape in the transient regime. We precise that in contrast to invisibility cloaks for waves, the temperature (or mass concentration) inside a diffusion cloak crucially depends upon time, its distance from the source, and the diffusivity of the invisibility region. However, heat (or mass) diffusion outside cloaks, concentrators and rotators is unaffected by their presence, whatever their shape or position. Finally, we propose simplified designs of layered cylindrical and spherical diffusion cloaks that might foster experimental efforts in thermal and biochemical metamaterials
A semi-analytical finite element method for a class of time-fractional diffusion equations
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...
Innovation diffusion equations on correlated scale-free networks
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.
Innovation diffusion equations on correlated scale-free networks
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.
Transition probabilities for diffusion equations by means of path integrals
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...
Transition probabilities for diffusion equations by means of path integrals.
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...
Application of The Full-Sweep AOR Iteration Concept for Space-Fractional Diffusion Equation
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.
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....
Non-probabilistic solutions of imprecisely defined fractional-order diffusion equations
International Nuclear Information System (INIS)
The fractional diffusion equation is one of the most important partial differential equations (PDEs) to model problems in mathematical physics. These PDEs are more practical when those are combined with uncertainties. Accordingly, this paper investigates the numerical solution of a non-probabilistic viz. fuzzy fractional-order diffusion equation subjected to various external forces. A fuzzy diffusion equation having fractional order 0 < α ≤ 1 with fuzzy initial condition is taken into consideration. Fuzziness appearing in the initial conditions is modelled through convex normalized triangular and Gaussian fuzzy numbers. A new computational technique is proposed based on double parametric form of fuzzy numbers to handle the fuzzy fractional diffusion equation. Using the single parametric form of fuzzy numbers, the original fuzzy diffusion equation is converted first into an interval-based fuzzy differential equation. Next, this equation is transformed into crisp form by using the proposed double parametric form of fuzzy numbers. Finally, the same is solved by Adomian decomposition method (ADM) symbolically to obtain the uncertain bounds of the solution. Computed results are depicted in terms of plots. Results obtained by the proposed method are compared with the existing results in special cases. (general)
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.
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.
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.
THE CORNER LAYER SOLUTION TO ROBIN PROBLEM FOR REACTION DIFFUSION EQUATION
Institute of Scientific and Technical Information of China (English)
无
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.
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.
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.
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
Mixed, Nonsplit, Extended Stability, Stiff Integration of Reaction Diffusion Equations
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.
A modified diffusion equation for room-acoustic predication.
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
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.
Non-negative mixed finite element formulations for a tensorial diffusion equation
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 ...
On the stability of some exact solutions to the generalized convection-reaction-diffusion equation
International Nuclear Information System (INIS)
Highlights: → We investigate stability properties of a set of traveling wave solutions to the convection-reaction-diffusion equation taking into account the effects of memory. → We examine the analytical solution previously obtained within the direct method. → Our analytical studies are backed by the numerical simulations. - Abstract: Stability of a set of traveling wave solutions to the convection-reaction-diffusion equation taking into account the effects of memory is studied by means of the qualitative methods and numerical simulation.
Numerical methods for advection-diffusion-reaction equations and medical applications
Montecinos, Gino Ignacio
2014-01-01
The purpose of this thesis is twofold, firstly, the study of a relaxation procedure for numerically solving advection-diffusion-reaction equations, and secondly, a medical application. Concerning the first topic, we extend the applicability of the Cattaneo relaxation approach to reformulate time-dependent advection-diffusion-reaction equations, that may include stiff reactive terms, as hyperbolic balance laws with stiff source terms. The resulting systems of hyperbolic balance laws are solved...
Sound Field Modeling in Architectural Acoustics using a Diffusion Equation Based Model
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...
Sound Field Modeling in Architectural Acoustics using a Diffusion Equation Based Model
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...
Painleve analysis and exact solutions for the predator-prey system of equations with diffusion
International Nuclear Information System (INIS)
A system of reaction-diffusion equations describing predator-prey relations is considered. The mathematical model corresponds to the modified Lotka-Volterra system with the logistic growth of the prey and with both predator and prey dispersing by diffusion. The Painleve analysis of the system of equations is performed. Exact traveling wave solutions are found by means of the logistic-function method
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...
International Nuclear Information System (INIS)
Highlights: • Uncertain neutron diffusion equation of bare square homogeneous reactor is studied. • Proposed interval arithmetic is extended for fuzzy numbers. • The developed fuzzy arithmetic is used to handle uncertain parameters. • Governing differential equation is modelled by modified fuzzy finite element method. • Fuzzy critical eigenvalues and effective multiplication factors are investigated. - Abstract: The scattering of neutron collision inside a reactor depends upon geometry of the reactor, diffusion coefficient and absorption coefficient etc. In general these parameters are not crisp and hence we get uncertain neutron diffusion equation. In this paper we have investigated the above equation for a bare square homogeneous reactor. Here the uncertain governing differential equation is modelled by a modified fuzzy finite element method. Using modified fuzzy finite element method, obtained eigenvalues and effective multiplication factors are studied. Corresponding results are compared with the classical finite element method in special cases and various uncertain results have been discussed
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
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.
A discrete-ordinate discontinuous-streamline diffusion method for the radiative transfer equation
Wang, Cheng; Sheng, Qiwei; Han, Weimin
2016-01-01
The radiative transfer equation (RTE) arises in many different areas of science and engineering. In this paper, we propose and investigate a discrete-ordinate discontinuous-streamline diffusion (DODSD) method for solving the RTE, which is a combination of the discrete-ordinate technique and the discontinuous-streamline diffusion method. Different from the discrete-ordinate discontinuous Galerkin (DODG) method for the RTE, an artificial diffusion parameter is added to the test functions in the...
An equation describing diffusivity of liquid atoms by magnetic confinement
International Nuclear Information System (INIS)
In this work, we report an obvious low field-induced magnetic confinement effect on the diffusivity in binary metallic melts under a weak magnetic field. A quantitative description of this nontrivial dynamic behavior is given by a physical analytical model based on the Hall effect, which is in agreement with our experimental results. Meanwhile, a quadratic B dependence of the dynamic viscosity obtained in the same confined environment is observed. Our results show that one can effectively control the atomic diffusion process of metallic melts by the application of magnetic field. Meanwhile, this magnetic confinement effect at atomic scale should provide an important new ingredient to deeply understand the condensed matter physics under the external magnetic field. (paper)
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.
L1 Error Estimates for Difference Approximations Of Degenerate Convection-Diffusion Equations
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.
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.
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.
Strang-type preconditioners for solving fractional diffusion equations by boundary value methods
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
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)
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.
A class of Fokker-Planck equations with logarithmic factors in diffusion and drift terms
International Nuclear Information System (INIS)
The x2 ln (x) dependence of the diffusion coefficient in the Fokker-Planck equation is retrieved by means of symmetry arguments. Exact solutions of the equation with logarithmic factors in coefficients are presented. Algebraic and log-algebraic solutions are found. For some values of exponents they seem to be analogues (on an interval) of log-normal distributions. (author)
Ivanova, N M; Sophocleous, C
2007-01-01
This is the second part of the series of papers on symmetry properties of a class of variable coefficient (1+1)-dimensional nonlinear diffusion-convection equations of general form $f(x)u_t=(g(x)A(u)u_x)_x+h(x)B(u)u_x$. At first, we review the results of Part 1 of the series on equivalence transformations and group classification of the class under consideration. Investigation of non-trivial limits of parameterized subclasses of equations from the given class, which generate contractions of the corresponding maximal Lie invariance algebras, leads to the natural notion of contractions of systems of differential equations. After a brief discussion on contractions of symmetries, equations and solutions in general case, such types of contractions are studied for diffusion--convection equations. A detailed symmetry analysis of an interesting equation from the class under consideration is performed. Exact solutions of some subclasses of the considered class are also given.
Reaction-diffusion equation for quark-hadron transition in heavy-ion collisions
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...
Fractional Fokker-Planck Equation and Black-Scholes Formula in Composite-Diffusive Regime
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.
International Nuclear Information System (INIS)
A robust numerical solution of the nonlinear Poisson–Boltzmann equation for asymmetric polyelectrolyte solutions in discrete pore geometries is presented. Comparisons to the linearized approximation of the Poisson–Boltzmann equation reveal that the assumptions leading to linearization may not be appropriate for the electrochemical regime in many cementitious materials. Implications of the electric double layer on both partitioning of species and on diffusive release are discussed. The influence of the electric double layer on anion diffusion relative to cation diffusion is examined.
Lattice fractional diffusion equation in terms of a Riesz-Caputo difference
Wu, Guo-Cheng; Baleanu, Dumitru; Deng, Zhen-Guo; Zeng, Sheng-Da
2015-11-01
A fractional difference is defined by the use of the right and the left Caputo fractional differences. The definition is a two-sided operator of Riesz type and introduces back and forward memory effects in space difference. Then, a fractional difference equation method is suggested for anomalous diffusion in discrete finite domains. A lattice fractional diffusion equation is proposed and the numerical simulation of the diffusion process is discussed for various difference orders. The result shows that the Riesz difference model is particularly suitable for modeling complicated dynamical behaviors on discrete media.
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)
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.
Enhanced Group Analysis of Variable Coefficient Semilinear Diffusion Equations with a Power Source
Vaneeva, O O; Sophocleous, C
2007-01-01
A new approach to group classification problems and more general investigations on transformational properties of classes of differential equations are proposed. It is based on mappings between classes of differential equations, generated by families of point transformations. A class of variable coefficient (1+1)-dimensional semilinear reaction--diffusion equations of the general form $f(x)u_t=(g(x)u_x)_x+h(x)u^m$ ($m\
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 ...
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...
Multi-dimensional traveling fronts in bistable reaction-diffusion equations
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...
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.
Utilization of Weibull equation to obtain soil-water diffusivity in horizontal infiltration
International Nuclear Information System (INIS)
Water movement was studied in horizontal infiltration experiments using laboratory columns of air-dry and homogeneous soil to obtain a simple and suitable equation for soil-water diffusivity. Many water content profiles for each one of the ten soil columns utilized were obtained through gamma-ray attenuation technique using a 137Cs source. During the measurement of a particular water content profile, the soil column was held in the same position in order to measure changes in time and so to reduce the errors in water content determination. The Weibull equation utilized was excellent in fitting water content profiles experimental data. The use of an analytical function for ν, the Boltzmann variable, according to Weibull model, allowed to obtain a simple equation for soil water diffusivity. Comparisons among the equation here obtained for diffusivity and others solutions found in literature were made, and the unsuitability of a simple exponential variation of diffusivity with water content for the full range of the latter was shown. The necessity of admitting the time dependency for diffusivity was confirmed and also the possibility fixing that dependency on a well known value extended to generalized soil water infiltration studies was found. Finally, it was shown that the soil water diffusivity function given by the equation here proposed can be obtained just by the analysis of the wetting front advance as a function of time. (Author)
Vaneeva, O O; Sophocleous, C
2011-01-01
The group classification of a class of variable coefficient reaction-diffusion equations with exponential nonlinearities is carried out up to both the equivalence generated by the corresponding generalized equivalence group and the general point equivalence. The set of admissible transformations of this class is exhaustively described via finding the complete family of maximal normalized subclasses and associated conditional equivalence groups. Limit processes between variable coefficient reaction-diffusion equations with power nonlinearities and those with exponential nonlinearities are simultaneously studied with limit processes between objects related to these equations (including Lie symmetries, exact solutions and conservation laws).
Fractional diffusion equation for an n -dimensional correlated Lévy walk
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.
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.
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....
