Lie group invariant finite difference schemes for the neutron diffusion equation
Energy Technology Data Exchange (ETDEWEB)
Jaegers, P.J.
1994-06-01
Finite difference techniques are used to solve a variety of differential equations. For the neutron diffusion equation, the typical local truncation error for standard finite difference approximation is on the order of the mesh spacing squared. To improve the accuracy of the finite difference approximation of the diffusion equation, the invariance properties of the original differential equation have been incorporated into the finite difference equations. Using the concept of an invariant difference operator, the invariant difference approximations of the multi-group neutron diffusion equation were determined in one-dimensional slab and two-dimensional Cartesian coordinates, for multiple region problems. These invariant difference equations were defined to lie upon a cell edged mesh as opposed to the standard difference equations, which lie upon a cell centered mesh. Results for a variety of source approximations showed that the invariant difference equations were able to determine the eigenvalue with greater accuracy, for a given mesh spacing, than the standard difference approximation. The local truncation errors for these invariant difference schemes were found to be highly dependent upon the source approximation used, and the type of source distribution played a greater role in determining the accuracy of the invariant difference scheme than the local truncation error.
A Four Group Reference Code for Solving Neutron Diffusion Equation in a VVER-440 Core
Energy Technology Data Exchange (ETDEWEB)
Saarinen, Simo [Fortum Nuclear Services Ltd., P.O. Box 100, 00048 Fortum (Finland)
2008-07-01
Nuclear reactor core power calculation is essential in the analysis of the nuclear power plant and especially the core. Currently, the core power distribution in Loviisa VVER-440 core is calculated using nodal code HEXBU-3D and pin-power reconstruction code ELSI-1440 that solve the two group neutron diffusion equation. The computer power available has increased significantly during the last decades allowing us to develop a fine mesh code HEXRE for solving the four group diffusion equation. The diffusion equations are discretized using piecewise linear polynomials. The core is discretized using one node per fuel pin cell. The axial discretization can be chosen freely. The boundary conditions are described using diffusion theory and albedos. Burnup dependence is modelled by tabulating diffusion parameters at certain burnup values and using interpolation for the intermediate values. A two degree polynomial is used for the modelling of the feedback effects. Eigenvalue calculation for both boron concentration and multiplication factor control has been formulated. A possibility to perform fuel loading and shuffling operations is implemented. HEXRE has been thoroughly compared with HEXBU-3D and ELSI-1440. The effect of the different energy and space discretizations used is investigated. Some safety criteria for the core calculated with the HEXRE and HEXBU-3D/ELSI-1440 have been compared. From the calculations (e.g. the safety criteria) we can estimate whether there exists systematic deviations in HEXBU- 3D/ELSI-1440 calculations or not. (author)
Wu Zhuo Qun; Li Hui Lai; Zhao Jun Ning
2001-01-01
Nonlinear diffusion equations, an important class of parabolic equations, come from a variety of diffusion phenomena which appear widely in nature. They are suggested as mathematical models of physical problems in many fields, such as filtration, phase transition, biochemistry and dynamics of biological groups. In many cases, the equations possess degeneracy or singularity. The appearance of degeneracy or singularity makes the study more involved and challenging. Many new ideas and methods have been developed to overcome the special difficulties caused by the degeneracy and singularity, which
Simulate-HEX - The multi-group diffusion equation in hexagonal-z geometry
Energy Technology Data Exchange (ETDEWEB)
Lindahl, S. O. [Studsvik Scandpower, Stensborgsg. 4, SE-72132 Vaesteraes (Sweden)
2013-07-01
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)
Directory of Open Access Journals (Sweden)
K. Banoo
1998-01-01
equation in the discrete momentum space. This is shown to be similar to the conventional drift-diffusion equation except that it is a more rigorous solution to the Boltzmann equation because the current and carrier densities are resolved into M×1 vectors, where M is the number of modes in the discrete momentum space. The mobility and diffusion coefficient become M×M matrices which connect the M momentum space modes. This approach is demonstrated by simulating electron transport in bulk silicon.
Nonlocal electrical diffusion equation
Gómez-Aguilar, J. F.; Escobar-Jiménez, R. F.; Olivares-Peregrino, V. H.; Benavides-Cruz, M.; Calderón-Ramón, C.
2016-07-01
In this paper, we present an analysis and modeling of the electrical diffusion equation using the fractional calculus approach. This alternative representation for the current density is expressed in terms of the Caputo derivatives, the order for the space domain is 0numerical methods based on Fourier variable separation. The case with spatial fractional derivatives leads to Levy flight type phenomena, while the time fractional equation is related to sub- or super diffusion. We show that the mathematical concept of fractional derivatives can be useful to understand the behavior of semiconductors, the design of solar panels, electrochemical phenomena and the description of anomalous complex processes.
Fractional Chemotaxis Diffusion Equations
Langlands, T A M
2010-01-01
We introduce mesoscopic and macroscopic model equations of chemotaxis with anomalous subdiffusion for modelling chemically directed transport of biological organisms in changing chemical environments with diffusion hindered by traps or macro-molecular crowding. The mesoscopic models are formulated using Continuous Time Random Walk master equations and the macroscopic models are formulated with fractional order differential equations. Different models are proposed depending on the timing of the chemotactic forcing. Generalizations of the models to include linear reaction dynamics are also derived. Finally a Monte Carlo method for simulating anomalous subdiffusion with chemotaxis is introduced and simulation results are compared with numerical solutions of the model equations. The model equations developed here could be used to replace Keller-Segel type equations in biological systems with transport hindered by traps, macro-molecular crowding or other obstacles.
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.
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
The single-group,steadystate,isotropic for mofthe neutron transport equationis given by[1]Ω·+σtI-σsPψ(x,Ω)=q(x,Ω)(x,Ω)∈D×Sψ(x,Ω)=g(x,Ω)x∈Din={x∈D,γ(x)·Ω<0(1)whereσtis the total cross section,σSis the scatteringcross section,andψ(x,Ω)is the angular flux to bedeter mined for all pointsx∈D,D Rn(n=2,3)and all possible travel directionsΩ,ΩS(Sis a u-nit disk or a unit sphere),γ(x)denotes the out wardunit nor mal atx∈D,Idenotes the identity opera-tor,the operatorPis defined by[Pψ](x)=∫Sψ(x,Ω)dΩ(2)Whenσt→∞,andσσ...
Anomalous Fractional Diffusion Equation for Transport Phenomena
Institute of Scientific and Technical Information of China (English)
QiuhuaZENG; HouqiangLI; 等
1999-01-01
We derive the standard diffusion equation from the continuity equation and by discussing the defectiveness of earlier proposed equations,we get the generalized fractional diffusion equation for anomalous diffusion.
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.
Singular Renormalization Group Equations
Minoru, HIRAYAMA; Department of Physics, Toyama University
1984-01-01
The possible behaviour of the effective charge is discussed in Oehme and Zimmermann's scheme of the renormalization group equation. The effective charge in an example considered oscillates so violently in the ultraviolet limit that the bare charge becomes indefinable.
Fractional-calculus diffusion equation
Ajlouni, Abdul-Wali MS; Al-Rabai'ah, Hussam A
2010-01-01
Background Sequel to the work on the quantization of nonconservative systems using fractional calculus and quantization of a system with Brownian motion, which aims to consider the dissipation effects in quantum-mechanical description of microscale systems. Results The canonical quantization of a system represented classically by one-dimensional Fick's law, and the diffusion equation is carried out according to the Dirac method. A suitable Lagrangian, and Hamiltonian, describing the diffusive...
Energy Technology Data Exchange (ETDEWEB)
Ceolin, C.; Schramm, M.; Vilhena, M.T.; Bodmann, B.E.J., E-mail: celina.ceolin@gmail.com, E-mail: marceloschramm@hotmail.com, E-mail: vilhena@pq.cnpq.br, E-mail: bardo.bodmann@ufrgs.br [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica
2013-07-01
In this work the authors solved the steady state neutron diffusion equation for a multi-layer slab assuming the multi-group energy model. The method to solve the equation system is based on a expansion in Taylor Series, which was proven to be useful in [1] [2] [3]. The results obtained can be used as initial condition for neutron space kinetics problems. The neutron scalar flux was expanded in a power series, and the coefficients were found by using the ordinary differential equation and the boundary and interface conditions. The effective multiplication factor k was evaluated using the power method [4]. We divided the domain into several slabs to guarantee the convergence with a low truncation order. We present the formalism together with some numerical simulations. (author)
Energy Technology Data Exchange (ETDEWEB)
Ceolin, Celina; Schramm, Marcelo; Bodmann, Bardo Ernst Josef; Vilhena, Marco Tullio Mena Barreto de [Universidade Federal do Rio Grande do Sul, Porto Alegre (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica; Bogado Leite, Sergio de Queiroz [Comissao Nacional de Energia Nuclear, Rio de Janeiro (Brazil)
2014-11-15
In this work the authors solved the steady state neutron diffusion equation for a multi-layer slab assuming the multi-group energy model. The method to solve the equation system is based on an expansion in Taylor Series resulting in an analytical expression. The results obtained can be used as initial condition for neutron space kinetics problems. The neutron scalar flux was expanded in a power series, and the coefficients were found by using the ordinary differential equation and the boundary and interface conditions. The effective multiplication factor k was evaluated using the power method. We divided the domain into several slabs to guarantee the convergence with a low truncation order. We present the formalism together with some numerical simulations.
Riemann equation for prime number diffusion.
Chen, Wen; Liang, Yingjie
2015-05-01
This study makes the first attempt to propose the Riemann diffusion equation to describe in a manner of partial differential equation and interpret in physics of diffusion the classical Riemann method for prime number distribution. The analytical solution of this equation is the well-known Riemann representation. The diffusion coefficient is dependent on natural number, a kind of position-dependent diffusivity diffusion. We find that the diffusion coefficient of the Riemann diffusion equation is nearly a straight line having a slope 0.99734 in the double-logarithmic axis. Consequently, an approximate solution of the Riemann diffusion equation is obtained, which agrees well with the Riemann representation in predicting the prime number distribution. Moreover, we interpret the scale-free property of prime number distribution via a power law function with 1.0169 the scale-free exponent in respect to logarithmic transform of the natural number, and then the fractal characteristic of prime number distribution is disclosed.
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.
Symmetry classification of time-fractional diffusion equation
Naeem, I.; Khan, M. D.
2017-01-01
In this article, a new approach is proposed to construct the symmetry groups for a class of fractional differential equations which are expressed in the modified Riemann-Liouville fractional derivative. We perform a complete group classification of a nonlinear fractional diffusion equation which arises in fractals, acoustics, control theory, signal processing and many other applications. Introducing the suitable transformations, the fractional derivatives are converted to integer order derivatives and in consequence the nonlinear fractional diffusion equation transforms to a partial differential equation (PDE). Then the Lie symmetries are computed for resulting PDE and using inverse transformations, we derive the symmetries for fractional diffusion equation. All cases are discussed in detail and results for symmetry properties are compared for different values of α. This study provides a new way of computing symmetries for a class of fractional differential equations.
Group analysis of differential equations
Ovsiannikov, L V
1982-01-01
Group Analysis of Differential Equations provides a systematic exposition of the theory of Lie groups and Lie algebras and its application to creating algorithms for solving the problems of the group analysis of differential equations.This text is organized into eight chapters. Chapters I to III describe the one-parameter group with its tangential field of vectors. The nonstandard treatment of the Banach Lie groups is reviewed in Chapter IV, including a discussion of the complete theory of Lie group transformations. Chapters V and VI cover the construction of partial solution classes for the g
The Riccati Differential Equation and a Diffusion-Type Equation
Suazo, Erwin; Vega-Guzman, Jose M
2008-01-01
We construct an explicit solution of the Cauchy initial value problem for certain diffusion-type equation with variable coefficients on the entire real line. The corresponding Green function (heat kernel) is given in terms of elementary functions and certain integrals involving a characteristic function, which should be found as an analytic or numerical solution of the second order linear differential equation with time-dependent coefficients. Some special and limiting cases are outlined. Solution of the corresponding nonhomogeneous equation is also found.
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.
A Diffusion Equation for Quantum Adiabatic Systems
Jain, S R
1998-01-01
For ergodic adiabatic quantum systems, we study the evolution of energy distribution as the system evolves in time. Starting from the von Neumann equation for the density operator, we obtain the quantum analogue of the Smoluchowski equation on coarse-graining over the energy spectrum. This result brings out the precise notion of quantum diffusion.
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.
Fractional diffusion equation for heterogeneous medium
Energy Technology Data Exchange (ETDEWEB)
Polo L, M. A.; Espinosa M, E. G.; Espinosa P, G. [Universidad Autonoma Metropolitana, Unidad Iztapalapa, Area de Ingenieria en Recursos Energeticos, Av, San Rafael Atlixco 186, Col. Vicentina, 09340 Mexico D. F. (Mexico); Del Valle G, E., E-mail: plabarrios@hotmail.com [Instituto Politecnico Nacional, Escuela Superior de Fisica y Matematicas, Av. IPN s/n, Col. San Pedro Zacatenco, 07738 Mexico D. F. (Mexico)
2011-11-15
The asymptotic diffusion approximation for the Boltzmann (transport) equation was developed in 1950 decade in order to describe the diffusion of a particle in an isotropic medium, considers that the particles have a diffusion infinite velocity. In this work is developed a new approximation where is considered that the particles have a finite velocity, with this model is possible to describe the behavior in an anomalous medium. According with these ideas the model was obtained from the Fick law, where is considered that the temporal term of the current vector is not negligible. As a result the diffusion equation of fractional order which describes the dispersion of particles in a highly heterogeneous or disturbed medium is obtained, i.e., in a general medium. (Author)
Langevin Equations for Reaction-Diffusion Processes
Benitez, Federico; Duclut, Charlie; Chaté, Hugues; Delamotte, Bertrand; Dornic, Ivan; Muñoz, Miguel A.
2016-09-01
For reaction-diffusion processes with at most bimolecular reactants, we derive well-behaved, numerically tractable, exact Langevin equations that govern a stochastic variable related to the response field in field theory. Using duality relations, we show how the particle number and other quantities of interest can be computed. Our work clarifies long-standing conceptual issues encountered in field-theoretical approaches and paves the way for systematic numerical and theoretical analyses of reaction-diffusion problems.
Approximating parameters in nonlinear reaction diffusion equations
Directory of Open Access Journals (Sweden)
Robert R. Ferdinand
2001-07-01
Full Text Available We present a model describing population dynamics in an environment. The model is a nonlinear, nonlocal, reaction diffusion equation with Neumann boundary conditions. An inverse method, involving minimization of a least-squares cost functional, is developed to identify unknown model parameters. Finally, numerical results are presented which display estimates of these parameters using computationally generated data.
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.
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.
Diffusion-equation method for crystallographic figure of merits.
Markvardsen, Anders J; David, William I F
2010-09-01
Global optimization methods play a significant role in crystallography, particularly in structure solution from powder diffraction data. This paper presents the mathematical foundations for a diffusion-equation-based optimization method. The diffusion equation is best known for describing how heat propagates in matter. However, it has also attracted considerable attention as the basis for global optimization of a multimodal function [Piela et al. (1989). J. Phys. Chem. 93, 3339-3346]. The method relies heavily on available analytical solutions for the diffusion equation. Here it is shown that such solutions can be obtained for two important crystallographic figure-of-merit (FOM) functions that fully account for space-group symmetry and allow the diffusion-equation solution to vary depending on whether atomic coordinates are fixed or not. The resulting expression is computationally efficient, taking the same order of floating-point operations to evaluate as the starting FOM function measured in terms of the number of atoms in the asymmetric unit. This opens the possibility of implementing diffusion-equation methods for crystallographic global optimization algorithms such as structure determination from powder diffraction data.