Energy Technology Data Exchange (ETDEWEB)
Fa, Kwok Sau, E-mail: kwok@dfi.uem.br
2015-02-15
An integro-differential diffusion equation with linear force, based on the continuous time random walk model, is considered. The equation generalizes the ordinary and fractional diffusion equations, which includes short, intermediate and long-time memory effects described by the waiting time probability density function. Analytical expression for the correlation function is obtained and analyzed, which can be used to describe, for instance, internal motions of proteins. The result shows that the generalized diffusion equation has a broad application and it may be used to describe different kinds of systems. - Highlights: • Calculation of the correlation function. • The correlation function is connected to the survival probability. • The model can be applied to the internal dynamics of proteins.
International Nuclear Information System (INIS)
An integro-differential diffusion equation with linear force, based on the continuous time random walk model, is considered. The equation generalizes the ordinary and fractional diffusion equations, which includes short, intermediate and long-time memory effects described by the waiting time probability density function. Analytical expression for the correlation function is obtained and analyzed, which can be used to describe, for instance, internal motions of proteins. The result shows that the generalized diffusion equation has a broad application and it may be used to describe different kinds of systems. - Highlights: • Calculation of the correlation function. • The correlation function is connected to the survival probability. • The model can be applied to the internal dynamics of proteins
DEFF Research Database (Denmark)
Møller, Jan Kloppenborg; Madsen, Henrik
Lamperti transform. This note gives an example driven introduction to the Lamperti transform. The general applicability of the Lamperti transform is limited to univariate diffusion processes, but for a restricted class of multivariate diffusion processes Lamperti type transformations are available and the...
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...
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)
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.
A nonlinear equation for ionic diffusion in a strong binary electrolyte
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...
Simulate-HEX - The multi-group diffusion equation in hexagonal-z geometry
International Nuclear Information System (INIS)
The multigroup diffusion equation is solved for the hexagonal-z geometry by dividing each hexagon into 6 triangles. In each triangle, the Fourier solution of the wave equation is approximated by 8 plane waves to describe the intra-nodal flux accurately. In the end an efficient Finite Difference like equation is obtained. The coefficients of this equation depend on the flux solution itself and they are updated once per power/void iteration. A numerical example demonstrates the high accuracy of the method. (authors)
Non-Fickian delay reaction-diffusion equations: theoretical and numerical study
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...
Conservation Laws of a Family of Reaction-Diffusion-Convection Equations
Bruzón, M. S.; Gandarias, M. L.; de la Rosa, R.
Ibragimov introduced the concept of nonlinear self-adjoint equations. This definition generalizes the concept of self-adjoint and quasi-self-adjoint equations. Gandarias defined the concept of weak self-adjoint. In this paper, we found a class of nonlinear self-adjoint nonlinear reaction-diffusion-convection equations which are neither self-adjoint nor quasi-self-adjoint nor weak self-adjoint. From a general theorem on conservation laws proved by Ibragimov we obtain conservation laws for these equations.
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.
An Efficient Implicit FEM Scheme for Fractional-in-Space Reaction-Diffusion Equations
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.
Diffusive approximation of a time-fractional Burgers equation in nonlinear acoustics
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...
Second order time evolution of the multigroup diffusion and P1 equations for radiation transport
International Nuclear Information System (INIS)
Highlights: → An existing multigroup transport algorithm is extended to be second-order in time. → A new algorithm is presented that does not require a grey acceleration solution. → The two algorithms are tested with 2D, multi-material problems. → The two algorithms have comparable computational requirements. - Abstract: An existing solution method for solving the multigroup radiation equations, linear multifrequency-grey acceleration, is here extended to be second order in time. This method works for simple diffusion and for flux-limited diffusion, with or without material conduction. A new method is developed that does not require the solution of an averaged grey transport equation. It is effective solving both the diffusion and P1 forms of the transport equation. Two dimensional, multi-material test problems are used to compare the solution methods.
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.
Biomixing by chemotaxis and efficiency of biological reactions: the critical reaction case
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...
A Bloch-Torrey Equation for Diffusion in a Deforming Media
International Nuclear Information System (INIS)
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 is applied with success to the static organs such as brain. However, the diffusion measurement on the dynamically deformable organs such as the in-vivo heart is a complex problem that has however a great potential in the measurement of cardiac health. In order to understand the behavior of the Magnetic Resonance (MR)signal in a deforming media, the Bloch-Torrey equation that leads the MR behavior is expressed in general curvilinear coordinates. These coordinates enable to follow the heart geometry and deformations through time. The equation is finally discredited and presented in a numerical formulation using implicit methods, in order to get a stable scheme that can be applied to any smooth deformations. Diffusion process enables the link between the macroscopic behavior of molecules and the microscopic structure in which they evolve. The measurement of diffusion in biological tissues is therefore of major importance in understanding the complex underlying structure that cannot be studied directly. The Diffusion Tensor Magnetic Resonance Imaging(DTMRI) technique enables the measurement of diffusion parameters and therefore provides information on the structure of the biological tissue. This technique has been applied with success to static organs such as the brain. However, diffusion measurement of dynamically deformable organs such as the in-vivo heart remains a complex problem, which holds great potential in determining cardiac health. In order to understand the behavior of the magnetic resonance (MR) signal in a deforming media, the Bloch-Torrey equation that defines the MR behavior is expressed in general curvilinear coordinates. These coordinates enable us to follow the heart geometry and deformations through time. The equation is finally discredited and presented in a
Non-local reaction-diffusion equations modelling predator-prey coevolution
Calsina, Ángel; Calsina i Ballesta, Àngel; Perelló Valls, Carles
1994-01-01
In this paper we examine a prey-predator system with a characteristic of the predator subject to mutation. The ultimate equilibrium of the system is found by Maynard-Smith et al. by the so called ESS (Evolutionary Stable Strategy). Using a system of reaction-diffusion equations with non local terms, we conclude the ESS result for the diffusion coefficient tending to zero, without resorting to any optimization criterion.
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.
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.
An analytical solution for the two-group kinetic neutron diffusion equation in cylindrical geometry
International Nuclear Information System (INIS)
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)
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)
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)
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)
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
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
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.
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.
A moving mesh finite difference method for equilibrium radiation diffusion equations
International Nuclear Information System (INIS)
An efficient moving mesh finite difference method is developed for the numerical solution of equilibrium radiation diffusion equations in two dimensions. The method is based on the moving mesh partial differential equation approach and moves the mesh continuously in time using a system of meshing partial differential equations. The mesh adaptation is controlled through a Hessian-based monitor function and the so-called equidistribution and alignment principles. Several challenging issues in the numerical solution are addressed. Particularly, the radiation diffusion coefficient depends on the energy density highly nonlinearly. This nonlinearity is treated using a predictor–corrector and lagged diffusion strategy. Moreover, the nonnegativity of the energy density is maintained using a cutoff method which has been known in literature to retain the accuracy and convergence order of finite difference approximation for parabolic equations. Numerical examples with multi-material, multiple spot concentration situations are presented. Numerical results show that the method works well for radiation diffusion equations and can produce numerical solutions of good accuracy. It is also shown that a two-level mesh movement strategy can significantly improve the efficiency of the computation
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.
A moving mesh finite difference method for equilibrium radiation diffusion equations
Yang, Xiaobo; Huang, Weizhang; Qiu, Jianxian
2015-10-01
An efficient moving mesh finite difference method is developed for the numerical solution of equilibrium radiation diffusion equations in two dimensions. The method is based on the moving mesh partial differential equation approach and moves the mesh continuously in time using a system of meshing partial differential equations. The mesh adaptation is controlled through a Hessian-based monitor function and the so-called equidistribution and alignment principles. Several challenging issues in the numerical solution are addressed. Particularly, the radiation diffusion coefficient depends on the energy density highly nonlinearly. This nonlinearity is treated using a predictor-corrector and lagged diffusion strategy. Moreover, the nonnegativity of the energy density is maintained using a cutoff method which has been known in literature to retain the accuracy and convergence order of finite difference approximation for parabolic equations. Numerical examples with multi-material, multiple spot concentration situations are presented. Numerical results show that the method works well for radiation diffusion equations and can produce numerical solutions of good accuracy. It is also shown that a two-level mesh movement strategy can significantly improve the efficiency of the computation.
A moving mesh finite difference method for equilibrium radiation diffusion equations
Energy Technology Data Exchange (ETDEWEB)
Yang, Xiaobo, E-mail: xwindyb@126.com [Department of Mathematics, College of Science, China University of Mining and Technology, Xuzhou, Jiangsu 221116 (China); Huang, Weizhang, E-mail: whuang@ku.edu [Department of Mathematics, University of Kansas, Lawrence, KS 66045 (United States); Qiu, Jianxian, E-mail: jxqiu@xmu.edu.cn [School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing, Xiamen University, Xiamen, Fujian 361005 (China)
2015-10-01
An efficient moving mesh finite difference method is developed for the numerical solution of equilibrium radiation diffusion equations in two dimensions. The method is based on the moving mesh partial differential equation approach and moves the mesh continuously in time using a system of meshing partial differential equations. The mesh adaptation is controlled through a Hessian-based monitor function and the so-called equidistribution and alignment principles. Several challenging issues in the numerical solution are addressed. Particularly, the radiation diffusion coefficient depends on the energy density highly nonlinearly. This nonlinearity is treated using a predictor–corrector and lagged diffusion strategy. Moreover, the nonnegativity of the energy density is maintained using a cutoff method which has been known in literature to retain the accuracy and convergence order of finite difference approximation for parabolic equations. Numerical examples with multi-material, multiple spot concentration situations are presented. Numerical results show that the method works well for radiation diffusion equations and can produce numerical solutions of good accuracy. It is also shown that a two-level mesh movement strategy can significantly improve the efficiency of the computation.
An Exact Formal Solution to Reaction-Diffusion Equations from Biomathematics
Institute of Scientific and Technical Information of China (English)
无
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.
Mixed time discontinuous space-time finite element method for convection diffusion equations
Institute of Scientific and Technical Information of China (English)
无
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.
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...
Nefedov, N. N.; Ni, Minkang
2015-12-01
A singularly perturbed boundary value problem for a second-order ordinary differential equation known in applications as a stationary reaction-diffusion equation is studied. A new class of problems is considered, namely, problems with nonlinearity having discontinuities localized in some domains, which leads to the formation of sharp transition layers in these domains. The existence of solutions with an internal transition layer is proved, and their asymptotic expansion is constructed.
L1 Error Estimates for Difference Approximations Of Degenerate Convection-Diffusion Equations
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.
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.
Strang-type preconditioners for solving fractional diffusion equations by boundary value methods
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...
Solving the time-fractional diffusion equation using a lie group integrator
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Hashemi Mir Sajjad
2015-01-01
Full Text Available In this paper, we propose a numerical method to approximate the solutions of time fractional diffusion equation which is in the class of Lie group integrators. Our utilized method, namely fictitious time integration method transforms the unknown dependent variable to a new variable with one dimension more. Then the group preserving scheme is used to integrate the new fractional partial differential equations in the augmented space R3+1. Effectiveness and validity of method demonstrated using two examples.
Portilheiro, Manuel; Tzavaras, Athanasios E.
2005-01-01
We consider a class of kinetic equations, equipped with a single conservation law, which generate L1-contractions. We discuss the hydrodynamic limit to a scalar conservation law and the diffusive limit to a (possibly) degenerate parabolic equation. The limits are obtained in the "dissipative" sense, equivalent to the notion of entropy solutions for conservation laws, which permits the use of the perturbed test function method and allows for simple proofs. A general compactness framework is ob...