Logarithmic diffusion and porous media equations: a unified description.
Pedron, I T; Mendes, R S; Buratta, T J; Malacarne, L C; Lenzi, E K
2005-09-01
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 superdiffusion like a Lévy kind. In addition an equation that unifies the porous media and the logarithmic diffusion equations, including a generalized diffusion equation in fractal dimension, is obtained. This unification is performed in the nonextensive thermostatistics context and increases the possibilities about the description of anomalous diffusive processes.
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.
Amplitude equations for isothermal double diffusive convection
Energy Technology Data Exchange (ETDEWEB)
Becerril, R.; Swift, J.B. [Center for Nonlinear Dynamics and Department of Physics, University of Texas, Austin, Texas 78712 (United States)
1997-05-01
Amplitude equations are derived for isothermal double diffusive convection near threshold for both the stationary and oscillatory instabilities as well as in the vicinity of the codimension-2 point. The convecting fluid is contained in a thin Hele-Shaw cell that renders the system two dimensional, and convection is sustained by vertical concentration gradients of two species with different diffusion rates. The locations of the tricritical point for the stationary instability and the codimension-2 point are found. It is shown that these points can be made well separated (in the Rayleigh number R{sub s} of the slow diffusing species) as the Lewis number varies. Hence the behavior near these points should be experimentally accessible. {copyright} {ital 1997} {ital The American Physical Society}
The Lie algebra of infinitesimal symmetries of nonlinear diffusion equations
Kersten, Paul H.M.; Gragert, Peter K.H.
1983-01-01
By using developed software for solving overdetermined systems of partial differential equations, the authors establish the complete Lie algebra of infinitesimal symmetries of nonlinear diffusion equations.
Energy Technology Data Exchange (ETDEWEB)
Petersen, Claudio Z. [Universidade Federal de Pelotas, Capao do Leao (Brazil). Programa de Pos Graduacao em Modelagem Matematica; Bodmann, Bardo E.J.; Vilhena, Marco T. [Universidade Federal do Rio Grande do Sul, Porto Alegre, RS (Brazil). Programa de Pos-graduacao em Engenharia Mecanica; Barros, Ricardo C. [Universidade do Estado do Rio de Janeiro, Nova Friburgo, RJ (Brazil). Inst. Politecnico
2014-12-15
In the present work we solve in analytical representation the three dimensional neutron kinetic diffusion problem in rectangular Cartesian geometry for homogeneous and bounded domains for any number of energy groups and precursor concentrations. The solution in analytical representation is constructed using a hierarchical procedure, i.e. the original problem is reduced to a problem previously solved by the authors making use of a combination of the spectral method and a recursive decomposition approach. Time dependent absorption cross sections of the thermal energy group are considered with step, ramp and Chebyshev polynomial variations. For these three cases, we present numerical results and discuss convergence properties and compare our results to those available in the literature.
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.
FRACTIONAL DIFFUSION EQUATIONS WITH INTERNAL DEGREES OF FREEDOM
Institute of Scientific and Technical Information of China (English)
Luis Vázquez
2003-01-01
We present a generalization of the linear one-dimensional diffusion equation by combining the fractional derivatives and the internal degrees of freedom. The solutions are constructed from those of the scalar fractional diffusion equation. We analyze the interpolation between the standard diffusion and wave equations defined by the fractional derivatives. Our main result is that we can define a diffusion process depending on the internal degrees of freedom associated to the system.
Nonlocal diffusion second order partial differential equations
Benedetti, I.; Loi, N. V.; Malaguti, L.; Taddei, V.
2017-02-01
The paper deals with a second order integro-partial differential equation in Rn with a nonlocal, degenerate diffusion term. Nonlocal conditions, such as the Cauchy multipoint and the weighted mean value problem, are investigated. The existence of periodic solutions is also studied. The dynamic is transformed into an abstract setting and the results come from an approximation solvability method. It combines a Schauder degree argument with an Hartman-type inequality and it involves a Scorza-Dragoni type result. The compact embedding of a suitable Sobolev space in the corresponding Lebesgue space is the unique amount of compactness which is needed in this discussion. The solutions are located in bounded sets and they are limits of functions with values in finitely dimensional spaces.
Lattice Boltzmann model for nonlinear convection-diffusion equations.
Shi, Baochang; Guo, Zhaoli
2009-01-01
A lattice Boltzmann model for convection-diffusion equation with nonlinear convection and isotropic-diffusion terms is proposed through selecting equilibrium distribution function properly. The model can be applied to the common real and complex-valued nonlinear evolutionary equations, such as the nonlinear Schrödinger equation, complex Ginzburg-Landau equation, Burgers-Fisher equation, nonlinear heat conduction equation, and sine-Gordon equation, by using a real and complex-valued distribution function and relaxation time. Detailed simulations of these equations are performed, and it is found that the numerical results agree well with the analytical solutions and the numerical solutions reported in previous studies.
Distributed-order diffusion equations and multifractality: Models and solutions
Sandev, Trifce; Chechkin, Aleksei V.; Korabel, Nickolay; Kantz, Holger; Sokolov, Igor M.; Metzler, Ralf
2015-10-01
We study distributed-order time fractional diffusion equations characterized by multifractal memory kernels, in contrast to the simple power-law kernel of common time fractional diffusion equations. Based on the physical approach to anomalous diffusion provided by the seminal Scher-Montroll-Weiss continuous time random walk, we analyze both natural and modified-form distributed-order time fractional diffusion equations and compare the two approaches. The mean squared displacement is obtained and its limiting behavior analyzed. We derive the connection between the Wiener process, described by the conventional Langevin equation and the dynamics encoded by the distributed-order time fractional diffusion equation in terms of a generalized subordination of time. A detailed analysis of the multifractal properties of distributed-order diffusion equations is provided.
Entropy Solution Theory for Fractional Degenerate Convection-Diffusion Equations
Jakobsen, Simone Cifani And Espen R
2010-01-01
We study a class of degenerate convection diffusion equations with a fractional nonlinear diffusion term. These equations are natural generalizations of anomalous diffusion equations, fractional conservations laws, local convection diffusion equations, and some fractional Porous medium equations. In this paper we define weak entropy solutions for this class of equations and prove well-posedness under weak regularity assumptions on the solutions, e.g. uniqueness is obtained in the class of bounded integrable functions. Then we introduce a monotone conservative numerical scheme and prove convergence toward an Entropy solution in the class of bounded integrable functions of bounded variation. We then extend the well-posedness results to non-local terms based on general L\\'evy type operators, and establish some connections to fully non-linear HJB equations. Finally, we present some numerical experiments to give the reader an idea about the qualitative behavior of solutions of these equations.
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 ...
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
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)
QUENCHING PROBLEMS OF DEGENERATE FUNCTIONAL REACTION-DIFFUSION EQUATION
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
This paper is concerned with the quenching problem of a degenerate functional reaction-diffusion equation. The quenching problem and global existence of solution for the reaction-diffusion equation are derived and, some results of the positive steady state solutions for functional elliptic boundary value are also presented.
Perturbation theory and renormalisation group equations
Litim, Daniel F; Litim, Daniel F.; Pawlowski, Jan M.
2002-01-01
We discuss the perturbative expansion of several one-loop improved renormalisation group equations. It is shown that in general the integrated renormalisation group flows fail to reproduce perturbation theory beyond one loop.
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.
The finite element method solution of variable diffusion coefficient convection-diffusion equations
Aydin, Selçuk Han; ćiftçi, Canan
2012-08-01
Mathematical modeling of many physical and engineering problems is defined with convection-diffusion equation. Therefore, there are many analytic and numeric studies about convection-diffusion equation in literature. The finite element method is the most preferred numerical method in these studies since it can be applied to many problems easily. But, most of the studies in literature are about constant coefficient case of the convection-diffusion equation. In this study, the finite element formulation of the variable coefficient case of the convection-diffusion equation is given in both one and two dimensional cases. Accuracy of the obtained formulations are tested on some problems in one and two dimensions.
Renormalization group and linear integral equations
Klein, W.
1983-04-01
We develop a position-space renormalization-group transformation which can be employed to study general linear integral equations. In this Brief Report we employ our method to study one class of such equations pertinent to the equilibrium properties of fluids. The results of applying our method are in excellent agreement with known numerical calculations where they can be compared. We also obtain information about the singular behavior of this type of equation which could not be obtained numerically.
Energy Technology Data Exchange (ETDEWEB)
Mugica R, C.A. [IPN, ESFM, Depto. de Ingenieria Nuclear, 07738 Mexico D.F. (Mexico)
2004-07-01
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)
The Riccati System and a Diffusion-Type Equation
Directory of Open Access Journals (Sweden)
Erwin Suazo
2014-05-01
Full Text Available We discuss a method of constructing solutions of the initial value problem for diffusion-type equations in terms of solutions of certain Riccati and Ermakov-type systems. A nonautonomous Burgers-type equation is also considered. Examples include, but are not limited to the Fokker-Planck equation in physics, the Black-Scholes equation and the Hull-White model in finance.
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.
Monotone method for initial value problem for fractional diffusion equation
Institute of Scientific and Technical Information of China (English)
ZHANG Shuqin
2006-01-01
Using the method of upper and lower solutions and its associated monotone iterative, consider the existence and uniqueness of solution of an initial value problem for the nonlinear fractional diffusion equation.
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.
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.
BOUNDARY LAYER AND VANISHING DIFFUSION LIMIT FOR NONLINEAR EVOLUTION EQUATIONS
Institute of Scientific and Technical Information of China (English)
彭艳
2014-01-01
In this paper, we consider an initial-boundary value problem for some nonlinear evolution equations with damping and diffusion. The main purpose is to investigate the boundary layer effect and the convergence rates as the diffusion parameterαgoes to zero.
Fluctuating lattice Boltzmann method for the diffusion equation.
Wagner, Alexander J; Strand, Kyle
2016-09-01
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.
The nonexistence of the linear diffusion equation beyond Fick's law
Schepper, I.M. de; Beyeren, H. van
1974-01-01
The self-diffusion of a tagged particle in a 3-dimensional fluid of identical particles cannot be described by a linear diffusion equation which contains corrections to Fick's law proportional to 4n, 6n, … For long times a t divergence is found for the super-Burnett coefficient, the proportionality
The Nonexistence of the Linear Diffusion Equation Beyond Fick's Law
Schepper, I.M. de; Beijeren, H. van; Ernst, M.H.
1974-01-01
The self-diffusion of a tagged particle in a 3-dimensional fluid of identical particles cannot be described by a linear diffusion equation which contains corrections to Fick's law proportional to V4n, V6n .... For long times a t 1/2 divergence is found for the super-Burnett coefficient, the proporti
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.
Linear integral equations and renormalization group
Klein, W.; Haymet, A. D. J.
1984-08-01
A formulation of the position-space renormalization-group (RG) technique is used to analyze the singular behavior of solutions to a number of integral equations used in the theory of the liquid state. In particular, we examine the truncated Kirkwood-Salsburg equation, the Ornstein-Zernike equation, and a simple nonlinear equation used in the mean-field theory of liquids. We discuss the differences in applying the position-space RG to lattice systems and to fluids, and the need for an explicit free-energy rescaling assumption in our formulation of the RG for integral equations. Our analysis provides one natural way to define a "fractal" dimension at a phase transition.
Diffusions, superdiffusions and partial differential equations
Dynkin, E B
2002-01-01
Interactions between the theory of partial differential equations of elliptic and parabolic types and the theory of stochastic processes are beneficial for both probability theory and analysis. At the beginning, mostly analytic results were used by probabilists. More recently, analysts (and physicists) took inspiration from the probabilistic approach. Of course, the development of analysis in general and of the theory of partial differential equations in particular, was motivated to a great extent by problems in physics. A difference between physics and probability is that the latter provides
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.
TRAJECTORY ATTRACTORS FOR NONCLASSICAL DIFFUSION EQUATIONS WITH FADING MEMORY
Institute of Scientific and Technical Information of China (English)
Yonghai WANG; Lingzhi WANG
2013-01-01
In this article,we consider the existence of trajectory and global attractors for nonclassical diffusion equations with linear fading memory.For this purpose,we will apply the method presented by Chepyzhov and Miranville [7,8],in which the authors provide some new ideas in describing the trajectory attractors for evolution equations with memory.
A new technology for solving diffusion and heat equations
Directory of Open Access Journals (Sweden)
Yang Xiao-Jun
2017-01-01
Full Text Available In this paper, a new technology combing the variational iterative method and an integral transform similar to Sumudu transform is proposed for the first time for solutions of diffusion and heat equations. The method is accurate and efficient in development of approximate solutions for the partial differential equations.
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.
Non-Markovian diffusion equations and processes: analysis and simulations
Mura, Antonio; Mainardi, Francesco
2007-01-01
In this paper we introduce and analyze a class of diffusion type equations related to certain non-Markovian stochastic processes. We start from the forward drift equation which is made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation can be interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time-evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the memory kernel K(t). We develop several applications and derive the exact solutions. We consider different stochastic models for the given equations providing path simulations.
Langevin equation approach to diffusion magnetic resonance imaging.
Cooke, Jennie M; Kalmykov, Yuri P; Coffey, William T; Kerskens, Christian M
2009-12-01
The normal phase diffusion problem in magnetic resonance imaging (MRI) is treated by means of the Langevin equation for the phase variable using only the properties of the characteristic function of Gaussian random variables. The calculation may be simply extended to anomalous diffusion using a fractional generalization of the Langevin equation proposed by Lutz [E. Lutz, Phys. Rev. E 64, 051106 (2001)] pertaining to the fractional Brownian motion of a free particle coupled to a fractal heat bath. The results compare favorably with diffusion-weighted experiments acquired in human neuronal tissue using a 3 T MRI scanner.
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.
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.
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.
An explicit high order method for fractional advection diffusion equations
Sousa, Ercília
2014-12-01
We propose a high order explicit finite difference method for fractional advection diffusion equations. These equations can be obtained from the standard advection diffusion equations by replacing the second order spatial derivative by a fractional operator of order α with 1convergence of the numerical method through consistency and stability. The order of convergence varies between two and three and for advection dominated flows is close to three. Although the method is conditionally stable, the restrictions allow wide stability regions. The analysis is confirmed by numerical examples.
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.
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.
Singular solutions of the diffusion equation of population genetics.
McKane, A J; Waxman, D
2007-08-21
The forward diffusion equation for gene frequency dynamics is solved subject to the condition that the total probability is conserved at all times. This can lead to solutions developing singular spikes (Dirac delta functions) at the gene frequencies 0 and 1. When such spikes appear in solutions they signal gene loss or gene fixation, with the "weight" associated with the spikes corresponding to the probability of loss or fixation. The forward diffusion equation is thus solved for all gene frequencies, namely the absorbing frequencies of 0 and 1 along with the continuous range of gene frequencies on the interval (0,1) that excludes the frequencies of 0 and 1. Previously, the probabilities of the absorbing frequencies of 0 and 1 were found by appeal to the backward diffusion equation, while those in the continuous range (0,1) were found from the forward diffusion equation. Our unified approach does not require two separate equations for a complete dynamical treatment of all gene frequencies within a diffusion approximation framework. For cases involving mutation, migration and selection, it is shown that a property of the deterministic part of gene frequency dynamics determines when fixation and loss can occur. It is also shown how solution of the forward equation, at long times, leads to the standard result for the fixation probability.