International Nuclear Information System (INIS)
Starting from the radiation transport equation for homogeneous, refractive lossy media, we derive the corresponding time-dependent multifrequency diffusion equations. Zeroth and first moments of the transport equation couple the energy density, flux and pressure tensor. The system is closed by neglecting the temporal derivative of the flux and replacing the pressure tensor by its diagonal analogue. The radiation equations are coupled to a diffusion equation for the matter temperature. We are interested in modeling heating and cooling of silica (SiO2), at possibly rapid rates. Hence, in contrast to related work, we retain the temporal derivative of the radiation field. We derive boundary conditions at a planar air-silica interface taking account of reflectivities obtained from the Fresnel relations that include absorption. The spectral dimension is discretized into a finite number of intervals leading to a system of multigroup diffusion equations. Three simulations are presented. One models cooling of a silica slab, initially at 2500 K, for 10 s. The other two are 1D and 2D simulations of irradiating silica with a CO2 laser, λ = 10.59 μm. In 2D, a laser beam (Gaussian profile, r0 = 0.5 mm for 1/e decay) shines on a disk (radius = 0.4, thickness = 0.4 cm).
Diffusion synthetic acceleration methods for the diamond-differenced discrete-ordinates equations
International Nuclear Information System (INIS)
A class of acceleration schemes is investigated which resembles the conventional synthetic method in that they utilize the diffusion operator in the transport iteration schemes. The accelerated iteration involves alternate diffusion and transport solutions where coupling between the equations is achieved by using a correction term applied to either the diffusion coefficient, the removal cross section, or the source of the diffusion equation. The methods involving the modification of the diffusion coefficient and of the removal term yield nonlinear acceleration schemes and are used in k/sub eff/ calculations, while the source term modification approach is linear at least before discretization, and is used for inhomogeneous source problems. A careful analysis shows that there is a preferred differencing method which eliminates the previously observed instability of the conventional synthetic method. Use of this preferred difference scheme results in an acceleration method which is at the same time stable and efficient. This preferred difference approach renders the source correction scheme, which is linear in its continuous form, nonlinear in its differenced form. An additional feature of these approaches is that they may be used as schemes for obtaining improved diffusion solutions for approximately twice the cost of a diffusion calculation. Numerical experimentation on a wide range of problems in one and two dimensions indicates that improvement from a factor of two to ten over rebalance or Chebyshev acceleration is obtained. The improvement is most pronounced in problems with large regions of scattering material where the unaccelerated transport solutions converge very slowly
Metallothionein mediates leukocyte chemotaxis
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Lynes Michael A
2005-09-01
Full Text Available Abstract Background Metallothionein (MT is a cysteine-rich, metal-binding protein that can be induced by a variety of agents. Modulation of MT levels has also been shown to alter specific immune functions. We have noticed that the MT genes map close to the chemokines Ccl17 and Cx3cl1. Cysteine motifs that characterize these chemokines are also found in the MT sequence suggesting that MT might also act as a chemotactic factor. Results In the experiments reported here, we show that immune cells migrate chemotactically in the presence of a gradient of MT. This response can be specifically blocked by two different monoclonal anti-MT antibodies. Exposure of cells to MT also leads to a rapid increase in F-actin content. Incubation of Jurkat T cells with cholera toxin or pertussis toxin completely abrogates the chemotactic response to MT. Thus MT may act via G-protein coupled receptors and through the cyclic AMP signaling pathway to initiate chemotaxis. Conclusion These results suggest that, under inflammatory conditions, metallothionein in the extracellular environment may support the beneficial movement of leukocytes to the site of inflammation. MT may therefore represent a "danger signal"; modifying the character of the immune response when cells sense cellular stress. Elevated metallothionein produced in the context of exposure to environmental toxicants, or as a result of chronic inflammatory disease, may alter the normal chemotactic responses that regulate leukocyte trafficking. Thus, MT synthesis may represent an important factor in immunomodulation that is associated with autoimmune disease and toxicant exposure.
Domain decomposition parallel solution of the neutron diffusion equation on a transputer network
International Nuclear Information System (INIS)
The domain decomposition method (DDM) allows parallel computing with coarse-grain parallelism when solving a partial differential equation. In a DDM, the physical domain is decomposed into overlapping or nonoverlapping subdomains. The solution to the governing equation is then obtained iteratively by solving the problem associated with each subdomain and passing information between s. The basic idea of overlapping DDM is the Schwarz alternating procedure. In this paper, a parallel computer composed of four T-800 transputers is used to solve the steady-state one-group neutron diffusion equation with DDM. The machine is a message-passing type multiple instruction multiple data (MIMD) architecture
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.
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
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.
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.
Bifurcation Analysis of Gene Propagation Model Governed by Reaction-Diffusion Equations
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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.
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.
General Solution of a Fractional Diffusion-Advection Equation for Solar Cosmic-Ray Transport
Rocca, M C; Plastino, A; De Paoli, A L
2016-01-01
In this effort we exactly solve the fractional diffusion-advection equation for solar cosmic-ray transport proposed in \\cite{LE2014} and give its {\\it general solution} in terms of hypergeometric distributions. Also, we regain all the results and approximations given in \\cite{LE2014} as {\\it particular cases} of our general solution.
A wavelet-Galerkin method for inhomogeneous diffusion equations subject to mass specification
International Nuclear Information System (INIS)
A wavelet-Galerkin procedure is derived and implemented for the numerical solution of inhomogeneous diffusion equations subject to mass specification involving non-polynomial functions. It is demonstrated that the accuracy of approximation of these functions by polynomial interpolation at Chebyshev nodes is in good agreement with the exact solution for several numerical experiments
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.
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.
LONG-TIME BEHAVIOR OF A CLASS OF REACTION DIFFUSION EQUATIONS WITH TIME DELAYS
Institute of Scientific and Technical Information of China (English)
无
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.
International Nuclear Information System (INIS)
The neutron diffusion equation in reactor physics is solved in parallel on a transputer network. A parallel variant of Schwarz alternating procedure for overlapping subdomains is used in domain decomposition. The results of parallel computation for a 2-dimensional benchmark problem are reported and compared with those of serial computation
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.
Solution of the 1D kinetic diffusion equations using a reduced nodal cubic scheme
International Nuclear Information System (INIS)
In this work it is described a novel method to solve the multi-group time-dependent diffusion equations based on a nodal cubic space interpolation in addition to the application of quadrature rules simplifying the stiffness and mass matrices arising in a finite element procedure. Numerical results for a well known benchmark problem are also provided. (authors)
International Nuclear Information System (INIS)
Nonlinear diffusion acceleration (NDA) can improve the performance of a neutron transport solver significantly especially for the multigroup eigenvalue problems. The high-order transport equation and the transport-corrected low-order diffusion equation form a nonlinear system in NDA, which can be solved via a Picard iteration. The consistency of the correction of the low-order equation is important to ensure the stabilization and effectiveness of the iteration. It also makes the low-order equation preserve the scalar flux of the high-order equation. In this paper, the consistent correction for a particular discretization scheme, self-adjoint angular flux (SAAF) formulation with discrete ordinates method (SN) and continuous finite element method (CFEM) is proposed for the multigroup neutron transport equation. Equations with the anisotropic scatterings and a void treatment are included. The Picard iteration with this scheme has been implemented and tested with RattleSNake, a MOOSE-based application at INL. Convergence results are presented. (authors)
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 H^{1}-norm. For the convection-dominated case, the streamline diffusion stabilization (aka SUPG) is also discussed.
International Nuclear Information System (INIS)
The first-order neutron transport equation was solved by the least-squares finite element method based on the discrete ordinates discretization. For the traditional source iteration method is very slowly for the optically thick diffusive medium, sometime even divergent especially for the scattering ratio is close to unity, so the acceleration method should be proposed. There is only diffusive synthetical acceleration (DSA) for the discontinuous finite element method (DFEM) and almost no one for the least- squares finite element method. The additive angular dependent rebalance (AADR) acceleration arithmetic and its extrapolate method were given, in which the additive modification was used. It was applied to solve the transport equation with fixed source, fission source, in optically thick diffusive regions and with unstructured-mesh. The numerical results of benchmark problems demonstrate that the arithmetic can shorten the CPU time about 1.5-2 times and give high precise. (authors)
An asymptotic-preserving scheme for linear kinetic equation with fractional diffusion limit
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.
On separable Fokker-Planck equations with a constant diagonal diffusion matrix
International Nuclear Information System (INIS)
We classify (1+3)-dimensional Fokker-Planck equations with a constant diagonal diffusion matrix that are solvable by the method of separation of variables. As a result, we get possible forms of the drift coefficients B1(x-vector),B2(x-vector),B3(x-vector) providing separability of the corresponding Fokker-Planck equations and carry out variable separation in the latter. It is established, in particular, that the necessary condition for the Fokker-Planck equation to be separable is that the drift coefficients B-vector (x-vector) must be linear. We also find the necessary condition for R-separability of the Fokker-Planck equation. Furthermore, exact solutions of the Fokker-Planck equation with separated variables are constructed. (author)
International Nuclear Information System (INIS)
Single-file diffusion behaves as normal diffusion at small time and as subdiffusion at large time. These properties can be described in terms of fractional Brownian motion with variable Hurst exponent or multifractional Brownian motion. We introduce a new stochastic process called Riemann–Liouville step fractional Brownian motion which can be regarded as a special case of multifractional Brownian motion with a step function type of Hurst exponent tailored for single-file diffusion. Such a step fractional Brownian motion can be obtained as a solution of the fractional Langevin equation with zero damping. Various kinds of fractional Langevin equations and their generalizations are then considered in order to decide whether their solutions provide the correct description of the long and short time behaviors of single-file diffusion. The cases where the dissipative memory kernel is a Dirac delta function, a power-law function and a combination of these functions are studied in detail. In addition to the case where the short time behavior of single-file diffusion behaves as normal diffusion, we also consider the possibility of a process that begins as ballistic motion
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)
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.
International Nuclear Information System (INIS)
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
Dictyostelium Chemotaxis studied with fluorescence fluctuation spectroscopy
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
The second boundary value problem for equations of viscoelastic diffusion in polymers
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...
Reaction-diffusion equation for quark-hadron transition in heavy-ion collisions
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.
Singular solution of the Feller diffusion equation via a spectral decomposition
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.
Asymptotic-preserving projective integration schemes for kinetic equations in the diffusion limit
Lafitte, Pauline
2010-01-01
We investigate a projective integration scheme for a kinetic equation in the limit of vanishing mean free path, in which the kinetic description approaches a diffusion phenomenon. The scheme first takes a few small steps with a simple, explicit method, such as a spatial centered flux/forward Euler time integration, and subsequently projects the results forward in time over a large time step on the diffusion time scale. We show that, with an appropriate choice of the inner step size, the time-step restriction on the outer time step is similar to the stability condition for the diffusion equation, whereas the required number of inner steps does not depend on the mean free path. We also provide a consistency result. The presented method is asymptotic-preserving, in the sense that the method converges to a standard finite volume scheme for the diffusion equation in the limit of vanishing mean free path. The analysis is illustrated with numerical results, and we present an application to the Su-Olson test.
The breakdown of the reaction-diffusion master equation with non-elementary rates
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...
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.
International Nuclear Information System (INIS)
Monte Carlo method is widely used for solving neutron transport equation. Basically Monte Carlo method treats continuous angle, space and energy. It gives very accurate solution when enough many particle histories are used, but it takes too long computation time. To reduce computation time, discrete Monte Carlo method was proposed. It is called Discrete Transport Monte Carlo (DTMC) method. It uses discrete space but continuous angle in mono energy one dimension problem and uses lump, linear-discontinuous (LLD) equation to make probabilities of leakage, scattering, and absorption. LLD may cause negative angular fluxes in highly scattering problem, so two scatter variance reduction method is applied to DTMC and shows very accurate solution in various problems. In transport Monte Carlo calculation, the particle history does not end for scattering event. So it also takes much computation time in highly scattering problem. To further reduce computation time, Discrete Diffusion Monte Carlo (DDMC) method is implemented. DDMC uses diffusion equation to make probabilities and has no scattering events. So DDMC takes very short computation time comparing with DTMC and shows very well-agreed results with cell-centered diffusion results. It is known that diffusion result may not be good in boundaries. So in hybrid method of DTMC and DDMC, boundary regions are calculated by DTMC and the other regions are calculated by DDMC. In this thesis, DTMC, DDMC and hybrid methods and their results of several problems are presented. The results show that DDMC and DTMC are well agreed with deterministic diffusion and transport results, respectively. The hybrid method shows transport-like results in problems where diffusion results are poor. The computation time of hybrid method is between DDMC and DTMC, as expected
Jump-diffusion unravelling of a non-Markovian generalized Lindblad master equation
International Nuclear Information System (INIS)
The ''correlated-projection technique'' has been successfully applied to derive a large class of highly non-Markovian dynamics, the so called non-Markovian generalized Lindblad-type equations or Lindblad rate equations. In this article, general unravelings are presented for these equations, described in terms of jump-diffusion stochastic differential equations for wave functions. We show also that the proposed unraveling can be interpreted in terms of measurements continuous in time but with some conceptual restrictions. The main point in the measurement interpretation is that the structure itself of the underlying mathematical theory poses restrictions on what can be considered as observable and what is not; such restrictions can be seen as the effect of some kind of superselection rule. Finally, we develop a concrete example and discuss possible effects on the heterodyne spectrum of a two-level system due to a structured thermal-like bath with memory.
Dynamical invariants in a non-Markovian quantum-state-diffusion equation
Luo, Da-Wei; Pyshkin, P. V.; Lam, Chi-Hang; Yu, Ting; Lin, Hai-Qing; You, J. Q.; Wu, Lian-Ao
2015-12-01
We find dynamical invariants for open quantum systems described by the non-Markovian quantum-state-diffusion (QSD) equation. In stark contrast to closed systems where the dynamical invariant can be identical to the system density operator, these dynamical invariants no longer share the equation of motion for the density operator. Moreover, the invariants obtained with a biorthonormal basis can be used to render an exact solution to the QSD equation and the corresponding non-Markovian dynamics without using master equations or numerical simulations. Significantly we show that we can apply these dynamical invariants to reverse engineering a Hamiltonian that is capable of driving the system to the target state, providing a different way to design control strategy for open quantum systems.
International Nuclear Information System (INIS)
Numerous lattice Boltzmann (LB) methods have been proposed for solution of the convection-diffusion equations (CDE). For the 2D problem, D2Q9, D2Q5 or D2Q4 velocity models are usually used. When LB convection-diffusion models are used to solve a CDE coupled with Navier-Stokes equations, boundary conditions are found to be critically important for accurately solving the coupled simulations. Following the idea of a regularized scheme (Latt et al 2008 Phys. Rev.E 77 056703), a regularized boundary condition for solving a CDE is proposed. A simple extrapolation scheme is also proposed for the Neumann boundary condition. Spatial accuracies of three existing and the proposed boundary conditions are discussed in details. The numerical evaluations are based on simulations of steady and unsteady natural convection flows in a cavity and an unsteady Taylor-Couette flow. Our studies show that the simplest D2Q4 model with terms of O(u) in the equilibrium distribution function is capable of obtaining results of equal accuracy as D2Q5 or D2Q9 models for the CDE. A slightly revised LB equation for solving a CDE that is used to cancel some unwanted terms does not seem to be necessary for incompressible flows. The regularized boundary condition for solving the CDE has second-order spatial accuracy and it is the best one in terms of the spatial accuracy. The regularized scheme and non-equilibrium extrapolation scheme are applicable to handle both the Dirichlet and Neumann boundary conditions. For the Neumann boundary condition with zero flux, all the five boundary conditions are applicable to give accurate results and the bounce-back scheme is the simplest one.
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.
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
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.
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.
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
Turing-Hopf bifurcation in the reaction-diffusion equations and its applications
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.
The determination of an unknown boundary condition in a fractional diffusion equation
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.
A diffusive Fisher-KPP equation with free boundaries and time-periodic advections
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 $...
Vaneeva, O O; Sophocleous, C
2010-01-01
Reduction operators (called also nonclassical or $Q$-conditional symmetries) of variable coefficient semilinear reaction-diffusion equations with exponential source $f(x)u_t=(g(x)u_x)_x+h(x)e^{mu}$ are investigated using the algorithm involving a mapping between classes of differential equations, which is generated by a family of point transformations. A special attention is paid for checking whether reduction operators are inequivalent to Lie symmetry operators. The derived reduction operators are applied to construction of exact solutions.
General Reaction-Diffusion Processes With Separable Equations for Correlation Functions
Karimipour, Vahid
2002-01-01
We consider general multi-species models of reaction diffusion processes and obtain a set of constraints on the rates which give rise to closed systems of equations for correlation functions. Our results are valid in any dimension and on any type of lattice. We also show that under these conditions the evolution equations for two point functions at different times are also closed. As an example we introduce a class of two species models which may be useful for the description of voting proces...
International Nuclear Information System (INIS)
Recently, Khachaturyan's group proposed a new calculation method for phase decomposition on the basis of the Onsager equation. IN the present study, the authors modified the Khachaturyan diffusion equation to allow simulation of the phase decomposition in actual alloy systems. Two-dimensional (2-D) computer calculations are performed for the phase decompositions of Al-Zn, Cu-Co, and Fe-Mo binary systems by using the thermodynamic data related to the equilibrium phase diagrams. The calculated microstructures are very similar to the actual micrographs experimentally obtained for these alloys
Entropy methods for reaction-diffusion equations: slowly growing a-priori bounds
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.
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.
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)
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
Exact solutions of a modified fractional diffusion equation in the finite and semi-infinite domains
Guo, Gang; Li, Kun; Wang, Yuhui
2015-01-01
We investigate the solutions of a modified fractional diffusion equation which has a secondary fractional time derivative acting on a diffusion operator. We obtain analytical solutions for the modified equation in the finite and semi-infinite domains subject to absorbing boundary conditions. Most of the results have been derived by using the Laplace transform, the Fourier Cosine transform, the Mellin transform and the properties of Fox H function. We show that the semi-infinite solution can be expressed using an infinite series of Fox H functions similar to the infinite case, while the finite solution requires double infinite series including both Fox H functions and trigonometric functions instead of one infinite series. The characteristic crossover between more and less anomalous behaviour as well as the effect of absorbing boundary conditions are clearly demonstrated according to the analytical solutions.
Dynamical diffusion and renormalization group equation for the Fermi velocity in doped graphene
Ardenghi, J. S.; Bechthold, P.; Jasen, P.; Gonzalez, E.; Juan, A.
2014-11-01
The aim of this work is to study the electron transport in graphene with impurities by introducing a generalization of linear response theory for linear dispersion relations and spinor wave functions. Current response and density response functions are derived and computed in the Boltzmann limit showing that in the former case a minimum conductivity appears in the no-disorder limit. In turn, from the generalization of both functions, an exact relation can be obtained that relates both. Combining this result with the relation given by the continuity equation it is possible to obtain general functional behavior of the diffusion pole. Finally, a dynamical diffusion is computed in the quasistatic limit using the definition of relaxation function. A lower cutoff must be introduced to regularize infrared divergences which allow us to obtain a full renormalization group equation for the Fermi velocity, which is solved up to order O(ℏ2).
Ancey, C.; Bohorquez, P.; Heyman, J.
2015-12-01
The advection-diffusion equation is one of the most widespread equations in physics. It 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). Phenomenological laws are usually sufficient to derive this equation and interpret its terms. Stochastic models can also be used to derive it, with the significant advantage that they provide information on the statistical properties of particle activity. These models are quite useful when sediment transport exhibits large fluctuations (typically at low transport rates), making the measurement of mean values difficult. Among these stochastic models, the most common approach consists of random walk models. For instance, they have been used to model the random displacement of tracers in rivers. Here we explore an alternative approach, which involves monitoring the evolution of the number of particles moving within an array of cells of finite length. Birth-death Markov processes are well suited to this objective. While the topic has been explored in detail for diffusion-reaction systems, the treatment of advection has received no attention. We therefore look into the possibility of deriving the advection-diffusion equation (with a source term) within the framework of birth-death Markov processes. We show that in the continuum limit (when the cell size becomes vanishingly small), we can derive an advection-diffusion equation for particle activity. Yet while this derivation is formally valid in the continuum limit, it runs into difficulty in practical applications involving cells or meshes of finite length. Indeed, within our stochastic framework, 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
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)
Doubly nonlocal reaction-diffusion equation and the emergence of species
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...
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)
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.
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.
Hepperger, Peter
2011-01-01
This thesis is concerned with high-dimensional jump-diffusion models. Two applications serve as motivating examples: stock basket options and options on so called electricity swaps. A Hilbert space valued model is introduced and discussed, which can be applied to both settings. The focus of the thesis lies on numerical techniques for the solution of pricing and hedging problems. The corresponding partial integro-differential equations are derived. Using proper orthogonal decomposition, the di...
Institute of Scientific and Technical Information of China (English)
无
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.
International Nuclear Information System (INIS)
This work deals with the numerical solution of the neutron diffusion equation under conditions, which follow from engineering applications. The Finite Element Method (FEM) was chosen as the solution method. It is the aim of this work to demonstrate the potential of the FEM for solving problems resulting from nuclear engineering applications. From this a concept is derived which allowed the development of a program system. Calculational results achieved with this system confirm the theoretical considerations. (orig.)
A Spectral Study of the Linearized Boltzmann Equation for Diffusively Excited Granular Media
Rey, Thomas
2013-01-01
In this work, we are interested in the spectrum of the diffusively excited granular gases equation, in a space inhomogeneous setting, linearized around an homogeneous equilibrium. We perform a study which generalizes to a non-hilbertian setting and to the inelastic case the seminal work of Ellis and Pinsky about the spectrum of the linearized Boltzmann operator. We first give a precise localization of the spectrum, which consists in an essential part lying on the left of the imaginary axis an...
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.
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...
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.
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.
Jiang, Tian; Zhang, Yong-Tao
2016-04-01
Implicit integration factor (IIF) methods were developed in the literature for solving time-dependent stiff partial differential equations (PDEs). Recently, IIF methods were combined with weighted essentially non-oscillatory (WENO) schemes in Jiang and Zhang (2013) [19] to efficiently solve stiff nonlinear advection-diffusion-reaction equations. The methods can be designed for arbitrary order of accuracy. The stiffness of the system is resolved well and the methods are stable by using time step sizes which are just determined by the non-stiff hyperbolic part of the system. To efficiently calculate large matrix exponentials, Krylov subspace approximation is directly applied to the implicit integration factor (IIF) methods. So far, the IIF methods developed in the literature are multistep methods. In this paper, we develop Krylov single-step IIF-WENO methods for solving stiff advection-diffusion-reaction equations. The methods are designed carefully to avoid generating positive exponentials in the matrix exponentials, which is necessary for the stability of the schemes. We analyze the stability and truncation errors of the single-step IIF schemes. Numerical examples of both scalar equations and systems are shown to demonstrate the accuracy, efficiency and robustness of the new methods.
Group Analysis of Variable Coefficient Diffusion--Convection Equations. III. Conservation Laws
Ivanova, N M; Sophocleous, C
2007-01-01
The notions of generating sets of conservation laws of systems of differential equations with respect to symmetry groups and equivalence groups are introduced and applied. This allows us to generalize essentially the procedure of finding potential symmetries for the systems with multidimensional spaces of conservation laws. A class of variable coefficient (1+1)-dimensional nonlinear diffusion-convection equations of general form $f(x)u_t=(g(x)A(u)u_x)_x+h(x)B(u)u_x$ is investigated. Using the most direct method, we carry out two classifications of local conservation laws up to equivalence relations generated by both usual and enhanced equivalence groups. Equivalence with respect to $\\hat G^{\\sim}$ and correct choice of gauge coefficients of equations play the major role for simple and clear formulation of the final results. The notion of contractions of conservation laws and one of characteristics of conservation laws are introduced and contractions of conservation laws of diffusion-convection equations are f...