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.
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.
Stagewise generalized estimating equations with grouped variables.
Vaughan, Gregory; Aseltine, Robert; Chen, Kun; Yan, Jun
2017-02-13
Forward stagewise estimation is a revived slow-brewing approach for model building that is particularly attractive in dealing with complex data structures for both its computational efficiency and its intrinsic connections with penalized estimation. Under the framework of generalized estimating equations, we study general stagewise estimation approaches that can handle clustered data and non-Gaussian/non-linear models in the presence of prior variable grouping structure. As the grouping structure is often not ideal in that even the important groups may contain irrelevant variables, the key is to simultaneously conduct group selection and within-group variable selection, that is, bi-level selection. We propose two approaches to address the challenge. The first is a bi-level stagewise estimating equations (BiSEE) approach, which is shown to correspond to the sparse group lasso penalized regression. The second is a hierarchical stagewise estimating equations (HiSEE) approach to handle more general hierarchical grouping structure, in which each stagewise estimation step itself is executed as a hierarchical selection process based on the grouping structure. Simulation studies show that BiSEE and HiSEE yield competitive model selection and predictive performance compared to existing approaches. We apply the proposed approaches to study the association between the suicide-related hospitalization rates of the 15-19 age group and the characteristics of the school districts in the State of Connecticut.
Diffusion Equations, Quantum Fields and Fundamental Interactions
Directory of Open Access Journals (Sweden)
Tosto S.
2015-04-01
Full Text Available The paper concerns an “ab initio” theoretical model based on the space-time quantum uncertainty and aimed to identify the conceptual root common to all four fundamental interactions known in nature. The essential information that identifies unambiguously each kind of interaction is inferred in a straightforward way via simple considerations involving the diffusion laws. The conceptual frame of the model is still that introduced in previous papers, where the basic statements of the relativity and wave mechanics have been contextually obtained as corollaries of the quantum uncertainty.
Image segmentation and edge enhancement with stabilized inverse diffusion equations.
Pollak, I; Willsky, A S; Krim, H
2000-01-01
We introduce a family of first-order multidimensional ordinary differential equations (ODEs) with discontinuous right-hand sides and demonstrate their applicability in image processing. An equation belonging to this family is an inverse diffusion everywhere except at local extrema, where some stabilization is introduced. For this reason, we call these equations "stabilized inverse diffusion equations" (SIDEs). Existence and uniqueness of solutions, as well as stability, are proven for SIDEs. A SIDE in one spatial dimension may be interpreted as a limiting case of a semi-discretized Perona-Malik equation. In an experiment, SIDE's are shown to suppress noise while sharpening edges present in the input signal. Their application to image segmentation is also demonstrated.
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
Multi-diffusive nonlinear Fokker-Planck equation
Ribeiro, Mauricio S.; Casas, Gabriela A.; Nobre, Fernando D.
2017-02-01
Nonlinear Fokker-Planck equations, characterized by more than one diffusion term, have appeared recently in literature. Here, it is shown that these equations may be derived either from approximations in a master equation, or from a Langevin-type approach. An H-theorem is proven, relating these Fokker-Planck equations to an entropy composed by a sum of contributions, each of them associated with a given diffusion term. Moreover, the stationary state of the Fokker-Planck equation is shown to coincide with the equilibrium state, obtained by extremization of the entropy, in the sense that both procedures yield precisely the same equation. Due to the nonlinear character of this equation, the equilibrium probability may be obtained, in most cases, only by means of numerical approaches. Some examples are worked out, where the equilibrium probability distribution is computed for nonlinear Fokker-Planck equations presenting two diffusion terms, corresponding to an entropy characterized by a sum of two contributions. It is shown that the resulting equilibrium distribution, in general, presents a form that differs from a sum of the equilibrium distributions that maximizes each entropic contribution separately, although in some cases one may construct such a linear combination as a good approximation for the equilibrium distribution.
Construction of Difference Equations Using Lie Groups
Energy Technology Data Exchange (ETDEWEB)
Axford, R.A.
1998-08-01
The theory of prolongations of the generators of groups of point transformations to the grid point values of dependent variables and grid spacings is developed and applied to the construction of group invariant numerical algorithms. The concepts of invariant difference operators and generalized discrete sources are introduced for the discretization of systems of inhomogeneous differential equations and shown to produce exact difference equations. Invariant numerical flux functions are constructed from the general solutions of first order partial differential equations that come out of the evaluation of the Lie derivatives of conservation forms of difference schemes. It is demonstrated that invariant numerical flux functions with invariant flux or slope limiters can be determined to yield high resolution difference schemes. The introduction of an invariant flux or slope limiter can be done so as not to break the symmetry properties of a numerical flux-function.
DEFF Research Database (Denmark)
Møller, Jan Kloppenborg; Madsen, Henrik
This report describes methods to eliminate state dependent diffusion terms in Stochastic Differential Equations (SDEs). Transformations that leave the diffusion term of SDEs constant is important for simulation, and estimation. It is important for simulation because the Euler approximation...... convergence rate is faster, and for estimation because the Extended Kalman Filter equations are easier to implement than higher order filters needed in the case of state dependent diffusion terms. The general class of transformations which leaves the diffusion term independent of the state is called...
Smoothing and Decay Estimates for Nonlinear Diffusion Equations Equations of Porous Medium Type
Vázquez, Juan Luis
2006-01-01
This text is concerned with the quantitative aspects of the theory of nonlinear diffusion equations; equations which can be seen as nonlinear variations of the classical heat equation. They appear as mathematical models in different branches of Physics, Chemistry, Biology, and Engineering, and are also relevant in differential geometry and relativistic physics. Much of the modern theory of such equations is based on estimates and functional analysis.Concentrating on a class of equations with nonlinearities of power type that lead to degenerate or singular parabolicity ("equations of porou
Renormalization Group Equations for the CKM matrix
Kielanowski, P; Montes de Oca Y, J H
2008-01-01
We derive the one loop renormalization group equations for the Cabibbo-Kobayashi-Maskawa matrix for the Standard Model, its two Higgs extension and the minimal supersymmetric extension in a novel way. The derived equations depend only on a subset of the model parameters of the renormalization group equations for the quark Yukawa couplings so the CKM matrix evolution cannot fully test the renormalization group evolution of the quark Yukawa couplings. From the derived equations we obtain the invariant of the renormalization group evolution for three models which is the angle $\\alpha$ of the unitarity triangle. For the special case of the Standard Model and its extensions with $v_{1}\\approx v_{2}$ we demonstrate that also the shape of the unitarity triangle and the Buras-Wolfenstein parameters $\\bar{\\rho}=(1-{1/2}\\lambda^{2})\\rho$ and $\\bar{\\eta}=(1-{1/2}\\lambda^{2})\\eta$ are conserved. The invariance of the angles of the unitarity triangle means that it is not possible to find a model in which the CKM matrix mi...
Lie group analysis method for two classes of fractional partial differential equations
Chen, Cheng; Jiang, Yao-Lin
2015-09-01
In this paper we deal with two classes of fractional partial differential equation: n order linear fractional partial differential equation and nonlinear fractional reaction diffusion convection equation, by using the Lie group analysis method. The infinitesimal generators general formula of n order linear fractional partial differential equation is obtained. For nonlinear fractional reaction diffusion convection equation, the properties of their infinitesimal generators are considered. The four special cases are exhaustively investigated respectively. At the same time some examples of the corresponding case are also given. So it is very convenient to solve the infinitesimal generator of some fractional partial differential equation.
Analytic continuation of functional renormalization group equations
Floerchinger, Stefan
2012-01-01
Functional renormalization group equations are analytically continued from imaginary Matsubara frequencies to the real frequency axis. On the example of a scalar field with O(N) symmetry we discuss the analytic structure of the flowing action and show how it is possible to derive and solve flow equations for real-time properties such as propagator residues and particle decay widths. The formalism conserves space-time symmetries such as Lorentz or Galilei invariance and allows for improved, self-consistent approximations in terms of derivative expansions in Minkowski space.
Collapsing behaviour of a singular diffusion equation
Hui, Kin Ming
2009-01-01
Let $0\\le u_0(x)\\in L^1(\\R^2)\\cap L^{\\infty}(\\R^2)$ be such that $u_0(x) =u_0(|x|)$ for all $|x|\\ge r_1$ and is monotone decreasing for all $|x|\\ge r_1$ for some constant $r_1>0$ and ${ess}\\inf_{\\2{B}_{r_1}(0)}u_0\\ge{ess} \\sup_{\\R^2\\setminus B_{r_2}(0)}u_0$ for some constant $r_2>r_1$. Then under some mild decay conditions at infinity on the initial value $u_0$ we will extend the result of P. Daskalopoulos, M.A. del Pino and N. Sesum \\cite{DP2}, \\cite{DS}, and prove the collapsing behaviour of the maximal solution of the equation $u_t=\\Delta\\log u$ in $\\R^2\\times (0,T)$, $u(x,0)=u_0(x)$ in $\\R^2$, near its extinction time $T=\\int_{R^2}u_0dx/4\\pi$.
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.
A comparison principle for singular parabolic equations in the Heisenberg group
Directory of Open Access Journals (Sweden)
Pablo Ochoa
2015-04-01
Full Text Available In this work, we prove a comparison principle for singular parabolic equations with boundary conditions in the context of the Heisenberg group. In particular, this result applies to interesting equations, such as the parabolic infinite Laplacian, the mean curvature flow equation and more general homogeneous diffusions.
DISCONTINUOUS FINITE ELEMENT METHOD FOR CONVECTION-DIFFUSION EQUATIONS
Institute of Scientific and Technical Information of China (English)
Abdellatif Agouzal
2000-01-01
A discontinuous finite element method for convection-diffusion equations is proposed and analyzed. This scheme is designed to produce an approximate solution which is completely discontinuous. Optimal order of convergence is obtained for model problem. This is the same convergence rate known for the classical methods.
Local exact controllability of the diffusion equation in one dimension
Directory of Open Access Journals (Sweden)
Marius Beceanu
2003-01-01
Full Text Available This paper establishes the local exact null controllability of the diffusion equation in one dimension using distributed controls in the case of the Dirichlet boundary value problem. Most of the techniques used in the course of the proof are borrowed from Barbu (2002.
Explicit solutions of fractional diffusion equations via Generalized Gamma Convolution
D'Ovidio, Mirko
2010-01-01
In this paper we deal with Mellin convolution of generalized Gamma densities which brings to integrals of modified Bessel functions of the second kind. Such convolutions allow us to write explicitly the solutions of the time-fractional diffusion equations involving the adjoint operators of a square Bessel process and a Bessel process.
Fundamental solution of the tempered fractional diffusion equation
Liemert, André; Kienle, Alwin
2015-11-01
In this paper, we consider the space-time fractional diffusion equation Dt β u ( x , t ) + K ( - ∞ Dx α , λ ) u ( x , t ) = 0 , x ∈ R , t > 0 , with the tempered Riemann-Liouville derivative of order 0 Mainardi function Mα(x) of order 0 < α < 1 and arguments x ∈ R0 + .
A numerical solution for the diffusion equation in hydrogeologic systems
Ishii, A.L.; Healy, R.W.; Striegl, R.G.
1989-01-01
The documentation of a computer code for the numerical solution of the linear diffusion equation in one or two dimensions in Cartesian or cylindrical coordinates is presented. Applications of the program include molecular diffusion, heat conduction, and fluid flow in confined systems. The flow media may be anisotropic and heterogeneous. The model is formulated by replacing the continuous linear diffusion equation by discrete finite-difference approximations at each node in a block-centered grid. The resulting matrix equation is solved by the method of preconditioned conjugate gradients. The conjugate gradient method does not require the estimation of iteration parameters and is guaranteed convergent in the absence of rounding error. The matrixes are preconditioned to decrease the steps to convergence. The model allows the specification of any number of boundary conditions for any number of stress periods, and the output of a summary table for selected nodes showing flux and the concentration of the flux quantity for each time step. The model is written in a modular format for ease of modification. The model was verified by comparison of numerical and analytical solutions for cases of molecular diffusion, two-dimensional heat transfer, and axisymmetric radial saturated fluid flow. Application of the model to a hypothetical two-dimensional field situation of gas diffusion in the unsaturated zone is demonstrated. The input and output files are included as a check on program installation. The definition of variables, input requirements, flow chart, and program listing are included in the attachments. (USGS)
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.
Fractional diffusion equation and impedance spectroscopy of electrolytic cells.
Lenzi, E K; Evangelista, L R; Barbero, G
2009-08-20
The influence of the ions on the electrochemical impedance of a cell is calculated in the framework of a complete model in which the fractional drift-diffusion problem is analytically solved. The resulting distribution of the electric field inside the sample is determined by solving Poisson's equation. The theoretical model to determine the electrical impedance we are proposing here is based on the fractional derivative of distributed order on the diffusion equation. We argue that this is the more convenient and physically significant approach to account for the enormous variety of the diffusive regimes in a real cell. The frequency dependence of the real and imaginary parts of the impedance are shown to be very similar to the ones experimentally obtained in a large variety of electrolytic samples.
Langevin and diffusion equation of turbulent fluid flow
Brouwers, J. J. H.
2010-08-01
A derivation of the Langevin and diffusion equations describing the statistics of fluid particle displacement and passive admixture in turbulent flow is presented. Use is made of perturbation expansions. The small parameter is the inverse of the Kolmogorov constant C 0 , which arises from Lagrangian similarity theory. The value of C 0 in high Reynolds number turbulence is 5-6. To achieve sufficient accuracy, formulations are not limited to terms of leading order in C0 - 1 including terms next to leading order in C0 - 1 as well. Results of turbulence theory and statistical mechanics are invoked to arrive at the descriptions of the Langevin and diffusion equations, which are unique up to truncated terms of O ( C0 - 2 ) in displacement statistics. Errors due to truncation are indicated to amount to a few percent. The coefficients of the presented Langevin and diffusion equations are specified by fixed-point averages of the Eulerian velocity field. The equations apply to general turbulent flow in which fixed-point Eulerian velocity statistics are non-Gaussian to a degree of O ( C0 - 1 ) . The equations provide the means to calculate and analyze turbulent dispersion of passive or almost passive admixture such as fumes, smoke, and aerosols in areas ranging from atmospheric fluid motion to flows in engineering devices.
Dynamical symmetries of semi-linear Schrodinger and diffusion equations
Energy Technology Data Exchange (ETDEWEB)
Stoimenov, Stoimen [Laboratoire de Physique des Materiaux , Laboratoire associe au CNRS UMR 7556, Universite Henri Poincare Nancy I, B.P. 239, F-54506 Vandoeuvre les Nancy Cedex (France); Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 Sofia (Bulgaria); Henkel, Malte [Laboratoire de Physique des Materiaux, Laboratoire associe au CNRS UMR 7556, Universite Henri Poincare Nancy I, B.P. 239, F-54506 Vandoeuvre les Nancy Cedex (France)]. E-mail: henkel@lpm.u-nancy.fr
2005-09-12
Conditional and Lie symmetries of semi-linear 1D Schrodinger and diffusion equations are studied if the mass (or the diffusion constant) is considered as an additional variable. In this way, dynamical symmetries of semi-linear Schrodinger equations become related to the parabolic and almost-parabolic subalgebras of a three-dimensional conformal Lie algebra (conf{sub 3}){sub C}. We consider non-hermitian representations and also include a dimensionful coupling constant of the non-linearity. The corresponding representations of the parabolic and almost-parabolic subalgebras of (conf{sub 3}){sub C} are classified and the complete list of conditionally invariant semi-linear Schrodinger equations is obtained. Possible applications to the dynamical scaling behaviour of phase-ordering kinetics are discussed.