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)
International Nuclear Information System (INIS)
In this paper, the strong approximation of a stochastic partial differential equation, whose differential operator is of advection-diffusion type and which is driven by a multiplicative, infinite dimensional, càdlàg, square integrable martingale, is presented. A finite dimensional projection of the infinite dimensional equation, for example a Galerkin projection, with nonequidistant time stepping is used. Error estimates for the discretized equation are derived in L2 and almost sure senses. Besides space and time discretizations, noise approximations are also provided, where the Milstein double stochastic integral is approximated in such a way that the overall complexity is not increased compared to an Euler–Maruyama approximation. Finally, simulations complete the paper.
Long-time behavior of a finite volume discretization for a fourth order diffusion equation
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.
Reynolds, A. M.
2010-06-01
Here, we report on numerical simulations showing that chemotaxis will take a body through a maze via the shortest possible route to the source of a chemoattractant. This is a robust finding that does not depend on the geometrical makeup of the maze. The predictions are supported by recent experimental studies which have shown that by moving down gradients in pH , a droplet of organic solvent can find the shortest of multiple possible paths through a maze to an acid-soaked exit. They are also consistent with numerical and experimental evidence that plant-parasitic nematodes take the shortest route through the labyrinth of air-filled pores within soil to preferred host plants that produce volatile chemoattractants. The predictions support the view that maze-solving is a robust property of chemotaxis and is not specific to particular kinds of maze or to the fractal structure of air-filled channels within soils.
Chemotaxis of large granular lymphocytes
International Nuclear Information System (INIS)
The hypothesis that large granular lymphocytes (LGL) are capable of directed locomotion (chemotaxis) was tested. A population of LGL isolated from discontinuous Percoll gradients migrated along concentration gradients of N-formyl-methionyl-leucyl-phenylalanine (f-MLP), casein, and C5a, well known chemoattractants for polymorphonuclear leukocytes and monocytes, as well as interferon-β and colony-stimulating factor. Interleukin 2, tuftsin, platelet-derived growth factor, and fibronectin were inactive. Migratory responses were greater in Percoll fractions with the highest lytic activity and HNK-1+ cells. The chemotactic response to f-MLP, casein, and C5a was always greater when the chemoattractant was present in greater concentration in the lower compartment of the Boyden chamber. Optimum chemotaxis was observed after a 1 hr incubation that made use of 12 μm nitrocellulose filters. LGL exhibited a high degree of nondirected locomotion when allowed to migrate for longer periods (> 2 hr), and when cultured in vitro for 24 to 72 hr in the presence or absence of IL 2 containing phytohemagluttinin-conditioned medium. LGL chemotaxis to f-MLP could be inhibited in a dose-dependent manner by the inactive structural analog CBZ-phe-met, and the RNK tumor line specifically bound f-ML(3H)P, suggesting that LGL bear receptors for the chemotactic peptide
Plante, Ianik
2016-01-01
The exact Green's function of the diffusion equation (GFDE) is often considered to be the gold standard for the simulation of partially diffusion-controlled reactions. As the GFDE with angular dependency is quite complex, the radial GFDE is more often used. Indeed, the exact GFDE is expressed as a Legendre expansion, the coefficients of which are given in terms of an integral comprising Bessel functions. This integral does not seem to have been evaluated analytically in existing literature. While the integral can be evaluated numerically, the Bessel functions make the integral oscillate and convergence is difficult to obtain. Therefore it would be of great interest to evaluate the integral analytically. The first term was evaluated previously, and was found to be equal to the radial GFDE. In this work, the second term of this expansion was evaluated. As this work has shown that the first two terms of the Legendre polynomial expansion can be calculated analytically, it raises the question of the possibility that an analytical solution exists for the other terms.
Neutrophil Chemotaxis Dysfunction in Human Periodontitis
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 ...
Chemotaxis of Azospirillum Species to Aromatic Compounds
Lopez-de-Victoria, Geralyne; Lovell, Charles R.
1993-01-01
Chemotaxis of Azospirillum lipoferum Sp 59b and Azospirillum brasilense Sp 7 and Sp CD to malate and to the aromatic substrates benzoate, protocatechuate, 4-hydroxybenzoate, and catechol was assayed by the capillary method and direct cell counts. A. lipoferum required induction by growth on 4-hydroxybenzoate for positive chemotaxis to this compound. Chemotaxis of Azospirillum spp. to all other substrates did not require induction. Maximum chemotactic responses for most aromatic compounds occu...
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.
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)
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.
International Nuclear Information System (INIS)
We consider one-dimensional nonlinear delay reaction-diffusion equations with varying transfer coefficients. We describe a few new methods for constructing exact solutions of such equations; these methods are based on using invariant subspaces for the corresponding nonlinear differential operators. A number of new exact generalized and functional separable solutions have been obtained. All of the equations and solutions involve several free parameters. The exact solutions obtained can be used to test approximate analytical and numerical methods for solving nonlinear delay reaction-diffusion equations
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)
Feeding ducks, bacterial chemotaxis, and the Gini index
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...
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.
Regularity and mass conservation for discrete coagulation–fragmentation equations with diffusion
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.
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.
Diffusive Mixing of Stable States in the Ginzburg-Landau Equation
Gallay, T; Gallay, Thierry; Mielke, Alexander
1998-01-01
For the time-dependent Ginzburg-Landau equation on the real line, we construct solutions which converge, as $x \\to \\pm\\infty$, to periodic stationary states with different wave-numbers $\\eta_\\pm$. These solutions are stable with respect to small perturbations, and approach as $t \\to +\\infty$ a universal diffusive profile depending only on the values of $\\eta_\\pm$. This extends a previous result of Bricmont and Kupiainen by removing the assumption that $\\eta_\\pm$ should be close to zero. The existence of the diffusive profile is obtained as an application of the theory of monotone operators, and the long-time behavior of our solutions is controlled by rewriting the system in scaling variables and using energy estimates involving an exponentially growing damping term.
Retinal Image Enhancement Using Robust Inverse Diffusion Equation and Self-Similarity Filtering
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
Analytical Solutions of a Fractional Diffusion-advection Equation for Solar Cosmic-Ray Transport
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.
Analytical solutions of a fractional diffusion-advection equation for solar cosmic-ray transport
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.
International Nuclear Information System (INIS)
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
Wielandt method applied to the diffusion equations discretized by finite element nodal methods
International Nuclear Information System (INIS)
Nowadays the numerical methods of solution to the diffusion equation by means of algorithms and computer programs result so extensive due to the great number of routines and calculations that should carry out, this rebounds directly in the execution times of this programs, being obtained results in relatively long times. This work shows the application of an acceleration method of the convergence of the classic method of those powers that it reduces notably the number of necessary iterations for to obtain reliable results, what means that the compute times they see reduced in great measure. This method is known in the literature like Wielandt method and it has incorporated to a computer program that is based on the discretization of the neutron diffusion equations in plate geometry and stationary state by polynomial nodal methods. In this work the neutron diffusion equations are described for several energy groups and their discretization by means of those called physical nodal methods, being illustrated in particular the quadratic case. It is described a model problem widely described in the literature which is solved for the physical nodal grade schemes 1, 2, 3 and 4 in three different ways: to) with the classic method of the powers, b) method of the powers with the Wielandt acceleration and c) method of the powers with the Wielandt modified acceleration. The results for the model problem as well as for two additional problems known as benchmark problems are reported. Such acceleration method can also be implemented to problems of different geometry to the proposal in this work, besides being possible to extend their application to problems in 2 or 3 dimensions. (Author)
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.
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.
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.
A study of quasilinear diffusion coefficient of the fokker-planck equation based on full wave method
International Nuclear Information System (INIS)
Based on the wave fields evaluated by full wave method, the various components of quasilinear diffusion coefficient in spherical velocity coordinates are obtained by numerically resolving. The quasilinear diffusion coefficient of Fokker-planck equation is presented for investigating fast wave current drive in the ion cyclotron range of frequencies, and the results will be used to solve bounce-averaged quasilinear Fokker-Planck equation for further study of fast wave current drive. (authors)
Harko, T.; Mak, M. K.
2015-11-01
We consider quasi-stationary (travelling wave type) solutions to a general nonlinear reaction-convection-diffusion equation with arbitrary, autonomous coefficients. The second order nonlinear equation describing one dimensional travelling waves can be reduced to a first kind first order Abel equation. By using two integrability conditions for the Abel equation (the Chiellini lemma and the Lemke transformation), several classes of exact travelling wave solutions of the general reaction-convection-diffusion equation are obtained, corresponding to different functional relations imposed between the diffusion, convection and reaction functions. In particular, we obtain travelling wave solutions for two non-linear second order partial differential equations, representing generalizations of the standard diffusion equation and of the classical Fisher-Kolmogorov equation, to which they reduce for some limiting values of the model parameters. The models correspond to some specific, power law type choices of the reaction and convection functions, respectively. The travelling wave solutions of these two classes of differential equation are investigated in detail by using both numerical and semi-analytical methods.
Study of an Advection-Reaction-Diffusion equation in a compressible flow field
Bianco, Federico; Prud'homme, Roger
2011-01-01
We have studied the front propagation in a one dimensional case of combustion by solving numerically an advection-reaction-diffusion equation. The physical model is simplified so that no coupling phenomena are considered and the reacting fluid is a binary mixture of gases. The compressible flow field is given analytically. We analyse the differences between popular models used in fundamental studies of compressible combustion and biological problems. Then, we investigate the effects of compressibility on the front interface dynamics for different reaction types and we characterise the conditions for which the reaction stops before its completion.
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.
Time adaptivity in the diffusive wave approximation to the shallow water equations
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.
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.
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.
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.
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.
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)
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)
Diffusion coefficients of Fokker-Planck equation for rotating dust grains in a fusion plasma
Energy Technology Data Exchange (ETDEWEB)
Bakhtiyari-Ramezani, M., E-mail: mahdiyeh.bakhtiyari@gmail.com; Alinejad, N., E-mail: nalinezhad@aeoi.org.ir [Plasma Physics and Nuclear Fusion Research School, Nuclear Science and Technology Research Institute (NSTRI), 14395-836 Tehran (Iran, Islamic Republic of); Mahmoodi, J., E-mail: mahmoodi@qom.ac.ir [Department of Physics, Faculty of Science, University of Qom, Qom (Iran, Islamic Republic of)
2015-11-15
In the fusion devices, ions, H atoms, and H{sub 2} 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 H{sub 2} molecules, and desorption of the recombined H{sub 2} 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.
Preconditioned iterative methods for space-time fractional advection-diffusion equations
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.
The Galerkin finite element method for a multi-term time-fractional diffusion equation
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.
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.
Symmetries of systems of stochastic differential equations with diffusion matrices of full rank
International Nuclear Information System (INIS)
Lie point symmetries of a system of stochastic differential equations (SDEs) with diffusion matrices of full rank are considered. It is proved that the maximal dimension of a symmetry group admitted by a system of n SDEs is n + 2. In addition, such systems cannot admit symmetry operators whose coefficients are proportional to a nonconstant coefficient of proportionality. These results are applied to compute the Lie group classification of a system of two SDEs. The classification is obtained with the help of non-equivalent realizations of real Lie algebras by fiber-preserving vector fields in 1 + 2 variables. Possibilities of using symmetries for integration of SDEs by quadratures are discussed.
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)
A solution of the neutron diffusion equation for a hemisphere containing a uniform source
International Nuclear Information System (INIS)
An analytic solution of the diffusion equation for a hemisphere of fissile or non-fissile material is presented which contains a spatially uniform neutron source. Numerical results are given for the flux distribution for one-speed fast neutrons in 235U and also for a non-fissile element of similar scattering properties. We use these results to check the accuracy of the finite element code EVENT. The procedure is also developed for multigroup calculations. In an Appendix we outline the procedure required when the hemisphere contains a source and is also irradiated by an external current of neutrons
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
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.
Stability and Bifurcation in a Delayed Reaction-Diffusion Equation with Dirichlet Boundary Condition
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.
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.
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.
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.
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.
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.
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)
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.
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.
Symmetries of the Fokker-Planck equation with a constant diffusion matrix in 2+1 dimensions
International Nuclear Information System (INIS)
We completely classify the symmetries of the Fokker-Planck equation in two spatial dimensions with a constant positive-definite diffusion matrix. We apply these results to construct group-invariant solutions for a physically interesting family of Fokker-Planck equations. (author)
Converged accelerated finite difference scheme for the multigroup neutron diffusion equation
International Nuclear Information System (INIS)
Computer codes involving neutron transport theory for nuclear engineering applications always require verification to assess improvement. Generally, analytical and semi-analytical benchmarks are desirable, since they are capable of high precision solutions to provide accurate standards of comparison. However, these benchmarks often involve relatively simple problems, usually assuming a certain degree of abstract modeling. In the present work, we show how semi-analytical equivalent benchmarks can be numerically generated using convergence acceleration. Specifically, we investigate the error behavior of a 1D spatial finite difference scheme for the multigroup (MG) steady-state neutron diffusion equation in plane geometry. Since solutions depending on subsequent discretization can be envisioned as terms of an infinite sequence converging to the true solution, extrapolation methods can accelerate an iterative process to obtain the limit before numerical instability sets in. The obtained results have been compared to the analytical solution to the 1D multigroup diffusion equation when available, using FORTRAN as the computational language. Finally, a slowing down problem has been solved using a cascading source update, showing how a finite difference scheme performs for ultra-fine groups (104 groups) in a reasonable computational time using convergence acceleration. (authors)
Tönjes, Ralf; Blasius, Bernd
2009-01-01
The Kuramoto phase-diffusion equation is a nonlinear partial differential equation which describes the spatiotemporal evolution of a phase variable in an oscillatory reaction-diffusion system. Synchronization manifests itself in a stationary phase gradient where all phases throughout a system evolve with the same velocity, the synchronization frequency. The formation of concentric waves can be explained by local impurities of higher frequency which can entrain their surroundings. Concentric waves in synchronization also occur in heterogeneous systems, where the local frequencies are distributed randomly. We present a perturbation analysis of the synchronization frequency where the perturbation is given by the heterogeneity of natural frequencies in the system. The nonlinearity in the form of dispersion leads to an overall acceleration of the oscillation for which the expected value can be calculated from the second-order perturbation terms. We apply the theory to simple topologies, like a line or sphere, and deduce the dependence of the synchronization frequency on the size and the dimension of the oscillatory medium. We show that our theory can be extended to include rotating waves in a medium with periodic boundary conditions. By changing a system parameter, the synchronized state may become quasidegenerate. We demonstrate how perturbation theory fails at such a critical point. PMID:19257112
Chai, Zhenhua; Guo, Zhaoli
2016-01-01
In this paper, based on the previous work [B. Shi, Z. Guo, Lattice Boltzmann model for nonlinear convection-diffusion equations, Phys. Rev. E 79 (2009) 016701], we develop a general multiple-relaxation-time (MRT) lattice Boltzmann model for nonlinear anisotropic convection-diffusion equation (NACDE), and show that the NACDE can be recovered correctly from the present model through the Chapman-Enskog analysis. We then test the MRT model through some classic CDEs, and find that the numerical results are in good agreement with analytical solutions or some available results. Besides, the numerical results also show that similar to the single-relaxation-time (SRT) lattice Boltzmann model or so-called BGK model, the present MRT model also has a second-order convergence rate in space. Finally, we also perform a comparative study on the accuracy and stability of the MRT model and BGK model by using two examples. In terms of the accuracy, both the theoretical analysis and numerical results show that a \\emph{numerical}...
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)
Diffuse fluorescence tomography based on the radiative transfer equation for small animal imaging
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.
On an adaptive time stepping strategy for solving nonlinear diffusion equations
International Nuclear Information System (INIS)
A new time step selection procedure is proposed for solving non- linear diffusion equations. It has been implemented in the ASWR finite element code of Lorenz and Svoboda [10] for 2D semiconductor process modelling diffusion equations. The strategy is based on equi-distributing the local truncation errors of the numerical scheme. The use of B-splines for interpolation (as well as for the trial space) results in a banded and diagonally dominant matrix. The approximate inverse of such a matrix can be provided to a high degree of accuracy by another banded matrix, which in turn can be used to work out the approximate finite difference scheme corresponding to the ASWR finite element method, and further to calculate estimates of the local truncation errors of the numerical scheme. Numerical experiments on six full simulation problems arising in semiconductor process modelling have been carried out. Results show that our proposed strategy is more efficient and better conserves the total mass. 18 refs., 6 figs., 2 tabs
Stationary states of quadratic diffusion equations with long-range attraction
Burger, M; Franek, M
2011-01-01
We study the existence and uniqueness of nontrivial stationary solutions to a nonlocal aggregation equation with quadratic diffusion arising in many contexts in population dynamics. The equation is the Wasserstein gradient flow generated by the energy E, which is the sum of a quadratic free energy and the interaction energy. The interaction kernel is taken radial and attractive, nonnegative and integrable, with further technical smoothness assumptions. The existence vs. nonexistence of such solutions is ruled by a threshold phenomenon, namely nontrivial steady states exist if and only if the diffusivity constant is strictly smaller than the total mass of the interaction kernel. In the one dimensional case we prove that steady states are unique up to translations and mass constraint. The strategy is based on a strong version of the Krein-Rutman theorem. The steady states are symmetric with respect to their center of mass x0, compactly supported on sets of the form [x0 -L, x0+L], C^2 on their support, strictly ...
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)
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)
Sound energy decay in coupled spaces using a parametric analytical solution of a diffusion equation.
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
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.
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.
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.
International Nuclear Information System (INIS)
Fractional diffusion equations have been the focus of modeling problems in hydrology, biology, viscoelasticity, physics, engineering, and other areas of applications. In this paper, a meshfree method based on the moving Kriging interpolation is developed for a two-dimensional time-fractional diffusion equation. The shape function and its derivatives are obtained by the moving Kriging interpolation technique. For possessing the Kronecker delta property, this technique is very efficient in imposing the essential boundary conditions. The governing time-fractional diffusion equations are transformed into a standard weak formulation by the Galerkin method. It is then discretized into a meshfree system of time-dependent equations, which are solved by the standard central difference method. Numerical examples illustrating the applicability and effectiveness of the proposed method are presented and discussed in detail. (general)
Bacterial strategies for chemotaxis response.
Celani, Antonio; Vergassola, Massimo
2010-01-26
Regular environmental conditions allow for the evolution of specifically adapted responses, whereas complex environments usually lead to conflicting requirements upon the organism's response. A relevant instance of these issues is bacterial chemotaxis, where the evolutionary and functional reasons for the experimentally observed response to chemoattractants remain a riddle. Sensing and motility requirements are in fact optimized by different responses, which strongly depend on the chemoattractant environmental profiles. It is not clear then how those conflicting requirements quantitatively combine and compromise in shaping the chemotaxis response. Here we show that the experimental bacterial response corresponds to the maximin strategy that ensures the highest minimum uptake of chemoattractants for any profile of concentration. We show that the maximin response is the unique one that always outcompetes motile but nonchemotactic bacteria. The maximin strategy is adapted to the variable environments experienced by bacteria, and we explicitly show its emergence in simulations of bacterial populations in a chemostat. Finally, we recast the contrast of evolution in regular vs. complex environments in terms of minimax vs. maximin game-theoretical strategies. Our results are generally relevant to biological optimization principles and provide a systematic possibility to get around the need to know precisely the statistics of environmental fluctuations. PMID:20080704
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
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
Three-dimensional h-adaptivity for the multigroup neutron diffusion equations
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.
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.
International Nuclear Information System (INIS)
A code called COMESH based on corner mesh finite difference scheme has been developed to solve multigroup diffusion theory equations. One can solve 1-D, 2-D or 3-D problems in Cartesian geometry and 1-D (r) or 2-D (r-z) problem in cylindrical geometry. On external boundary one can use either homogeneous Dirichlet (θ-specified) or Neumann (∇θ specified) type boundary conditions or a linear combination of the two. Internal boundaries for control absorber simulations are also tackled by COMESH. Many an acceleration schemes like successive line over-relaxation, two parameter Chebyschev acceleration for fission source, generalised coarse mesh rebalancing etc., render the code COMESH a very fast one for estimating eigenvalue and flux/power profiles in any type of reactor core configuration. 6 refs. (author)
Ivanova, N M; Sophocleous, C
2007-01-01
After discussing the classical statement of group classification problem and some its extensions in the general case, we carry out the complete extended group classification for a class of (1+1)-dimensional nonlinear diffusion-convection equations with coefficients depending on the space variable. At first, we construct the usual equivalence group and the extended one including transformations which are nonlocal with respect to arbitrary elements. The extended equivalence group has interesting structure since it contains a non-trivial subgroup of non-local gauge equivalence transformations. The complete group classification of the class under consideration is performed with respect to the extended equivalence group and with respect to the set of all local transformations. Usage of extended equivalence and correct choice of gauges of arbitrary elements play the major role for simple and clear formulation of the final results. The set of admissible transformations of this class is preliminary investigated.
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
International Nuclear Information System (INIS)
Numerical solution of the diffusion equation plays a key role in the study of inertial confinement fusion (ICF). In this paper, based on the global support operator method, a flux-based scheme is proposed. The scheme has local stencil with second-order accuracy both in space and time. For strongly distorted meshes, a procedure of normal direction fix is adopted with proper methods for the computation of corner volume weights, which obtains accurate discretization of the face flux. Numerical experiments show that the scheme can obtain accurate solution for linear problems on non-convex meshes. The method has second-order spatial and temporal accuracy on non-smooth meshes. The method can also preserve the symmetry well and can be extended to the three dimensional unstructured meshes. (authors)
Rapidity window dependences of higher order cumulants and diffusion master equation
International Nuclear Information System (INIS)
We study the rapidity window dependences of higher order cumulants of conserved charges observed in relativistic heavy ion collisions. The time evolution and the rapidity window dependence of the non-Gaussian fluctuations are described by the diffusion master equation. Analytic formulas for the time evolution of cumulants in a rapidity window are obtained for arbitrary initial conditions. We discuss that the rapidity window dependences of the non-Gaussian cumulants have characteristic structures reflecting the non-equilibrium property of fluctuations, which can be observed in relativistic heavy ion collisions with the present detectors. It is argued that various information on the thermal and transport properties of the hot medium can be revealed experimentally by the study of the rapidity window dependences, especially by the combined use, of the higher order cumulants. Formulas of higher order cumulants for a probability distribution composed of sub-probabilities, which are useful for various studies of non-Gaussian cumulants, are also presented
The dynamics of nonlinear reaction-diffusion equations with small Lévy noise
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.
International Nuclear Information System (INIS)
The neutron diffusion equation in reactor physics is solved on a multiple-instruction, multiple-data parallel computer network composed of five transputers. A parallel variant of the Schwarz alternating procedure for overlapping subdomains is used for domain decomposition. The parallel Schwartz algorithm with the concept of underrelaxation in pseudo-boundary conditions is applied to two types of reactor benchmark problems: fixed-source problems and eigenvalue problems. Results of parallel computation for these problems are reported and compared with results of sequential computation. The results show that a very high speedup can be achieved in fixed-source problems in spite of the small problem size and that a relatively high speedup, although lower than that of fixed-source problems, can be obtained in eigenvalue problems
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.