Traveling wavefront solutions to nonlinear reaction-diffusion-convection equations
Indekeu, Joseph O.; Smets, Ruben
2017-08-01
Physically motivated modified Fisher equations are studied in which nonlinear convection and nonlinear diffusion is allowed for besides the usual growth and spread of a population. It is pointed out that in a large variety of cases separable functions in the form of exponentially decaying sharp wavefronts solve the differential equation exactly provided a co-moving point source or sink is active at the wavefront. The velocity dispersion and front steepness may differ from those of some previously studied exact smooth traveling wave solutions. For an extension of the reaction-diffusion-convection equation, featuring a memory effect in the form of a maturity delay for growth and spread, also smooth exact wavefront solutions are obtained. The stability of the solutions is verified analytically and numerically.
Diffusion processes through social groups' dynamics
Apolloni, Andrea
2011-01-01
Axelrod's model describes the dissemination of a set of cultural traits in a society constituted by individual agents. In a social context, nevertheless, individual choices toward a specific attitude are also at the basis of the formation of communities, groups and parties. The membership in a group changes completely the behavior of single agents who start acting according to a social identity. Groups act and interact among them as single entities, but still conserve an internal dynamics. We show that, under certain conditions of social dynamics, the introduction of group dynamics in a cultural dissemination process avoids the flattening of the culture into a single entity and preserves the multiplicity of cultural attitudes. We also considered diffusion processes on this dynamical background, showing the conditions under which information as well as innovation can spread through the population in a scenario where the groups' choices determine the social structure.
Innovation diffusion equations on correlated scale-free networks
Energy Technology Data Exchange (ETDEWEB)
Bertotti, M.L., E-mail: marialetizia.bertotti@unibz.it [Free University of Bozen–Bolzano, Faculty of Science and Technology, Bolzano (Italy); Brunner, J., E-mail: johannes.brunner@tis.bz.it [TIS Innovation Park, Bolzano (Italy); Modanese, G., E-mail: giovanni.modanese@unibz.it [Free University of Bozen–Bolzano, Faculty of Science and Technology, Bolzano (Italy)
2016-07-29
Highlights: • The Bass diffusion model can be formulated on scale-free networks. • In the trickle-down version, the hubs adopt earlier and act as monitors. • We improve the equations in order to describe trickle-up diffusion. • Innovation is generated at the network periphery, and hubs can act as stiflers. • We compare diffusion times, in dependence on the scale-free exponent. - Abstract: 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.
Unstructured Grids and the Multigroup Neutron Diffusion Equation
Directory of Open Access Journals (Sweden)
German Theler
2013-01-01
Full Text Available The neutron diffusion equation is often used to perform core-level neutronic calculations. It consists of a set of second-order partial differential equations over the spatial coordinates that are, both in the academia and in the industry, usually solved by discretizing the neutron leakage term using a structured grid. This work introduces the alternatives that unstructured grids can provide to aid the engineers to solve the neutron diffusion problem and gives a brief overview of the variety of possibilities they offer. It is by understanding the basic mathematics that lie beneath the equations that model real physical systems; better technical decisions can be made. It is in this spirit that this paper is written, giving a first introduction to the basic concepts which can be incorporated into core-level neutron flux computations. A simple two-dimensional homogeneous circular reactor is solved using a coarse unstructured grid in order to illustrate some basic differences between the finite volumes and the finite elements method. Also, the classic 2D IAEA PWR benchmark problem is solved for eighty combinations of symmetries, meshing algorithms, basic geometric entities, discretization schemes, and characteristic grid lengths, giving even more insight into the peculiarities that arise when solving the neutron diffusion equation using unstructured grids.
Optimal prediction for moment models: crescendo diffusion and reordered equations
Seibold, Benjamin; Frank, Martin
2009-12-01
A direct numerical solution of the radiative transfer equation or any kinetic equation is typically expensive, since the radiative intensity depends on time, space and direction. An expansion in the direction variables yields an equivalent system of infinitely many moments. A fundamental problem is how to truncate the system. Various closures have been presented in the literature. We want to generally study the moment closure within the framework of optimal prediction, a strategy to approximate the mean solution of a large system by a smaller system, for radiation moment systems. We apply this strategy to radiative transfer and show that several closures can be re-derived within this framework, such as P N , diffusion, and diffusion correction closures. In addition, the formalism gives rise to new parabolic systems, the reordered P N equations, that are similar to the simplified P N equations. Furthermore, we propose a modification to existing closures. Although simple and with no extra cost, this newly derived crescendo diffusion yields better approximations in numerical tests.
On the entropy conditions for some flux limited diffusion equations
Caselles, V.
2011-04-01
In this paper we give a characterization of the notion of entropy solutions of some flux limited diffusion equations for which we can prove that the solution is a function of bounded variation in space and time. This includes the case of the so-called relativistic heat equation and some generalizations. For them we prove that the jump set consists of fronts that propagate at the speed given by Rankine-Hugoniot condition and we give on it a geometric characterization of the entropy conditions. Since entropy solutions are functions of bounded variation in space once the initial condition is, to complete the program we study the time regularity of solutions of the relativistic heat equation under some conditions on the initial datum. An analogous result holds for some other related equations without additional assumptions on the initial condition.
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.
Diffusive and dynamical radiating stars with realistic equations of state
Brassel, Byron P.; Maharaj, Sunil D.; Goswami, Rituparno
2017-03-01
We model the dynamics of a spherically symmetric radiating dynamical star with three spacetime regions. The local internal atmosphere is a two-component system consisting of standard pressure-free, null radiation and an additional string fluid with energy density and nonzero pressure obeying all physically realistic energy conditions. The middle region is purely radiative which matches to a third region which is the Schwarzschild exterior. A large family of solutions to the field equations are presented for various realistic equations of state. We demonstrate that it is possible to obtain solutions via a direct integration of the second order equations resulting from the assumption of an equation of state. A comparison of our solutions with earlier well known results is undertaken and we show that all these solutions, including those of Husain, are contained in our family. We then generalise our class of solutions to higher dimensions. Finally we consider the effects of diffusive transport and transparently derive the specific equations of state for which this diffusive behaviour is possible.
Critical Exponents for Fast Diffusion Equations with Nonlinear Boundary Sources
Institute of Scientific and Technical Information of China (English)
WANG LU-SHENG; WANG ZE-JIA
2011-01-01
In this paper, we study the large time behavior of solutions to a class of fast diffusion equations with nonlinear boundary sources on the exterior domain of the unit ball. We are interested in the critical global exponent q0 and the critical Fujita exponent qc for the problem considered, and show that q0 ＝ qc for the multidimensional Non-Newtonian polytropic filtration equation with nonlinear boundary sources, which is quite different from the known results that q0 ＜ qc for the onedimensional case; moreover, the value is different from the slow case.
Maximum Principles for Discrete and Semidiscrete Reaction-Diffusion Equation
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Petr Stehlík
2015-01-01
Full Text Available We study reaction-diffusion equations with a general reaction function f on one-dimensional lattices with continuous or discrete time ux′ (or Δtux=k(ux-1-2ux+ux+1+f(ux, x∈Z. We prove weak and strong maximum and minimum principles for corresponding initial-boundary value problems. Whereas the maximum principles in the semidiscrete case (continuous time exhibit similar features to those of fully continuous reaction-diffusion model, in the discrete case the weak maximum principle holds for a smaller class of functions and the strong maximum principle is valid in a weaker sense. We describe in detail how the validity of maximum principles depends on the nonlinearity and the time step. We illustrate our results on the Nagumo equation with the bistable nonlinearity.
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.
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.
Algorithm Refinement for Stochastic Partial Differential Equations. I. Linear Diffusion
Alexander, Francis J.; Garcia, Alejandro L.; Tartakovsky, Daniel M.
2002-10-01
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.
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.
Symmetries and Similarity Reductions of Nonlinear Diffusion Equation
Institute of Scientific and Technical Information of China (English)
LI Hui-Jun; RUAN Hang-Yu
2004-01-01
The inverse recursion operator, three new sets of symmetries, and infinite-dimensional Lie algebras for the nonlinear diffusion equation are given. Some nonlocal symmetries related to eigenvectors of the recursion operator Ф with the eigenvalue λi are also obtained with the help of the recursion operator Фi = Ф - λi. Using a part of these symmetries we get twelve types of nontrivial new similarity reduction.
Symmetries and Similarity Reductions of Nonlinear Diffusion Equation
Institute of Scientific and Technical Information of China (English)
LIHui-Jun; RUANHang-Yu
2004-01-01
The inverse recursion operator, three new sets of symmetries, and infinite-dimensional Lie algebras for the nonlinear diffusion equation are given. Some nonlocal symmetries related to eigenvectors of the recursion operator with the eigenvalue λi are also obtained with the help of the recursion operator φi=φ-λi. Using a part of these symmetries we get twelve types of nontrivial new similarity reduction.
Mixed and mixed-hybrid elements for the diffusion equation
Energy Technology Data Exchange (ETDEWEB)
Coulomb, F.; Fedon-Magnaud, C.
1988-11-01
Among the classical methods used for solving the neutron diffusion equation, the Lagrange finite element method can be efficiently implemented to provide a fast numerical treatment. Mixed elements are used because they allow simultaneous approximations for the flux and its gradient of the same order. Although the linear systems produced are not positive definite, a solution ca be achieved after eliminating some of the unknowns. Numerical results include core calculations of two types of reactors.
Blumenthal, Adrian
2015-01-01
Stochastic models that account for sudden, unforeseeable events play a crucial role in many different fields such as finance, economics, biology, chemistry, physics and so on. That kind of stochastic problems can be modeled by stochastic differential equations driven by jump-diffusion processes. In addition, there are situations, where a stochastic model is based on stochastic differential equations with multiple scales. Such stochastic problems are called stiff and lead for classical ex...
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.
Stability of planar diffusion wave for nonlinear evolution equation
Institute of Scientific and Technical Information of China (English)
无
2012-01-01
It is known that the one-dimensional nonlinear heat equation ut = f(u)x1x1,f'(u) 0,u(±∞,t) = u±,u+ = u_ has a unique self-similar solution u(x1/1+t).In multi-dimensional space,u(x1/1+t) is called a planar diffusion wave.In the first part of the present paper,it is shown that under some smallness conditions,such a planar diffusion wave is nonlinearly stable for the nonlinear heat equation:ut-△f(u) = 0,x ∈ Rn.The optimal time decay rate is obtained.In the second part of this paper,it is further shown that this planar diffusion wave is still nonlinearly stable for the quasilinear wave equation with damping:utt + utt+ △f(u) = 0,x ∈ Rn.The time decay rate is also obtained.The proofs are given by an elementary energy method.
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.
Analytical solutions for the fractional diffusion-advection equation describing super-diffusion
Directory of Open Access Journals (Sweden)
Gómez Francisco
2016-01-01
Full Text Available This paper presents the alternative construction of the diffusion-advection equation in the range (1; 2. The fractional derivative of the Liouville-Caputo type is applied. Analytical solutions are obtained in terms of Mittag-Leffler functions. In the range (1; 2 the concentration exhibits the superdiffusion phenomena and when the order of the derivative is equal to 2 ballistic diffusion can be observed, these behaviors occur in many physical systems such as semiconductors, quantum optics, or turbulent diffusion. This mathematical representation can be applied in the description of anomalous complex processes.
Characterization of Cocycle Attractors for Nonautonomous Reaction-Diffusion Equations
Cardoso, C. A.; Langa, J. A.; Obaya, R.
In this paper, we describe in detail the global and cocycle attractors related to nonautonomous scalar differential equations with diffusion. In particular, we investigate reaction-diffusion equations with almost-periodic coefficients. The associated semiflows are strongly monotone which allow us to give a full characterization of the cocycle attractor. We prove that, when the upper Lyapunov exponent associated to the linear part of the equations is positive, the flow is persistent in the positive cone, and we study the stability and the set of continuity points of the section of each minimal set in the global attractor for the skew product semiflow. We illustrate our result with some nontrivial examples showing the richness of the dynamics on this attractor, which in some situations shows internal chaotic dynamics in the Li-Yorke sense. We also include the sublinear and concave cases in order to go further in the characterization of the attractors, including, for instance, a nonautonomous version of the Chafee-Infante equation. In this last case we can show exponentially forward attraction to the cocycle (pullback) attractors in the positive cone of solutions.
Optimal prediction for moment models: Crescendo diffusion and reordered equations
Seibold, Benjamin
2009-01-01
A direct numerical solution of the radiative transfer equation or any kinetic equation is typically expensive, since the radiative intensity depends on time, space and direction. An expansion in the direction variables yields an equivalent system of infinitely many moments. A fundamental problem is how to truncate the system. Various closures have been presented in the literature. We want to study moment closure generally within the framework of optimal prediction, a strategy to approximate the mean solution of a large system by a smaller system, for radiation moment systems. We apply this strategy to radiative transfer and show that several closures can be re-derived within this framework, e.g. $P_N$, diffusion, and diffusion correction closures. In addition, the formalism gives rise to new parabolic systems, the reordered $P_N$ equations, that are similar to the simplified $P_N$ equations. Furthermore, we propose a modification to existing closures. Although simple and with no extra cost, this newly derived...
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.
The Real-Space Renormalization Group Applied to Diffusion in Inhomogeneous Media
Kawasaki, Mitsuhiro
2002-01-01
The real-space renormalization group technique is introduced to evaluate the effective diffusion constant for diffusion in inhomogeneous media, which has been obtained by singular perturbation methods. Our method is formulated on a discretized real space and hence it can be easily combined with numerical studies for partial differential equations.
Ohmori, Shousuke; Yamazaki, Yoshihiro
2016-01-01
Ultradiscrete equations are derived from a set of reaction-diffusion partial differential equations, and cellular automaton rules are obtained on the basis of the ultradiscrete equations. Some rules reproduce the dynamical properties of the original reaction-diffusion equations, namely, bistability and pulse annihilation. Furthermore, other rules bring about soliton-like preservation and periodic pulse generation with a pacemaker, which are not obtained from the original reaction-diffusion equations.
Anticipative Stochastic Differential Equations with Non-smooth Diffusion Coefficient
Institute of Scientific and Technical Information of China (English)
Zong Xia LIANG
2006-01-01
In this paper we prove the existence and uniqueness of the solutions to the one-dimensional linear stochastic differential equation with Skorohod integralXt(ω) = η(ω) + ∫t0 asXs(ω)dWs + bsXs(ω)ds, t ∈ [0, 1],where (Ws) is the canonical Wiener process defined on the standard Wiener space ((W), (H),μ), a is non-smooth and adapted, but η and b may be anticipating to the filtration generated by (Ws). The intention of the paper is to eliminate the regularity of the diffusion coefficient a in the Malliavin sense, in the existing literature. The idea is to approach the non-smooth diffusion coefficient a by smooth ones.
Partial Differential Equations of an Epidemic Model with Spatial Diffusion
Directory of Open Access Journals (Sweden)
El Mehdi Lotfi
2014-01-01
Full Text Available The aim of this paper is to study the dynamics of a reaction-diffusion SIR epidemic model with specific nonlinear incidence rate. The global existence, positivity, and boundedness of solutions for a reaction-diffusion system with homogeneous Neumann boundary conditions are proved. The local stability of the disease-free equilibrium and endemic equilibrium is obtained via characteristic equations. By means of Lyapunov functional, the global stability of both equilibria is investigated. More precisely, our results show that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than or equal to unity, which leads to the eradication of disease from population. When the basic reproduction number is greater than unity, then disease-free equilibrium becomes unstable and the endemic equilibrium is globally asymptotically stable; in this case the disease persists in the population. Numerical simulations are presented to illustrate our theoretical results.
Reaction-diffusion master equation in the microscopic limit
Hellander, Stefan; Hellander, Andreas; Petzold, Linda
2012-04-01
Stochastic modeling of reaction-diffusion kinetics has emerged as a powerful theoretical tool in the study of biochemical reaction networks. Two frequently employed models are the particle-tracking Smoluchowski framework and the on-lattice reaction-diffusion master equation (RDME) framework. As the mesh size goes from coarse to fine, the RDME initially becomes more accurate. However, recent developments have shown that it will become increasingly inaccurate compared to the Smoluchowski model as the lattice spacing becomes very fine. Here we give a general and simple argument for why the RDME breaks down. Our analysis reveals a hard limit on the voxel size for which no local RDME can agree with the Smoluchowski model and lets us quantify this limit in two and three dimensions. In this light we review and discuss recent work in which the RDME has been modified in different ways in order to better agree with the microscale model for very small voxel sizes.
On the Reaction Diffusion Master Equation in the Microscopic Limit
Hellander, Stefan; Petzold, Linda
2011-01-01
Stochastic modeling of reaction-diffusion kinetics has emerged as a powerful theoretical tool in the study of biochemical reaction networks. Two frequently employed models are the particle-tracking Smoluchowski framework and the on-lattice Reaction-Diffusion Master Equation (RDME) framework. As the mesh size goes from coarse to fine, the RDME initially becomes more accurate. However, recent developments have shown that it will become increasingly inaccurate compared to the Smoluchowski model as the lattice spacing becomes very fine. In this paper we give a new, general and simple argument for why the RDME breaks down. Our analysis reveals a hard limit on the voxel size for which no local RDME can agree with the Smoluchowski model.
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.
The influence of fractional diffusion in Fisher-KPP equations
Cabre, Xavier
2012-01-01
We study the Fisher-KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with power decaying kernel, an important example being the fractional Laplacian. In contrast with the case of the stan- dard Laplacian where the stable state invades the unstable one at constant speed, we prove that with fractional diffusion, generated for instance by a stable L\\'evy process, the front position is exponential in time. Our results provide a mathe- matically rigorous justification of numerous heuristics about this model.
On the solutions of fractional reaction-diffusion equations
Directory of Open Access Journals (Sweden)
Jagdev Singh
2013-05-01
Full Text Available In this paper, we obtain the solution of a fractional reaction-diffusion equation associated with the generalized Riemann-Liouville fractional derivative as the time derivative and Riesz-Feller fractional derivative as the space-derivative. The results are derived by the application of the Laplace and Fourier transforms in compact and elegant form in terms of Mittag-Leffler function and H-function. The results obtained here are of general nature and include the results investigated earlier by many authors.
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.
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
Energy Technology Data Exchange (ETDEWEB)
Hamada, Souad, E-mail: souadhamada@yahoo.fr [LSP-IE: Research Laboratory, Electrical Engineering Department, University of Batna, 05000 Batna (Algeria); Louai, Fatima Zohra, E-mail: fz_louai@yahoo.com [LSP-IE: Research Laboratory, Electrical Engineering Department, University of Batna, 05000 Batna (Algeria); Nait-Said, Nasreddine, E-mail: n_naitsaid@yahoo.com [LSP-IE: Research Laboratory, Electrical Engineering Department, University of Batna, 05000 Batna (Algeria); Benabou, Abdelkader, E-mail: Abdelkader.Benabou@univ-lille1.fr [L2EP, Université de Lille1, 59655 Villeneuve d’Ascq (France)
2016-07-15
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.
ENERGY ESTIMATES FOR DELAY DIFFUSION-REACTION EQUATIONS
Institute of Scientific and Technical Information of China (English)
J.A.Ferreira; P.M.da Silva
2008-01-01
In this paper we consider nonlinear delay diffusion-reaction equations with initial and Dirichlet boundary conditions.The behaviour and the stability of the solution of such initial boundary value problems(IBVPs)are studied using the energy method.Simple numerical methods are considered for the computation of numerical approximations to the solution of the nonlinear IBVPs.Using the discrete energy method we study the stability and convergence of the numerical approximations.Numerical experiments are carried out to illustrate our theoretical results.
New variable separation solutions for the generalized nonlinear diffusion equations
Fei-Yu, Ji; Shun-Li, Zhang
2016-03-01
The functionally generalized variable separation of the generalized nonlinear diffusion equations ut = A(u,ux)uxx + B(u,ux) is studied by using the conditional Lie-Bäcklund symmetry method. The variant forms of the considered equations, which admit the corresponding conditional Lie-Bäcklund symmetries, are characterized. To construct functionally generalized separable solutions, several concrete examples defined on the exponential and trigonometric invariant subspaces are provided. Project supported by the National Natural Science Foundation of China (Grant Nos. 11371293, 11401458, and 11501438), the National Natural Science Foundation of China, Tian Yuan Special Foundation (Grant No. 11426169), and the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2015JQ1014).
Mimetic discretizations for Maxwell equations and the equations of magnetic diffusion
Energy Technology Data Exchange (ETDEWEB)
Hyman, J.M.; Shashkov, M.
1998-03-01
The authors construct reliable finite difference methods for approximating the solutions Maxwell`s equations and equations of magnetic field diffusion using discrete analogs of differential operators that satisfy the identities and theorems of vector and tensor calculus in discrete form. These methods mimic many fundamental properties of the underlying physical problem including the conservation laws, the symmetries in the solution, the nondivergence of particular vector fields and they do not have spurious modes. The constructed method can be applied in case of strongly discontinuous properties of the media for nonorthogonal and nonsmooth computational grids.
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.
PERTURBATIONAL FINITE DIFFERENCE SCHEME OF CONVECTION-DIFFUSION EQUATION
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
The Perturbational Finite Difference (PFD) method is a kind of high-order-accurate compact difference method, But its idea is different from the normal compact method and the multi-nodes method. This method can get a Perturbational Exact Numerical Solution (PENS) scheme for locally linearlized Convection-Diffusion (CD) equation. The PENS scheme is similar to the Finite Analytical (FA) scheme and Exact Difference Solution (EDS) scheme, which are all exponential schemes, but PENS scheme is simpler and uses only 3, 5 and 7 nodes for 1-, 2- and 3-dimensional problems, respectively. The various approximate schemes of PENS scheme are also called Perturbational-High-order-accurate Difference (PHD) scheme. The PHD schemes can be got by expanding the exponential terms in the PENS scheme into power series of grid Renold number, and they are all upwind schemes and remain the concise structure form of first-order upwind scheme. For 1-dimensional (1-D) CD equation and 2-D incompressible Navier-Stokes equation, their PENS and PHD schemes were constituted in this paper, they all gave highly accurate results for the numerical examples of three 1-D CD equations and an incompressible 2-D flow in a square cavity.
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.
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.
High order schemes for the tempered fractional diffusion equations
Li, Can
2014-01-01
L\\'{e}vy flight models whose jumps have infinite moments are mathematically used to describe the superdiffusion in complex systems. Exponentially tempering the probability of large jumps of L\\'{e}vy flights leads to the tempered stable L\\'{e}vy processes which combine both the $\\alpha$-stable and Gaussian trends; and the very large jumps are unlikely and all their moments exist. The probability density functions of the tempered stable L\\'{e}vy processes solve the tempered fractional diffusion equation. This paper focuses on designing the high order difference schemes for the tempered fractional diffusion equation on bounded domain. The high order difference approximations, called the tempered and weighted and shifted Gr\\"{u}nwald difference (tempered-WSGD) operators, in space are obtained by using the properties of the tempered fractional calculus and weighting and shifting their first order Gr\\"{u}nwald type difference approximations. And the Crank-Nicolson discretization is used in the time direction. The s...
On the Stability of Jensen's Functional Equation on Groups
Indian Academy of Sciences (India)
Valeriĭ A Faĭziev; Prasanna K Sahoo
2007-02-01
In this paper we establish the stability of Jensen’s functional equation on some classes of groups. We prove that Jensen equation is stable on noncommutative groups such as metabelian groups and (2,), where is an arbitrary commutative field with characteristic different from two. We also prove that any group can be embedded into some group such that the Jensen functional equation is stable on .
Energy Technology Data Exchange (ETDEWEB)
Nadler, Boaz [Department of Mathematics, Yale University, New-Haven, CT 06520 (United States); Schuss, Zeev [Department of Applied Mathematics, Tel-Aviv University, Ramat-Aviv 69978, Tel-Aviv (Israel); Singer, Amit [Department of Applied Mathematics, Tel-Aviv University, Ramat-Aviv 69978, Tel-Aviv (Israel); Eisenberg, R S [Department of Molecular Biophysics and Physiology, Rush Medical Center, 1750 Harrison Street, Chicago, IL 60612 (United States)
2004-06-09
Ionic diffusion through and near small domains is of considerable importance in molecular biophysics in applications such as permeation through protein channels and diffusion near the charged active sites of macromolecules. The motion of the ions in these settings depends on the specific nanoscale geometry and charge distribution in and near the domain, so standard continuum type approaches have obvious limitations. The standard machinery of equilibrium statistical mechanics includes microscopic details, but is also not applicable, because these systems are usually not in equilibrium due to concentration gradients and to the presence of an external applied potential, which drive a non-vanishing stationary current through the system. We present a stochastic molecular model for the diffusive motion of interacting particles in an external field of force and a derivation of effective partial differential equations and their boundary conditions that describe the stationary non-equilibrium system. The interactions can include electrostatic, Lennard-Jones and other pairwise forces. The analysis yields a new type of Poisson-Nernst-Planck equations, that involves conditional and unconditional charge densities and potentials. The conditional charge densities are the non-equilibrium analogues of the well studied pair correlation functions of equilibrium statistical physics. Our proposed theory is an extension of equilibrium statistical mechanics of simple fluids to stationary non-equilibrium problems. The proposed system of equations differs from the standard Poisson-Nernst-Planck system in two important aspects. First, the force term depends on conditional densities and thus on the finite size of ions, and second, it contains the dielectric boundary force on a discrete ion near dielectric interfaces. Recently, various authors have shown that both of these terms are important for diffusion through confined geometries in the context of ion channels.
Kengne, Emmanuel; Saydé, Michel; Ben Hamouda, Fathi; Lakhssassi, Ahmed
2013-11-01
Analytical entire traveling wave solutions to the 1+1 density-dependent nonlinear reaction-diffusion equation via the extended generalized Riccati equation mapping method are presented in this paper. This equation can be regarded as an extension case of the Fisher-Kolmogoroff equation, which is used for studying insect and animal dispersal with growth dynamics. The analytical solutions are then used to investigate the effect of equation parameters on the population distribution.
Energy Technology Data Exchange (ETDEWEB)
Zhou, Xiafeng, E-mail: zhou-xf11@mails.tsinghua.edu.cn; Guo, Jiong, E-mail: guojiong12@tsinghua.edu.cn; Li, Fu, E-mail: lifu@tsinghua.edu.cn
2015-12-15
Highlights: • NEMs are innovatively applied to solve convection diffusion equation. • Stability, accuracy and numerical diffusion for NEM are analyzed for the first time. • Stability and numerical diffusion depend on the NEM expansion order and its parity. • NEMs have higher accuracy than both second order upwind and QUICK scheme. • NEMs with different expansion orders are integrated into a unified discrete form. - Abstract: The traditional finite difference method or finite volume method (FDM or FVM) is used for HTGR thermal-hydraulic calculation at present. However, both FDM and FVM require the fine mesh sizes to achieve the desired precision and thus result in a limited efficiency. Therefore, a more efficient and accurate numerical method needs to be developed. Nodal expansion method (NEM) can achieve high accuracy even on the coarse meshes in the reactor physics analysis so that the number of spatial meshes and computational cost can be largely decreased. Because of higher efficiency and accuracy, NEM can be innovatively applied to thermal-hydraulic calculation. In the paper, NEMs with different orders of basis functions are successfully developed and applied to multi-dimensional steady convection diffusion equation. Numerical results show that NEMs with three or higher order basis functions can track the reference solutions very well and are superior to second order upwind scheme and QUICK scheme. However, the false diffusion and unphysical oscillation behavior are discovered for NEMs. To explain the reasons for the above-mentioned behaviors, the stability, accuracy and numerical diffusion properties of NEM are analyzed by the Fourier analysis, and by comparing with exact solutions of difference and differential equation. The theoretical analysis results show that the accuracy of NEM increases with the expansion order. However, the stability and numerical diffusion properties depend not only on the order of basis functions but also on the parity of
Derivation of a volume-averaged neutron diffusion equation; Atomos para el desarrollo de Mexico
Energy Technology Data Exchange (ETDEWEB)
Vazquez R, R.; Espinosa P, G. [UAM-Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, Mexico D.F. 09340 (Mexico); Morales S, Jaime B. [UNAM, Laboratorio de Analisis en Ingenieria de Reactores Nucleares, Paseo Cuauhnahuac 8532, Jiutepec, Morelos 62550 (Mexico)]. e-mail: rvr@xanum.uam.mx
2008-07-01
This paper presents a general theoretical analysis of the problem of neutron motion in a nuclear reactor, where large variations on neutron cross sections normally preclude the use of the classical neutron diffusion equation. A volume-averaged neutron diffusion equation is derived which includes correction terms to diffusion and nuclear reaction effects. A method is presented to determine closure-relationships for the volume-averaged neutron diffusion equation (e.g., effective neutron diffusivity). In order to describe the distribution of neutrons in a highly heterogeneous configuration, it was necessary to extend the classical neutron diffusion equation. Thus, the volume averaged diffusion equation include two corrections factor: the first correction is related with the absorption process of the neutron and the second correction is a contribution to the neutron diffusion, both parameters are related to neutron effects on the interface of a heterogeneous configuration. (Author)
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.
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...
Directory of Open Access Journals (Sweden)
Chih-Chun Hsieh
2012-01-01
Full Text Available This study performs a precipitation examination of the phase using the general diffusion equation with comparison to the Vitek model in dissimilar stainless steels during multipass welding. Experimental results demonstrate that the diffusivities (, , and of Cr, Ni, and Si are higher in -ferrite than (, , and in the phase, and that they facilitate the precipitation of the σ phase in the third pass fusion zone. The Vitek diffusion equation can be modified as follows: .
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.
Dynamics of stochastic nonclassical diffusion equations on unbounded domains
Directory of Open Access Journals (Sweden)
Wenqiang Zhao
2015-11-01
Full Text Available This article concerns the dynamics of stochastic nonclassical diffusion equation on $\\mathbb{R}^N$ perturbed by a $\\epsilon$-random term, where $\\epsilon\\in(0,1]$ is the intension of noise. By using an energy approach, we prove the asymptotic compactness of the associated random dynamical system, and then the existence of random attractors in $H^1(\\mathbb{R}^N$. Finally, we show the upper semi-continuity of random attractors at $\\epsilon=0$ in the sense of Hausdorff semi-metric in $H^1(\\mathbb{R}^N$, which implies that the obtained family of random attractors indexed by $\\epsilon$ converge to a deterministic attractor as $\\epsilon$ vanishes.