A Radiation Chemistry Code Based on the Greens Functions of the Diffusion Equation
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
Hybrid nodal methods in the solution of the diffusion equations in X Y geometry
International Nuclear Information System (INIS)
In 1979, Hennart and collaborators applied several schemes of classic finite element in the numerical solution of the diffusion equations in X Y geometry and stationary state. Almost two decades then, in 1996, himself and other collaborators carried out a similar work but using nodal schemes type finite element. Continuing in this last direction, in this work a group it is described a set of several Hybrid Nodal schemes denominated (NH) as well as their application to solve the diffusion equations in multigroup in stationary state and X Y geometry. The term hybrid nodal it means that such schemes interpolate not only Legendre moments of face and of cell but also the values of the scalar flow of neutrons in the four corners of each cell or element of the spatial discretization of the domain of interest. All the schemes here considered are polynomials like they were it their predecessors. Particularly, its have developed and applied eight different hybrid nodal schemes that its are very nearby related with those developed by Hennart and collaborators in the past. It is treated of schemes in those that nevertheless that decreases the number of interpolation parameters it is conserved the accurate in relation to the bi-quadratic and bi-cubic schemes. Of these eight, three were described and applied in a previous work. It is the bi-lineal classic scheme as well as the hybrid nodal schemes, bi-quadratic and bi-cubic for that here only are described the other 5 hybrid nodal schemes although they are provided numerical results for several test problems with all them. (Author)
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...
Lattice Boltzmann method for convection-diffusion equations with general interfacial conditions
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.
Non-dispersive carrier transport in molecularly doped polymers and the convection-diffusion equation
Tyutnev, A. P.; Parris, P. E.; Saenko, V. S.
2015-08-01
We reinvestigate the applicability of the concept of trap-free carrier transport in molecularly doped polymers and the possibility of realistically describing time-of-flight (TOF) current transients in these materials using the classical convection-diffusion equation (CDE). The problem is treated as rigorously as possible using boundary conditions appropriate to conventional time of flight experiments. Two types of pulsed carrier generation are considered. In addition to the traditional case of surface excitation, we also consider the case where carrier generation is spatially uniform. In our analysis, the front electrode is treated as a reflecting boundary, while the counter electrode is assumed to act either as a neutral contact (not disturbing the current flow) or as an absorbing boundary at which the carrier concentration vanishes. As expected, at low fields transient currents exhibit unusual behavior, as diffusion currents overwhelm drift currents to such an extent that it becomes impossible to determine transit times (and hence, carrier mobilities). At high fields, computed transients are more like those typically observed, with well-defined plateaus and sharp transit times. Careful analysis, however, reveals that the non-dispersive picture, and predictions of the CDE contradict both experiment and existing disorder-based theories in important ways, and that the CDE should be applied rather cautiously, and even then only for engineering purposes.
Competing computational approaches to reaction-diffusion equations in clusters of cells
International Nuclear Information System (INIS)
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
Finite difference solution of the time dependent neutron group diffusion equations
International Nuclear Information System (INIS)
In this thesis two unrelated topics of reactor physics are examined: the prompt jump approximation and alternating direction checkerboard methods. In the prompt jump approximation it is assumed that the prompt and delayed neutrons in a nuclear reactor may be described mathematically as being instantaneously in equilibrium with each other. This approximation is applied to the spatially dependent neutron diffusion theory reactor kinetics model. Alternating direction checkerboard methods are a family of finite difference alternating direction methods which may be used to solve the multigroup, multidimension, time-dependent neutron diffusion equations. The reactor mesh grid is not swept line by line or point by point as in implicit or explicit alternating direction methods; instead, the reactor mesh grid may be thought of as a checkerboard in which all the ''red squares'' and '' black squares'' are treated successively. Two members of this family of methods, the ADC and NSADC methods, are at least as good as other alternating direction methods. It has been found that the accuracy of implicit and explicit alternating direction methods can be greatly improved by the application of an exponential transformation. This transformation is incompatible with checkerboard methods. Therefore, a new formulation of the exponential transformation has been developed which is compatible with checkerboard methods and at least as good as the former transformation for other alternating direction methods
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.
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)
Fourier spectral methods for fractional-in-space reaction-diffusion equations
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.
A CNN-based approach to integrate the 3-D turbolent diffusion equation
Nunnari, G.
2003-04-01
The paper deals with the integration of the 3-D turbulent diffusion equation. This problem is relevant in several application fields including fluid dynamics, air/water pollution, volcanic ash emissions and industrial hazard assessment. As it is well known numerical solution of such a kind of equation is very time consuming even by using modern digital computers and this represents a short-coming for on-line applications. To overcome this drawback a Cellular Neural Network Approach is proposed in this paper. CNN's proposed by Chua and Yang in 1988 are massive parallel analog non-linear circuits with local interconnections between the computing elements that allow very fast distributed computations. Nowadays several producers of semiconductors such as SGS-Thomson are producing on chip CNN's so that their massive use for heavy computing applications is expected in the near future. In the paper the methodological background of the proposed approach will be outlined. Further some results both in terms of accuracy and computation time will be presented also in comparison with traditional three-dimensional computation schemes. Some results obtained to model 3-D pollution problems in the industrial area of Siracusa (Italy), characterised by a large concentration of petrol-chemical plants, will be presented.
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.
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.
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
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...
Cisternas, Jaime; Descalzi, Orazio; Albers, Tony; Radons, Günter
2016-05-01
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.
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...
International Nuclear Information System (INIS)
Most of thermal hydraulic processes in nuclear engineering can be described by general convection-diffusion equations that are often can be simulated numerically with finite-difference method (FDM). An effective scheme for finite-difference discretization of such equations is presented in this report. The derivation of this scheme is based on analytical solutions of a simplified one-dimensional equation written for every control volume of the finite-difference mesh. These analytical solutions are constructed using linearized representations of both diffusion coefficient and source term. As a result, the Efficient Finite-Differencing (EFD) scheme makes it possible to significantly improve the accuracy of numerical method even using mesh systems with fewer grid nodes that, in turn, allows to speed-up numerical simulation. EFD has been carefully verified on the series of sample problems for which either analytical or very precise numerical solutions can be found. EFD has been compared with other popular FDM schemes including novel, accurate (as well as sophisticated) methods. Among the methods compared were well-known central difference scheme, upwind scheme, exponential differencing and hybrid schemes of Spalding. Also, newly developed finite-difference schemes, such as the the quadratic upstream (QUICK) scheme of Leonard, the locally analytic differencing (LOAD) scheme of Wong and Raithby, the flux-spline scheme proposed by Varejago and Patankar as well as the latest LENS discretization of Sakai have been compared. Detailed results of this comparison are given in this report. These tests have shown a high efficiency of the EFD scheme. For most of sample problems considered EFD has demonstrated the numerical error that appeared to be in orders of magnitude lower than that of other discretization methods. Or, in other words, EFD has predicted numerical solution with the same given numerical error but using much fewer grid nodes. In this report, the detailed
Singularity formation in chemotaxis systems with volume-filling effect
International Nuclear Information System (INIS)
A parabolic–elliptic model of chemotaxis which takes into account volume-filling effects is considered under the assumption that there is an a priori threshold for the cell density. For a wide range of nonlinear diffusion operators including singular and degenerate ones it is proved that if the taxis force is strong enough with respect to diffusion and the initial data are chosen properly then there exists a classical solution which reaches the threshold at the maximal time of its existence, no matter whether the latter is finite or infinite. Moreover, we prove that the threshold may even be reached in finite time provided the diffusion of cells is non-degenerate
Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis
Wang, Zhi-An; Xiang, Zhaoyin; Yu, Pei
2016-02-01
The asymptotic behavior of solutions to a singular chemotaxis system modeling the onset of tumor angiogenesis in two and three dimensional whole spaces is investigated in the paper. By a Cole-Hopf type transformation, the singular chemotaxis is converted into a non-singular hyperbolic system. Then we study the transformed system and establish the global existence, asymptotic decay rates and diffusion convergence rate of solutions by the method of energy estimates. The main novelty of our results is the finding of a hidden interactive dissipation structure in the system by which the energy dissipation is established.
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)
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 ...
International Nuclear Information System (INIS)
In this paper, a detailed Lie symmetry analysis of the (2+1)-dimensional coupled nonlinear extension of the reaction-diffusion equation is presented. The general finite transformation group is derived via a simple direct method, which is equivalent to Lie point symmetry group actually. Similarity reduction and some exact solutions of the original equation are obtained based on the optimal system of one-dimensional subalgebras. In addition, conservation laws are constructed by employing the new conservation theorem. (general)
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}.
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
The intrinsic periodic fluctuation of forest: a theoretical model based on diffusion equation
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.
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.
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.
Datta, Dibakar
2013-01-01
In the present study, an advection-diffusion problem has been considered for the numerical solution. The continuum equation is discretized using both upwind and centered scheme. The linear system is solved using the ILU preconditioned BiCGSTAB method. Both Dirichlet and Neumann boundary condition has been considered. The obtained results have been compared for different cases.
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Galerkin/least-squares finite element methods are presented for advective-diffusive equations. Galerkin/least-squares represents a conceptual simplification of SUPG, and is in fact applicable to a wide variety of other problem types. A convergence analysis and error estimates are presented. (author)
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.
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Research highlights: This article concentrates the phase transformation in δ → σ in dissimilar stainless steels using the Vitek equation and thermodynamics simulation during the multi-pass welding. The phase transformation in δ → σ is very important to the properties of stainless steel composites. In this study, the diffusion behavior of Cr, Ni and Si in the δ, σ, and γ phases were discussed using the DSC analysis and diffusion equation calculation. This method has a novelty for discussing the phase transformation in δ → σ in the dissimilar stainless steel. We hope that we can give a scientific contribution for the phase transformation of the dissimilar stainless steels during the multi-pass welding. - Abstract: This study performed a precipitation examination of the σ phase using the Vitek diffusion equation and thermodynamic simulation in dissimilar stainless steels during multi-pass welding. The results of the experiment demonstrate that the diffusion rates (DCrδ and DNiδ) of Cr and Ni are higher in δ-ferrite than (DCrγ and DNiγ) in the γ phase and that they facilitate the precipitation of σ phase in the third pass fusion zone. When the diffusion activation energy of Cr in δ-ferrite is equal to that of Ni in δ-ferrite (QdCrδ=QdNiδ), phase transformation of the δ → σ can be occurred.
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.
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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)
Radiative transfer equation modeling by streamline diffusion modified continuous Galerkin method
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.
Simulation of the radiolysis of water using Green's functions of the diffusion equation
International Nuclear Information System (INIS)
Radiation chemistry is of fundamental importance in the understanding of the effects of ionising radiation, notably with regard to DNA damage by indirect effect (e.g. damage by .OH radicals created by the radiolysis of water). In the recent years, Green's functions of the diffusion equation (GFDEs) have been used extensively in biochemistry, notably to simulate biochemical networks in time and space. In the present work, an approach based on the GFDE will be used to refine existing models on the indirect effect of ionising radiation on DNA. As a starting point, the code RITRACKS (relativistic ion tracks) will be used to simulate the radiation track structure and calculate the position of all radiolytic species formed during irradiation. The chemical reactions between these radiolytic species and with DNA will be done by using an efficient Monte Carlo sampling algorithm for the GFDE of reversible reactions with an intermediate state that has been developed recently. These simulations should help the understanding of the contribution of the indirect effect in the formation of DNA damage, particularly with regards to the formation of double-strand breaks. (authors)
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.