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.
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.
Renormalization group flow equations from the 4PI equations of motion
Carrington, M E
2013-01-01
The 4PI effective action provides a a hierarchy of integral equations which have the form of Bethe-Salpeter equations. The vertex functions obtained from these equations can be used to truncate the exact renormalization group flow equations. This truncation has the property that the flow is a total derivative with respect to the flow parameter and is equivalent to solving the nPI equations of motion. This result establishes a direct connection between two non-perturbative methods.
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...
A New Contraction Family for Porous Medium and Fast Diffusion Equations
Chmaycem, G.; Jazar, M.; Monneau, R.
2016-08-01
In this paper, we present a surprising two-dimensional contraction family for porous medium and fast diffusion equations. This approach provides new a priori estimates on the solutions, even for the standard heat equation.
The First Integral Method to Study a Class of Reaction-Diffusion Equations
Institute of Scientific and Technical Information of China (English)
KE Yun-Quan; YU Jun
2005-01-01
In this letter, a class of reaction-diffusion equations, which arise in chemical reaction or ecology and other fields of physics, are investigated. A more general analytical solution of the equation is obtained by using the first integral method.
The constructive technique and its application in solving a nonlinear reaction diffusion equation
Institute of Scientific and Technical Information of China (English)
Lai Shao-Yong; Guo Yun-Xi; Qing Yin; Wu Yong-Hong
2009-01-01
A mathematical technique based on the consideration of a nonlinear partial differential equation together with an additional condition in the form of an ordinary differential equation is employed to study a nonlinear reaction diffusion equation which describes a real process in physics and in chemistry. Several exact solutions for the equation are acquired under certain circumstances.
The Semiclassical Limit in the Quantum Drift-Diffusion Equations with Isentropic Pressure
Institute of Scientific and Technical Information of China (English)
Li CHEN; Qiangchang JU
2008-01-01
The semiclassical limit in the transient quantum drift-diffusion equations with isentropic pressure in one space dimension is rigorously proved. The equations are supple- mented with homogeneous Neumann boundary conditions. It is shown that the semiclas- sical limit of this solution solves the classical drift-diffusion model. In the meanwhile, the global existence of weak solutions is proved.
Diffusion phenomenon for linear dissipative wave equations in an exterior domain
Ikehata, Ryo
Under the general condition of the initial data, we will derive the crucial estimates which imply the diffusion phenomenon for the dissipative linear wave equations in an exterior domain. In order to derive the diffusion phenomenon for dissipative wave equations, the time integral method which was developed by Ikehata and Matsuyama (Sci. Math. Japon. 55 (2002) 33) plays an effective role.
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.
Traveling waves and the renormalization group improvedBalitsky-Kovchegov equation
Energy Technology Data Exchange (ETDEWEB)
Enberg, Rikard
2006-12-01
I study the incorporation of renormalization group (RG)improved BFKL kernels in the Balitsky-Kovchegov (BK) equation whichdescribes parton saturation. The RG improvement takes into accountimportant parts of the next-to-leading and higher order logarithmiccorrections to the kernel. The traveling wave front method for analyzingthe BK equation is generalized to deal with RG-resummed kernels,restricting to the interesting case of fixed QCD coupling. The resultsshow that the higher order corrections suppress the rapid increase of thesaturation scale with increasing rapidity. I also perform a "diffusive"differential equation approximation, which illustrates that someimportant qualitative properties of the kernel change when including RGcorrections.
Fa, Kwok Sau
2015-02-01
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.
Numerical approximation of Levy-Feller diffusion equation and its probability interpretation
Zhang, H.; Liu, F.; Anh, V.
2007-09-01
In this paper, we consider the Levy-Feller fractional diffusion equation, which is obtained from the standard diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order and skewness [theta] ([theta][less-than-or-equals, slant]min{[alpha],2-[alpha]}). We construct two new discrete schemes of the Cauchy problem for the above equation with 0Feller fractional diffusion equation with 1<[alpha]<2 in a bounded spatial domain. Finally, we present a numerical example to evaluate our theoretical analysis.
Sweilam, N. H.; Abou Hasan, M. M.
2016-08-01
This paper reports a new spectral algorithm for obtaining an approximate solution for the Lévy-Feller diffusion equation depending on Legendre polynomials and Chebyshev collocation points. The Lévy-Feller diffusion equation is obtained from the standard diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative. A new formula expressing explicitly any fractional-order derivatives, in the sense of Riesz-Feller operator, of Legendre polynomials of any degree in terms of Jacobi polynomials is proved. Moreover, the Chebyshev-Legendre collocation method together with the implicit Euler method are used to reduce these types of differential equations to a system of algebraic equations which can be solved numerically. Numerical results with comparisons are given to confirm the reliability of the proposed method for the Lévy-Feller diffusion equation.
Preliminary group classification of quasilinear third-order evolution equations
Institute of Scientific and Technical Information of China (English)
Ding-jiang HUANG; Hong-qing ZHANG
2009-01-01
Group classification of quasilinear third-order evolution equations is given by using the classical infinitesimal Lie method, the technique of equivalence transfor-mations, and the theory of classification of abstract low-dimensional Lie algebras. We show that there are three equations admitting simple Lie algebras of dimension three. All non-equivalent equations admitting simple Lie algebras are nothing but these three. Furthermore, we also show that there exist two, five, twenty-nine and twenty-six non-equivalent third-order nonlinear evolution equations admitting one-, two-, three-, and four-dimensional solvable Lie algebras, respectively.
Akimoto, Takuma; Yamamoto, Eiji
2016-06-01
We consider the Langevin equation with dichotomously fluctuating diffusivity, where the diffusion coefficient changes dichotomously over time, in order to study fluctuations of time-averaged observables in temporally heterogeneous diffusion processes. We find that the time-averaged mean-square displacement (TMSD) can be represented by the occupation time of a state in the asymptotic limit of the measurement time and hence occupation time statistics is a powerful tool for calculating the TMSD in the model. We show that the TMSD increases linearly with time (normal diffusion) but the time-averaged diffusion coefficients are intrinsically random when the mean sojourn time for one of the states diverges, i.e., intrinsic nonequilibrium processes. Thus, we find that temporally heterogeneous environments provide anomalous fluctuations of time-averaged diffusivity, which have relevance to large fluctuations of the diffusion coefficients obtained by single-particle-tracking trajectories in experiments.
Numerical method for solving the three-dimensional time-dependent neutron diffusion equation
Energy Technology Data Exchange (ETDEWEB)
Khaled, S.M. [Institute of Nuclear Techniques, Budapest University of Technology and Economics, Budapest (Hungary)]. E-mail: K_S_MAHMOUD@hotmail.com; Szatmary, Z. [Institute of Nuclear Techniques, Budapest University of Technology and Economics, Budapest (Hungary)]. E-mail: szatmary@reak.bme.hu
2005-07-01
A numerical time-implicit method has been developed for solving the coupled three-dimensional time-dependent multi-group neutron diffusion and delayed neutron precursor equations. The numerical stability of the implicit computation scheme and the convergence of the iterative associated processes have been evaluated. The computational scheme requires the solution of large linear systems at each time step. For this purpose, the point over-relaxation Gauss-Seidel method was chosen. A new scheme was introduced instead of the usual source iteration scheme. (author)
Horowitz, Jordan M.
2015-01-01
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 s...
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.
Polchinski's equation for group field theory
Energy Technology Data Exchange (ETDEWEB)
Krajewski, T. [Centre de Physique Theorique, Campus de Luminy, 13 288 Marseille cedex 9 (France); Laboratoire d' Informatique de Paris Nord, 99 avenue Jean Baptiste Clement, 93 430 Villetaneuse (France); Toriumi, R. [Centre de Physique Theorique, Campus de Luminy, 13 288 Marseille cedex 9 (France)
2014-09-11
We derive an exact renormalization group equation in the context of (colored) group field theories. This equation describes the variation of the effective action as some of the modes of the fields are integrated out. From a combinatorial point of view, the effective action can be expressed using a boundary triangulation and the corresponding renormalization group equation identifies some of its simplexes, In group field theory, terms in the effective action are parametrized by spin networks, while the group field theory Feynman graphs correspond to spin foams. This provides a formulation of group field theories that only involves boundary graphs. (Copyright copyright 2014 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)
Algorithmic derivation of functional renormalization group equations and Dyson-Schwinger equations
Huber, Markus Q
2011-01-01
We present the Mathematica application DoFun which allows to derive Dyson-Schwinger equations and renormalization group flow equations for n-point functions in a simple manner. DoFun offers several tools which considerably simplify the derivation of these equations from a given physical action. We discuss the application of DoFun by means of two different types of quantum field theories, namely a bosonic O(N) theory and the Gross-Neveu model.
Ji, Fei-Yu; Zhang, Shun-Li
2013-11-01
In this paper, the generalized diffusion equation with perturbation ut = A(u;ux)uII+eB(u;ux) is studied in terms of the approximate functional variable separation approach. A complete classification of these perturbed equations which admit approximate functional separable solutions is presented. Some approximate solutions to the resulting perturbed equations are obtained by examples.
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.
Ramsey, Scott; Baty, Roy
2015-11-01
This work considers the group invariance properties of the inviscid compressible flow equations (Euler equations) under the assumptions of one-dimensional symmetry and a modified Tait equation of state (EOS) closure model. When written in terms of an adiabatic bulk modulus, a transformed version of these equations is found to be identical to that for an ideal gas EOS. As a result, the Lie group invariance structure of these equations - and their subsequent reduction to a lower-order system - is identical to the published results for the ideal gas case. Following the reduction of the Euler equations to a system of ordinary differential equations, a variety of elementary closed-form solutions are derived. These solutions are then used in conjunction with the Rankine-Hugoniot conditions to construct discontinuous shock wave and free surface solutions that are analogous to the classical Noh, Sedov, Guderley, and Hunter similarity solutions of the Euler equations for an ideal gas EOS. The versions of these problems for the modified Tait EOS are found to be semi-analytic in that a transcendental root extraction (and in some cases numerical integration of ordinary differential equations) enables solution of the relevant equations.
Transformed Fourier and Fick equations for the control of heat and mass diffusion
Directory of Open Access Journals (Sweden)
S. Guenneau
2015-05-01
Full Text Available 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.
Transformed Fourier and Fick equations for the control of heat and mass diffusion
Guenneau, S.; Petiteau, D.; Zerrad, M.; Amra, C.; Puvirajesinghe, T.
2015-05-01
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.
Numerical Solution of Fractional Diffusion Equation Model for Freezing in Finite Media
Directory of Open Access Journals (Sweden)
R. S. Damor
2013-01-01
Full Text Available Phase change problems play very important role in engineering sciences including casting of nuclear waste materials, vivo freezing of biological tissues, solar collectors and so forth. In present paper, we propose fractional diffusion equation model for alloy solidification. A transient heat transfer analysis is carried out to study the anomalous diffusion. Finite difference method is used to solve the fractional differential equation model. The temperature profiles, the motion of interface, and interface velocity have been evaluated for space fractional diffusion equation.
A high-order splitting scheme for the advection-diffusion equation of pollutants
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
A high-order splitting scheme for the advection-diffusion equation of pollutants is proposed in this paper. The multidimensional advection-diffusion equation is splitted into several one-dimensional equations that are solved by the scheme. Only three spatial grid points are needed in each direction and the scheme has fourth-order spatial accuracy. Several typically pure advection and advection-diffusion problems are simulated. Numerical results show that the accuracy of the scheme is much higher than that of the classical schemes and the scheme can be efficiently solved with little programming effort.
Umarov, Sabir
2009-01-01
In this paper diffusion processes with changing modes are studied involving the variable order partial differential equations. We prove the existence and uniqueness theorem of a solution of the Cauchy problem for fractional variable order (with respect to the time derivative) pseudo-differential equations. Depending on the parameters of variable order derivatives short or long range memories may appear when diffusion modes change. These memory effects are classified and studied in detail. Processes that have distinctive regimes of different types of diffusion depending on time are ubiquitous in the nature. Examples include diffusion in a heterogeneous media and protein movement in cell biology.
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...
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.
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.
Comparison principle for parabolic equations in the Heisenberg group
Directory of Open Access Journals (Sweden)
Thomas Bieske
2005-09-01
Full Text Available We define two notions of viscosity solutions to parabolic equations in the Heisenberg group, depending on whether the test functions concern only the past or both the past and the future. We then exploit the Heisenberg geometry to prove a comparison principle for a class of parabolic equations and show the sufficiency of considering the test functions that concern only the past.
Cho, Guangsup; Uhm, Han Sup
2016-10-01
The time-dependent solution of diffusion equation by the Fourier integration provides the axial diffusion velocity of a plasma packet, which is a key element of the plasma propagation in a plasma jet operated by the several tens of kHz. The plasma diffusion velocity is higher than the order of un ˜ 10 m/s at a high electric-field region of plasma generation and it is about the order of un ˜ 10 m/s at the plasma column of a low field region in a jet-nozzle inside. Meanwhile, the diffusion velocity is slower than the order of un ˜ 10 m/s in the open-air space where the plasma density flattens due to its radial expansion. Using these diffusion velocity data, the group-velocity of plasma diffusion wave-packet is given by ug ˜ cs2/un, a combination of the diffusion velocity un and the acoustic velocity cs. The experimental results of the plasma propagation can be verified with the plasma propagation in a form of the wave-packet whose propagation velocity is 104 m/s in a tube inside and is as fast as 105 m/s in the open-air space, thereby reconfirming that the theory of a plasma diffusion-wave is the origin of the plasma propagation in a plasma jet.
Harnack inequality and strong Feller property for stochastic fast-diffusion equations
Liu, Wei; Wang, Feng-Yu
2008-06-01
As a continuation to [F.-Y. Wang, Harnack inequality and applications for stochastic generalized porous media equations, Ann. Probab. 35 (2007) 1333-1350], where the Harnack inequality and the strong Feller property are studied for a class of stochastic generalized porous media equations, this paper presents analogous results for stochastic fast-diffusion equations. Since the fast-diffusion equation possesses weaker dissipativity than the porous medium one does, some technical difficulties appear in the study. As a compensation to the weaker dissipativity condition, a Sobolev-Nash inequality is assumed for the underlying self-adjoint operator in applications. Some concrete examples are constructed to illustrate the main results.
Institute of Scientific and Technical Information of China (English)
Xiao-jing LIU; Ji-zeng WANG; Xiao-min WANG; You-he ZHOU
2014-01-01
General exact solutions in terms of wavelet expansion are obtained for multi-term time-fractional diffusion-wave equations with Robin type boundary conditions. By proposing a new method of integral transform for solving boundary value problems, such fractional partial differential equations are converted into time-fractional ordinary differ-ential equations, which are further reduced to algebraic equations by using the Laplace transform. Then, with a wavelet-based exact formula of Laplace inversion, the resulting exact solutions in the Laplace transform domain are reversed to the time-space domain. Three examples of wave-diffusion problems are given to validate the proposed analytical method.
Solutions of fractional reaction-diffusion equations in terms of the H-function
Haubold, H. J.; Mathai, A. M.; Saxena, R. K.
2007-12-01
This paper deals with the investigation of the solution of an unified fractional reaction-diffusion equation associated with the Caputo derivative as the time-derivative and Riesz-Feller fractional derivative as the space-derivative. The solution is derived by the application of the Laplace and Fourier transforms in closed form in terms of the H-function. The results derived are of general nature and include the results investigated earlier by many authors, notably by Mainardi et al. (2001, 2005) for the fundamental solution of the space-time fractional diffusion equation, and Saxena et al. (2006a, b) for fractional reaction-diffusion equations. The advantage of using Riesz-Feller derivative lies in the fact that the solution of the fractional reaction-diffusion equation containing this derivative includes the fundamental solution for space-time fractional diffusion, which itself is a generalization of neutral fractional diffusion, space-fractional diffusion, and time-fractional diffusion. These specialized types of diffusion can be interpreted as spatial probability density functions evolving in time and are expressible in terms of the H-functions in compact form.
Non-probabilistic solutions of imprecisely defined fractional-order diffusion equations
Chakraverty, S.; Smita, Tapaswini
2014-12-01
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 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.
A CLASS OF REACTION-DIFFUSION EQUATIONS WITH HYSTERESIS DIFFERENTIAL OPERATOR
Institute of Scientific and Technical Information of China (English)
XuLongfeng
2002-01-01
In this paper, the classical and weak derivatives with respect to spatial variable of a class of hysteresis functional are discussed. Some conclusions about solutions of a class of reaction-diffusion equations with hysteresis differential operator are given.
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.
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.
Application of linear-extended neutron diffusion equation in a semi-infinite homogeneous medium
Energy Technology Data Exchange (ETDEWEB)
Espinosa-Paredes, Gilberto, E-mail: gepe@xanum.uam.mx [Area de Ingenieria en Recursos Energeticos, Universidad Autonoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186 Col. Vicentina, Mexico 09340, D.F. (Mexico); Vazquez-Rodriguez, Rodolfo [Area de Ingenieria en Recursos Energeticos, Universidad Autonoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186 Col. Vicentina, Mexico 09340, D.F. (Mexico)
2011-02-15
The linear-extended neutron diffusion equation (LENDE) is the volume-averaged neutron diffusion equation (VANDE) which includes two correction terms: the first correction is related with the absorption process of the neutron and the second is a contribution to the neutron diffusion, both parameters are related to neutron effects on the interface of a heterogeneous configuration. In this work an analysis of a plane source in a semi-infinite homogeneous medium was considered to study the effects of the correction terms and the results obtained with the linear-extended neutron diffusion equation were compared against a semi-analytical benchmark for the same case. The comparison of the results demonstrate the excellent approach between the linear-extended diffusion theory and the selected benchmark, which means that the correction terms of the VANDE are physically acceptable.
Numerical Solutions for Convection-Diffusion Equation through Non-Polynomial Spline
Directory of Open Access Journals (Sweden)
Ravi Kanth A.S.V.
2016-01-01
Full Text Available In this paper, numerical solutions for convection-diffusion equation via non-polynomial splines are studied. We purpose an implicit method based on non-polynomial spline functions for solving the convection-diffusion equation. The method is proven to be unconditionally stable by using Von Neumann technique. Numerical results are illustrated to demonstrate the efficiency and stability of the purposed method.
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.
Solving the Advection-Diffusion Equations in Biological Contexts using the Cellular Potts Model
Dan, D; Chen, K; Glazier, J A; Dan, Debasis; Mueller, Chris; Chen, Kun; Glazier, James A.
2005-01-01
The Cellular Potts Model (CPM) is a robust, cell-level methodology for simulation of biological tissues and morphogenesis. Both tissue physiology and morphogenesis depend on diffusion of chemical morphogens in the extra-cellular fluid or matrix (ECM). Standard diffusion solvers applied to the cellular potts model use finite difference methods on the underlying CPM lattice. However, these methods produce a diffusing field tied to the underlying lattice, which is inaccurate in many biological situations in which cell or ECM movement causes advection rapid compared to diffusion. Finite difference schemes suffer numerical instabilities solving the resulting advection-diffusion equations. To circumvent these problems we simulate advection-diffusion within the framework of the CPM using off-lattice finite-difference methods. We define a set of generalized fluid particles which detach advection and diffusion from the lattice. Diffusion occurs between neighboring fluid particles by local averaging rules which approxi...
Caserta, A; Salusti, E
2016-01-01
In this paper we reconsider the classical nonlinear diffusivity equation of real gas in an heterogenous porous medium in light of the recent studies about the generalized fractional equation of conservation of mass. We first recall the physical meaning of the fractional conservation of mass recently studied by Wheatcraft and Meerschaert (2008) and then consider the implications in the classical model of diffusion of a real gas in a porous medium. Then we show that the obtained equation can be simply linearized into a classical space-fractional diffusion equation, widely studied in the literature. We also consider the case of a power-law pressure-dependence of the permeability coefficient. In this case we provide some useful exact analytical results. In particular, we are able to find a Barenblatt-type solution for a space-fractional Boussinesq equation, arising in this context.
Energy Technology Data Exchange (ETDEWEB)
Horowitz, Jordan M., E-mail: jordan.horowitz@umb.edu [Department of Physics, University of Massachusetts at Boston, Boston, Massachusetts 02125 (United States)
2015-07-28
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.
Horowitz, Jordan M
2015-07-28
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.
Renormalization group analysis of anisotropic diffusion in turbulent shear flows
Rubinstein, Robert; Barton, J. Michael
1991-01-01
The renormalization group is applied to compute anisotropic corrections to the scalar eddy diffusivity representation of turbulent diffusion of a passive scalar. The corrections are linear in the mean velocity gradients. All model constants are computed theoretically. A form of the theory valid at arbitrary Reynolds number is derived. The theory applies only when convection of the velocity-scalar correlation can be neglected. A ratio of diffusivity components, found experimentally to have a nearly constant value in a variety of shear flows, is computed theoretically for flows in a certain state of equilibrium. The theoretical value is well within the fairly narrow range of experimentally observed values. Theoretical predictions of this diffusivity ratio are also compared with data from experiments and direct numerical simulations of homogeneous shear flows with constant velocity and scalar gradients.
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.
Quasilinear equations with source terms on Carnot groups
Phuc, Nguyen Cong
2012-01-01
In this paper we give necessary and sufficient conditions for the existence of solutions to quasilinear equations of Lane--Emden type with measure data on a Carnot group $\\mathbb G$ of arbitrary step. The quasilinear part involves operators of the $p$-Laplacian type $\\Delta_{\\mathbb G,\\,p}\\,$, $1
equations with source terms which has been known only for equations involving the sub-Laplacian ($p=2$) on the Heisenberg group.
Variational iteration method for solving the time-fractional diffusion equations in porous medium
Institute of Scientific and Technical Information of China (English)
Wu Guo-Cheng
2012-01-01
The variational iteration method is successfully extended to the case of solving fractional differential equations,and the Lagrange multiplier of the method is identified in a more accurate way.Some diffusion models with fractional derivatives are investigated analytically,and the results show the efficiency of the new Lagrange multiplier for fractional differential equations of arbitrary order.
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
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.
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.
Lie symmetries and differential galois groups of linear equations
Oudshoorn, W.R.; Put, M. van der
2002-01-01
For a linear ordinary differential equation the Lie algebra of its infinitesimal Lie symmetries is compared with its differential Galois group. For this purpose an algebraic formulation of Lie symmetries is developed. It turns out that there is no direct relation between the two above objects. In co
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...
Directory of Open Access Journals (Sweden)
Bashar Zogheib
2017-01-01
Full Text Available A numerical approach is proposed for solving multidimensional parabolic diffusion and hyperbolic wave equations subject to the appropriate initial and boundary conditions. The considered numerical solutions of the these equations are considered as linear combinations of the shifted Bernoulli polynomials with unknown coefficients. By collocating the main equations together with the initial and boundary conditions at some special points (i.e., CGL collocation points, equations will be transformed into the associated systems of linear algebraic equations which can be solved by robust Krylov subspace iterative methods such as GMRES. Operational matrices of differentiation are implemented for speeding up the operations. In both of the one-dimensional and two-dimensional diffusion and wave equations, the geometrical distributions of the collocation points are depicted for clarity of presentation. Several numerical examples are provided to show the efficiency and spectral (exponential accuracy of the proposed method.
On the Fractional Nagumo Equation with Nonlinear Diffusion and Convection
Directory of Open Access Journals (Sweden)
Abdon Atangana
2014-01-01
Full Text Available We presented the Nagumo equation using the concept of fractional calculus. With the help of two analytical techniques including the homotopy decomposition method (HDM and the new development of variational iteration method (NDVIM, we derived an approximate solution. Both methods use a basic idea of integral transform and are very simple to be used.
Spherically Symmetric Waves of a Reaction-Diffusion Equation.
1980-02-01
The space A c M x [0,1] of chapter 4, section II inherited it.- semiflow from M x [0.11 which comes from the equation in cuestion . In this chapter we...have the property that an open neighbourhood of the solution whose stability is in cuestion with the inherited topology should be effectively the same
An Application of Equivalence Transformations to Reaction Diffusion Equations
Directory of Open Access Journals (Sweden)
Mariano Torrisi
2015-10-01
Full Text Available In this paper, we consider a quite general class of advection reaction diffusion systems. By using an equivalence generator, derived in a previous paper, the authors apply a projection theorem to determine some special forms of the constitutive functions that allow the extension by one of the two-dimensional principal Lie algebra. As an example, a special case is discussed at the end of the paper.
A two-phase free boundary problem for a nonlinear diffusion-convection equation
Energy Technology Data Exchange (ETDEWEB)
De Lillo, S; Lupo, G [Dipartimento di Matematica e Informatica, Universita degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia (Italy)], E-mail: silvana.delillo@pg.infn.it
2008-04-11
A two-phase free boundary problem associated with a diffusion-convection equation is considered. The problem is reduced to a system of nonlinear integral equations, which admits a unique solution for small times. The system admits an explicit two-component solution corresponding to a two-component shock wave of the Burgers equation. The stability of such a solution is also discussed.
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.
Directory of Open Access Journals (Sweden)
Xiang-Chao Shi
2016-02-01
Full Text Available The fractional reaction diffusion equation is one of the popularly used fractional partial differential equations in recent years. The fast Adomian decomposition method is used to obtain the solution of the Cauchy problem. Also, the analytical scheme is extended to the fractional one where the Taylor series is employed. In comparison with the classical Adomian decomposition method, the ratio of the convergence is increased. The method is more reliable for the fractional partial differential equations.
Malacarne, L C; Mendes, R S; Pedron, I T; Lenzi, E K
2001-03-01
The nonlinear diffusion equation partial delta rho/delta t=D Delta rho(nu) is analyzed here, where Delta[triple bond](1/r(d-1))(delta/delta r)r(d-1-theta) delta/delta r, and d, theta, and nu are real parameters. This equation unifies the anomalous diffusion equation on fractals (nu=1) and the spherical anomalous diffusion for porous media (theta=0). An exact point-source solution is obtained, enabling us to describe a large class of subdiffusion [ theta>(1-nu)d], "normal" diffusion [theta=(1-nu)d] and superdiffusion [theta<(1-nu)d]. Furthermore, a thermostatistical basis for this solution is given from the maximum entropic principle applied to the Tsallis entropy.
Approximate self-similar solutions to a nonlinear diffusion equation with time-fractional derivative
Płociniczak, Łukasz; Okrasińska, Hanna
2013-10-01
In this paper, we consider a fractional nonlinear problem for anomalous diffusion. The diffusion coefficient we use is of power type, and hence the investigated problem generalizes the porous-medium equation. A generalization is made by introducing a fractional time derivative. We look for self-similar solutions for which the fractional setting introduces other than classical space-time scaling. The resulting similarity equations are of nonlinear integro-differential type. We approximate these equations by an expansion of the integral operator and by looking for solutions in a power function form. Our method can be easily adapted to solve various problems in self-similar diffusion. The approximations obtained give very good results in numerical analysis. Their simplicity allows for easy use in applications, as our fitting with experimental data shows. Moreover, our derivation justifies theoretically some previously used empirical models for anomalous diffusion.
The three-layered mismatched media diffusion equation in frequency domain
Wang, Xichang; Wang, Shumei; Meng, Zhaokun; Yang, Shangming
2006-09-01
Near-IR radiation has great potential in medical diagnosis and therapy because of the non-invasive nature of light and the selectively poisonous effect to tumors of photodynarnic treatment. Therefore, Near-IR light propagation in highly scattering biological tissue must be understudied for basic research and clinical application of biomedical optics. A tissue is multi-layered mismatched medium, but many investigators only study the diffusion equation of matched medium. they take the tissue as the same refractive index. In order to understand the light transport in tissue, We analyze the diffusion of photons three-layered mismatched medium and set up the solution of Green's function in frequency domain, we employ the extrapolated boundary condition to set up a solution of the diffusion equation. At the same time, we utilize the diffuse equation to calculate the phase in different situation
Lectures on the theory of group properties of differential equations
Ovsyannikov, LV
2013-01-01
These lecturers provide a clear introduction to Lie group methods for determining and using symmetries of differential equations, a variety of their applications in gas dynamics and other nonlinear models as well as the author's remarkable contribution to this classical subject. It contains material that is useful for students and teachers but cannot be found in modern texts. For example, the theory of partially invariant solutions developed by Ovsyannikov provides a powerful tool for solving systems of nonlinear differential equations and investigating complicated mathematical models. Readers
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.
Space-time fractional diffusion equation using a derivative with nonsingular and regular kernel
Gómez-Aguilar, J. F.
2017-01-01
In this paper, using the fractional operators with Mittag-Leffler kernel in Caputo and Riemann-Liouville sense the space-time fractional diffusion equation is modified, the fractional equation will be examined separately; with fractional spatial derivative and fractional temporal derivative. For the study cases, the order considered is 0 < β , γ ≤ 1 respectively. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional diffusion equation, these parameters related to equation results in a fractal space-time geometry provide a new family of solutions for the diffusive processes. The proposed mathematical representation can be useful to understand electrochemical phenomena, propagation of energy in dissipative systems, viscoelastic materials, material heterogeneities and media with different scales.
A Domain Decomposition Method for Time Fractional Reaction-Diffusion Equation
Directory of Open Access Journals (Sweden)
Chunye Gong
2014-01-01
Full Text Available The computational complexity of one-dimensional time fractional reaction-diffusion equation is O(N2M compared with O(NM for classical integer reaction-diffusion equation. Parallel computing is used to overcome this challenge. Domain decomposition method (DDM embodies large potential for parallelization of the numerical solution for fractional equations and serves as a basis for distributed, parallel computations. A domain decomposition algorithm for time fractional reaction-diffusion equation with implicit finite difference method is proposed. The domain decomposition algorithm keeps the same parallelism but needs much fewer iterations, compared with Jacobi iteration in each time step. Numerical experiments are used to verify the efficiency of the obtained algorithm.
Greenshields, Christopher J
2007-01-01
Howard Brenner has recently proposed modifications to the Navier-Stokes equations that relate to a diffusion of fluid volume that would be significant for flows with high density gradients. In a previous paper (Greenshields & Reese, 2007), we found these modifications gave good predictions of the viscous structure of shock waves in argon in the range Mach 1.0-12.0 (while conventional Navier-Stokes equations are known to fail above about Mach 2). However, some areas of concern with this model were a somewhat arbitrary choice of modelling coefficient, and potentially unphysical and unstable solutions. In this paper, we therefore present slightly different modifications to include molecule mass diffusion fully in the Navier-Stokes equations. These modifications are shown to be stable and produce physical solutions to the shock problem of a quality broadly similar to those from the family of extended hydrodynamic models that includes the Burnett equations. The modifications primarily add a diffusion term to t...