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The lowest order Nodal Integral Method (NIM) which belongs to a large class of nodal methods, the Lawrence-Dorning class, is written in a five-point, weighted-difference form and contrasted against the edge-centered Finite Difference Method (FDM). The final equations for the two methods exhibit three differences: the NIM employs almost three times as many discrete-variables (which are node- and surface-averaged values of the flux) as the FDM; the spatial weights in the NIM include hyperbolic functions opposed to the algebraic weights in the FDM; the NIM explicitly imposes continuity of the net current across cell edges. A homogeneous model problem is devised to enable an analytical study of the numerical solutions accuracy. The analysis shows that on a given mesh the FDM calculated fundamental mode eigenvalue is more accurate than that calculated by the NIM. However, the NIM calculated flux distribution is more accurate, especially when the problem size is several times as thick as the diffusion length. Numerical results for a nonhomogeneous test problem indicate the very high accuracy of the NIM for fixed source problems in such cases. 18 refs., 1 fig., 1 tab
Radiative transfer equation modeling by streamline diffusion modified continuous Galerkin method.
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
Vectorized and multi-tasked solutions of the two-group neutron diffusion equations
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A numerical algorithm with parallelism has been employed to solve the two-group, two-dimensional [X-Y] neutron diffusion equations on a FPS-164 attached scientific processor, IBM-3090/Model 200 with vector facilities mainframe computer and a CRAY X/MP-48 supercomputer. All three machines employ vector pipelines and the latter two computers also contain respectively 2 and 4 CPUs. Utilizing a parallel numerical algorithm, the FPS-164, IBM-3090/Model 200 and CRAY X/MP-48 [using 1 CPU] execute, respectively 5.5, 1.7 and 11.6 times faster in vector versus scalar mode for pin-wise mesh. Run time reductions versus an IBM-3081/Model K for the FPS-164, IBM-3090/Model 200 and CRAY X/MP-48 [using 1 CPU], respectively are 1.0, 4.7 and 27.5. The results from the preliminary work completed for multi-tasking on 2 CPUs of the CRAY X/MP-48 indicate a speedup of about 1.62 for just the Inner iteration code segment that was multi-tasked. The theoretical speedup of 2.0 is not obtained mainly because of vector pipelines' performance degradation due to the decrease in vector lengths associated with multi-tasking the Inner iteration
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In the context of nuclear waste repositories, we consider the numerical discretization of the non stationary convection diffusion equation. Discontinuous physical parameters and heterogeneous space and time scales lead us to use different space and time discretizations in different parts of the domain. In this work, we choose the discrete duality finite volume (DDFV) scheme and the discontinuous Galerkin scheme in time, coupled by an optimized Schwarz waveform relaxation (OSWR) domain decomposition method, because this allows the use of non-conforming space-time meshes. The main difficulty lies in finding an upwind discretization of the convective flux which remains local to a sub-domain and such that the multi domain scheme is equivalent to the mono domain one. These difficulties are first dealt with in the one-dimensional context, where different discretizations are studied. The chosen scheme introduces a hybrid unknown on the cell interfaces. The idea of up winding with respect to this hybrid unknown is extended to the DDFV scheme in the two-dimensional setting. The well-posedness of the scheme and of an equivalent multi domain scheme is shown. The latter is solved by an OSWR algorithm, the convergence of which is proved. The optimized parameters in the Robin transmission conditions are obtained by studying the continuous or discrete convergence rates. Several test-cases, one of which inspired by nuclear waste repositories, illustrate these results. (author)
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)
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.
Qualitative behavior of numerical traveling solutions for reaction–diffusion equations with memory
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.
On Fick's first law and the equations of quasi-static motion governing the molecular diffusion
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...
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)
Fundamental constraints on the abundances of chemotaxis proteins
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...
Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations
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.
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
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...
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...
Directory of Open Access Journals (Sweden)
Ahmad Golbabai
2011-12-01
Full Text Available In this paper, a family of high-order compact finite difference methods in combination preconditioned methods are used for solution of the Diffusion-Convection equation. We developed numerical methods by replacing the time and space derivatives by compact finite-difference approximations. The system of resulting nonlinear finite difference equations are solved by preconditioned Krylov subspace methods. Numerical results are given to verify the behavior of high-order compact approximations in combination preconditioned methods for stability, convergence. Also, the accuracy and efficiency of the proposed scheme are considered.
Feeding ducks, bacterial chemotaxis, and the Gini index
Peaudecerf, François J.; Goldstein, Raymond E.
2015-08-01
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 phenomenon we term "chemotactic levelling," and the influence of these results on population fitness are briefly considered.
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
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.
Convergence to a propagating front in a degenerate Fisher-KPP equation with advection
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.
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A semi linear model of weakly coupled parabolic p.d.e. with reaction-diffusion is investigated. The system describes fission gas transfer from grain interior of UO2 to grain boundaries. The problem is studied in a bounded domain. Using the upper-lower solutions method, two monotone sequences for the finite differences equations are constructed. Reasons are mentioned that allow to affirm that in the proposed functional sector the algorithm converges to the unique solution of the differential system. (author)
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This paper introduces a generic artificial viscosity method based on diffusing along iso-values (curves in 2D and surfaces in 3D). The construction and a study of properties of the method are presented. Application to FEM for the Euler and Navier-Stokes equations is established. The performance of the proposed method is demonstrated through numerical tests and comparison to other classical methods
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It is proposed to apply the fractional spatial derivatives for describing the effect of the fast electrons anomalous diffusion in the stochastic magnetic field on the distribution function form/ The self-simulating form of the kinetic equation is considered. Application of the self-simulating variables makes it possible to determine the velocities range, wherein the distribution function distortion becomes extremely strong. The accomplished calculations make it possible to evaluate the values, connected with the stochasticity of the magnetic power lines
International Nuclear Information System (INIS)
The modified Waldmann equation (MWE) has been applied to the separation of binary gas mixtures in a thermal diffusion column. However, the application is difficult for a mixture where the molecular weight and viscosity of each component differs greatly from each other. Therefore, a column was divided into 50 theoretical local columns, and the MWE calculation in each local column was repeated for the stack of 50. The composition distribution along the column can be determined by this method. (author)
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.
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.
International Nuclear Information System (INIS)
A computer program has been developed which uses a technique of synthetic acceleration by diffusion by analytical schemes. Both in the diffusion equation as in that of transport, analytical schemes were used which allowed a substantial time saving in the number of iterations required by source iteration method to obtain the Keff. The program developed ASD (Synthetic Diffusion Acceleration) by diffusion was written in FORTRAN and can be executed on a personal computer with a hard disc and mathematical O-processor. The program is unlimited as to the number of regions and energy groups. The results obtained by the ASD program for Keff is nearly completely concordant with those of obtained utilizing the ANISN-PC code for different analytical type problems in this work. The ASD program allowed obtention of an approximate solution of the neutron transport equation with a relatively low number of internal reiterations with good precision. One of its applications would be in the direct determinations of axial distribution neutronic flow in a fuel assembly as well as in the obtention of the effective multiplication factor. (Author)
Al-Chalabi, Rifat M. Khalil
1997-09-01
reconstruction methodology. The relaxation method, which is the power method, utilizes a constant coefficient matrix within the NEM non-linear iterative strategy. The choice of the MG nesting within the nested iterative strategy enables the incorporation of other non-linear effects with no additional coding effort. In addition, if an eigenvalue problem is being solved, it remains an eigenvalue problem at all grid levels, simplifying coding implementation. The merit of the developed MG method was tested by incorporating it into the NESTLE iterative solver, and employing it to solve four different benchmark problems. In addition to the base cases, three different sensitivity studies are performed, examining the effects of number of MG levels, homogenized coupling coefficients correction (i.e. restriction operator), and fine-mesh reconstruction algorithm (i.e. prolongation operator). The multilevel acceleration scheme developed in this research provides the foundation for developing adaptive multilevel acceleration methods for steady-state and transient NEM nodal neutron diffusion equations. (Abstract shortened by UMI.)
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.
Asymptotic Bayesian Estimation of a First Order Equation with Small Diffusion
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.
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...
International Nuclear Information System (INIS)
The objective of this work is to obtain an analytical solution of the neutron diffusion kinetic equation in one-dimensional cartesian geometry, to monoenergetic and multigroup problems. These equations are of the type stiff, due to large differences in the orders of magnitude of the time scales of the physical phenomena involved, which make them difficult to solve. The basic idea of the proposed method is applying the spectral expansion in the scalar flux and in the precursor concentration, taking moments and solving the resulting matrix problem by the Laplace transform technique. Bearing in mind that the equation for the precursor concentration is a first order linear differential equation in the time variable, to enable the application of the spectral method we introduce a fictitious diffusion term multiplied by a positive value which tends to zero. This procedure opened the possibility to find an analytical solution to the problem studied. We report numerical simulations and analysis of the results obtained with the precision controlled by the truncation order of the series. (author)
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
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.
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.
Positivity and the attractor dimension in a fourth-order reaction-diffusion equation
Bartuccelli, M.V.; Gourley, S.A.; A. A. Ilyin
2002-01-01
In this paper we investigate the semilinear partial differential equation ut = -fuxxxx - uxx + u(1 - u2) with a view, particularly, to obtaining some insight into how one might establish positivity preservation results for equations containing fourth-order spatial derivatives. The maximum principle cannot be applied to such equations. However, progress can be made by employing some very recent 'best possible' interpolation inequalities, due to the third-named author, in which the inte...
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.
The domain dependence of chemotaxis in two-dimensional turbulence
Tang, Wenbo; Jones, Kimberly; Walker, Phillip
2015-11-01
Coherent structures are ubiquitous in environmental and geophysical flows and they affect reaction-diffusion processes in profound ways. In this presentation, we show an example of the domain dependence of chemotaxis process in a two-dimensional turbulent flow. The flow has coherent structures that form barriers that prohibit long-range transport of tracers. Accordingly, the uptake advantage of nutrient by motile and nonmotile species differs significantly if the process start in different locations of the flow. Interestingly, the conventional diagnostic of Finite-time Lyapunov exponents alone is not sufficient to explain the variability -- methods to extract elliptic transport barriers are essential to relate to the explanation. We also offer some explanations of the observed scalar behaviors via analyses of bulk quantities. Support: NSF-DMS-1212144.
Toward Synthetic Spatial Patterns in Engineered Cell Populations with Chemotaxis.
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
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.
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
Lu, Benzhuo; Zhou, Y C
2011-05-18
The effects of finite particle size on electrostatics, density profiles, and diffusion have been a long existing topic in the study of ionic solution. The previous size-modified Poisson-Boltzmann and Poisson-Nernst-Planck models are revisited in this article. In contrast to many previous works that can only treat particle species with a single uniform size or two sizes, we generalize the Borukhov model to obtain a size-modified Poisson-Nernst-Planck (SMPNP) model that is able to treat nonuniform particle sizes. The numerical tractability of the model is demonstrated as well. The main contributions of this study are as follows. 1), We show that an (arbitrarily) size-modified PB model is indeed implied by the SMPNP equations under certain boundary/interface conditions, and can be reproduced through numerical solutions of the SMPNP. 2), The size effects in the SMPNP effectively reduce the densities of highly concentrated counterions around the biomolecule. 3), The SMPNP is applied to the diffusion-reaction process for the first time, to our knowledge. In the case of low substrate density near the enzyme reactive site, it is observed that the rate coefficients predicted by SMPNP model are considerably larger than those by the PNP model, suggesting both ions and substrates are subject to finite size effects. 4), An accurate finite element method and a convergent Gummel iteration are developed for the numerical solution of the completely coupled nonlinear system of SMPNP equations. PMID:21575582
Negative chemotaxis does not control quail neural crest cell dispersion.
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