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....
Asymptotic behaviour for a diffusion equation governed by nonlocal interactions
Ovono, Armel Andami
2010-01-01
In this paper we study the asymptotic behaviour of a nonlocal nonlinear parabolic equation governed by a parameter. After giving the existence of unique branch of solutions composed by stable solutions in stationary case, we gives for the parabolic problem $L^\\infty $ estimates of solution based on using the Moser iterations and existence of global attractor. We finish our study by the issue of asymptotic behaviour in some cases when $t\\to \\infty$.
Renormalization group analysis of the gluon mass equation
Aguilar, A C; Papavassiliou, J
2014-01-01
In the present work we carry out a systematic study of the renormalization properties of the integral equation that determines the momentum evolution of the effective gluon mass. A detailed, all-order analysis of the complete kernel appearing in this particular equation reveals that the renormalization procedure may be accomplished through the sole use of ingredients known from the standard perturbative treatment of the theory, with no additional assumptions. However, the subtle interplay of terms operating at the level of the exact equation gets distorted by the approximations usually employed when evaluating the aforementioned kernel. This fact is reflected in the form of the obtained solutions, whose deviations from the correct behavior are best quantified by resorting to appropriately defined renormalization-group invariant quantities. This analysis, in turn, provides a solid guiding principle for improving the form of the kernel, and furnishes a well-defined criterion for discriminating between various p...
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)
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)
Modified definition of group velocity and electromagnetic energy conservation equation
Wang, Changbiao
2013-01-01
The classical definition of group velocity has two flaws: (a) the group velocity can be greater than the phase velocity in a non-dispersive medium; (b) the definition is not consistent with the principle of relativity. To remove the flaws, a modified definition is proposed. A criterion is set up to identify the justification of group velocity definition. A "superluminal power flow" is constructed to show that the electromagnetic energy conservation equation cannot uniquely define the power flow if the principle of Fermat is not taken into account.
Linearized pseudo-Einstein equations on the Heisenberg group
Barletta, Elisabetta; Dragomir, Sorin; Jacobowitz, Howard
2017-02-01
We study the pseudo-Einstein equation R11bar = 0 on the Heisenberg group H1 = C × R. We consider first order perturbations θɛ =θ0 + ɛ θ and linearize the pseudo-Einstein equation about θ0 (the canonical Tanaka-Webster flat contact form on H1 thought of as a strictly pseudoconvex CR manifold). If θ =e2uθ0 the linearized pseudo-Einstein equation is Δb u - 4 | Lu|2 = 0 where Δb is the sublaplacian of (H1 ,θ0) and L bar is the Lewy operator. We solve the linearized pseudo-Einstein equation on a bounded domain Ω ⊂H1 by applying subelliptic theory i.e. existence and regularity results for weak subelliptic harmonic maps. We determine a solution u to the linearized pseudo-Einstein equation, possessing Heisenberg spherical symmetry, and such that u(x) → - ∞ as | x | → + ∞.
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.
Simultaneous contrast improvement and denoising via diffusion-related equations
Sapiro, Guillermo; Caselles, Vicent
1995-08-01
The explicit use of partial differential equations (PDE's) in image processing became a major topic of study in the last years. In this work we present an algorithm for histogram modification via PDE's. We show that the histogram can be modified to achieve any given distribution. The modification can be performed while simultaneously reducing noise. This avoids the noise sharpening effect in classical algorithms. The approach is extended to local contrast enhancement as well. A variational interpretation of the flow is presented and theoretical results on the existence of solutions are given.
Energy Technology Data Exchange (ETDEWEB)
Shumaker, D E; Woodward, C S
2005-05-03
In this paper, the authors investigate performance of a fully implicit formulation and solution method of a diffusion-reaction system modeling radiation diffusion with material energy transfer and a fusion fuel source. In certain parameter regimes this system can lead to a rapid conversion of potential energy into material energy. Accuracy in time integration is essential for a good solution since a major fraction of the fuel can be depleted in a very short time. Such systems arise in a number of application areas including evolution of a star and inertial confinement fusion. Previous work has addressed implicit solution of radiation diffusion problems. Recently Shadid and coauthors have looked at implicit and semi-implicit solution of reaction-diffusion systems. In general they have found that fully implicit is the most accurate method for difficult coupled nonlinear equations. In previous work, they have demonstrated that a method of lines approach coupled with a BDF time integrator and a Newton-Krylov nonlinear solver could efficiently and accurately solve a large-scale, implicit radiation diffusion problem. In this paper, they extend that work to include an additional heating term in the material energy equation and an equation to model the evolution of the reactive fuel density. This system now consists of three coupled equations for radiation energy, material energy, and fuel density. The radiation energy equation includes diffusion and energy exchange with material energy. The material energy equation includes reaction heating and exchange with radiation energy, and the fuel density equation includes its depletion due to the fuel consumption.
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...
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.
A fractional diffusion equation model for cancer tumor
Iyiola, Olaniyi Samuel; Zaman, F. D.
2014-10-01
In this article, we consider cancer tumor models and investigate the need for fractional order derivative as compared to the classical first order derivative in time. Three different cases of the net killing rate are taken into account including the case where net killing rate of the cancer cells is dependent on the concentration of the cells. At first, we use a relatively new analytical technique called q-Homotopy Analysis Method on the resulting time-fractional partial differential equations to obtain analytical solution in form of convergent series with easily computable components. Our numerical analysis enables us to give some recommendations on the appropriate order (fractional) of derivative in time to be used in modeling cancer tumor.
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...
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.
Energy Technology Data Exchange (ETDEWEB)
Holden, Helge; Karlsen, Kenneth H.; Lie, Knut-Andreas
1999-10-01
We present and analyze a numerical method for the solution of a class of scalar, multi-dimensional, nonlinear degenerate convection-diffusion equations. The method is based on operator splitting to separate the convective and the diffusive terms in the governing equation. The nonlinear, convective part is solved using front tracking and dimensional splitting, while the nonlinear diffusion equation is solved by a suitable difference scheme. We verify L{sup 1} compactness of the corresponding set of approximate solutions and derive precise entropy estimates. In particular, these results allow us to pass to the limit in our approximations and recover an entropy solution of the problem in question. The theory presented covers a large class of 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. A thorough numerical investigation of the method analyzed in this paper (and similar methods) is presented in a companion paper. (author)
Directory of Open Access Journals (Sweden)
Gianni Pagnini
2012-01-01
inhomogeneity and nonstationarity properties of the medium. For instance, when this superposition is applied to the time-fractional diffusion process, the resulting Master Equation emerges to be the governing equation of the Erdélyi-Kober fractional diffusion, that describes the evolution of the marginal distribution of the so-called generalized grey Brownian motion. This motion is a parametric class of stochastic processes that provides models for both fast and slow anomalous diffusion: it is made up of self-similar processes with stationary increments and depends on two real parameters. The class includes the fractional Brownian motion, the time-fractional diffusion stochastic processes, and the standard Brownian motion. In this framework, the M-Wright function (known also as Mainardi function emerges as a natural generalization of the Gaussian distribution, recovering the same key role of the Gaussian density for the standard and the fractional Brownian motion.
Asymptotic Properties of Solutions of Parabolic Equations Arising from Transient Diffusions
Institute of Scientific and Technical Information of China (English)
A.M. Il'in; R.Z. Khasminskii; G. Yin
2002-01-01
This work is concerned with asymptotic properties of a class of parabolic systems arising from singularly perturbed diffusions. The underlying system has a fast varying component and a slowly changing component. One of the distinct features is that the fast varying diffusion is transient. Under such a setup, this paper presents an asymptotic analysis of the solutions of such parabolic equations. Asymptotic expansions of functional satisfying the parabolic system are obtained. Error bounds are derived.
Renormalization group methods for the Reynolds stress transport equations
Rubinstein, R.
1992-01-01
The Yakhot-Orszag renormalization group is used to analyze the pressure gradient-velocity correlation and return to isotropy terms in the Reynolds stress transport equations. The perturbation series for the relevant correlations, evaluated to lowest order in the epsilon-expansion of the Yakhot-Orszag theory, are infinite series in tensor product powers of the mean velocity gradient and its transpose. Formal lowest order Pade approximations to the sums of these series produce a rapid pressure strain model of the form proposed by Launder, Reece, and Rodi, and a return to isotropy model of the form proposed by Rotta. In both cases, the model constants are computed theoretically. The predicted Reynolds stress ratios in simple shear flows are evaluated and compared with experimental data. The possibility is discussed of deriving higher order nonlinear models by approximating the sums more accurately. The Yakhot-Orszag renormalization group provides a systematic procedure for deriving turbulence models. Typical applications have included theoretical derivation of the universal constants of isotropic turbulence theory, such as the Kolmogorov constant, and derivation of two equation models, again with theoretically computed constants and low Reynolds number forms of the equations. Recent work has applied this formalism to Reynolds stress modeling, previously in the form of a nonlinear eddy viscosity representation of the Reynolds stresses, which can be used to model the simplest normal stress effects. The present work attempts to apply the Yakhot-Orszag formalism to Reynolds stress transport modeling.
Finite element method for nonlinear Riesz space fractional diffusion equations on irregular domains
Yang, Z.; Yuan, Z.; Nie, Y.; Wang, J.; Zhu, X.; Liu, F.
2017-02-01
In this paper, we consider two-dimensional Riesz space fractional diffusion equations with nonlinear source term on convex domains. Applying Galerkin finite element method in space and backward difference method in time, we present a fully discrete scheme to solve Riesz space fractional diffusion equations. Our breakthrough is developing an algorithm to form stiffness matrix on unstructured triangular meshes, which can help us to deal with space fractional terms on any convex domain. The stability and convergence of the scheme are also discussed. Numerical examples are given to verify accuracy and stability of our scheme.
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.
Anomalous diffusion in nonhomogeneous media: time-subordinated Langevin equation approach.
Srokowski, Tomasz
2014-03-01
Diffusion in nonhomogeneous media is described by a dynamical process driven by a general Lévy noise and subordinated to a random time; the subordinator depends on the position. This problem is approximated by a multiplicative process subordinated to a random time: it separately takes into account effects related to the medium structure and the memory. Density distributions and moments are derived from the solutions of the corresponding Langevin equation and compared with the numerical calculations for the exact problem. Both subdiffusion and enhanced diffusion are predicted. Distribution of the process satisfies the fractional Fokker-Planck equation.
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.
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.
A moving mesh finite difference method for equilibrium radiation diffusion equations
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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.
Ferrari, Leonardo
2008-07-28
The problem of the derivation of the diffusion equation exactly following from the Fokker-Planck (or Klein-Kramers) equation for heavy (or large) particles in a fluid in an external force field is solved in the case in which the particles are ions subject to a uniform (but in general time-varying) electric field. It is found that such a diffusion equation maintains memory of the initial ion velocity distribution, unless sufficiently large values of time are considered. In such temporal asymptotic limit, the diffusion equation exactly becomes (i) the Smoluchowski equation when the electric field is constant in time, and (ii) a new equation generalizing the Smoluchowski equation, when the electric field is arbitrarily time varying. Finally, it is shown that the obtained exact (or asymptotic) results make questionable the procedures and the results of approximate theories developed in the past to get a "corrected" Smoluchowski equation when the external force can also be, in general, position dependent.
On anomalous diffusion and the fractional generalized Langevin equation for a harmonic oscillator
Figueiredo Camargo, R.; Capelas de Oliveira, E.; Vaz, J.
2009-12-01
The fractional generalized Langevin equation (FGLE) is proposed to discuss the anomalous diffusive behavior of a harmonic oscillator driven by a two-parameter Mittag-Leffler noise. The solution of this FGLE is discussed by means of the Laplace transform methodology and the kernels are presented in terms of the three-parameter Mittag-Leffler functions. Recent results associated with a generalized Langevin equation are recovered.
Jiang, Song; 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.
VARIATIONAL DISCRETIZATION FOR OPTIMAL CONTROL GOVERNED BY CONVECTION DOMINATED DIFFUSION EQUATIONS
Institute of Scientific and Technical Information of China (English)
Michael Hinze; Ningning Yan; Zhaojie Zhou
2009-01-01
In this paper, we study variational discretization for the constrained optimal control problem governed by convection dominated diffusion equations, where the state equation is approximated by the edge stabilization Galerkin method. A priori error estimates are derived for the state, the adjoint state and the control. Moreover, residual type a posteriori error estimates in the L2-norm are obtained. Finally, two numerical experiments are presented to illustrate the theoretical results.
Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains
Li, Dingshi; Wang, Bixiang; Wang, Xiaohu
2017-02-01
This paper deals with the limiting behavior of stochastic reaction-diffusion equations driven by multiplicative noise and deterministic non-autonomous terms defined on thin domains. We first prove the existence, uniqueness and periodicity of pullback tempered random attractors for the equations in an (n + 1)-dimensional narrow domain, and then establish the upper semicontinuity of these attractors when a family of (n + 1)-dimensional thin domains collapses onto an n-dimensional domain.
D'Ovidio, Mirko
2012-01-01
We consider fractional directional derivatives and establish some connection with stable densities. Solutions to advection equations involving fractional directional derivatives are presented and some properties investigated. In particular we obtain solutions written in terms of Wright functions by exploiting operational rules involving the shift operator. We also consider fractional advection diffusion equations involving fractional powers of the negative Laplace operator and directional derivatives of fractional order and discuss the probabilistic interpretations of solutions.
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.
Renormalization Group Equation for Low Momentum Effective Nuclear Interactions
Bogner, S K; Kuo, T T S; Brown, G E
2001-01-01
We consider two nonperturbative methods originally used to derive shell model effective interactions in nuclei. These methods have been applied to the two nucleon sector to obtain an energy independent effective interaction V_{low k}, which preserves the low momentum half-on-shell T matrix and the deuteron pole, with a sharp cutoff imposed on all intermediate state momenta. We show that V_{low k} scales with the cutoff precisely as one expects from renormalization group arguments. This result is a step towards reformulating traditional model space many-body calculations in the language of effective field theories and the renormalization group. The numerical scaling properties of V_{low k} are observed to be in excellent agreement with our exact renormalization group equation.
Summation of Higher Order Effects using the Renormalization Group Equation
Elias, V; Sherry, T N
2004-01-01
The renormalization group (RG) is known to provide information about radiative corrections beyond the order in perturbation theory to which one has calculated explicitly. We first demonstrate the effect of the renormalization scheme used on these higher order effects determined by the RG. Particular attention is payed to the relationship between bare and renormalized quantities. Application of the method of characteristics to the RG equation to determine higher order effects is discussed, and is used to examine the free energy in thermal field theory, the relationship between the bare and renormalized coupling and the effective potential in massless scalar electrodynamics.
A Liouville Theorem for Nonlocal Equations in the Heisenberg Group
Directory of Open Access Journals (Sweden)
Eleonora Cinti
2014-12-01
Full Text Available We establish a Liouville-type theorem for a subcritical nonlinear problem, involving a fractional power of the sub-Laplacian in the Heisenberg group. To prove our result we will use the local realization of fractional CR covariant operators, which can be constructed as the Dirichlet-to-Neumann operator of a degenerate elliptic equation in the spirit of Caffarelli and Silvestre [8], as established in [14]. The main tools in our proof are the CR inversion and the moving plane method, applied to the solution of the lifted problem in the half-space ℍn × ℝ+.
Lie groups, differential equations, and geometry advances and surveys
2017-01-01
This book collects a series of contributions addressing the various contexts in which the theory of Lie groups is applied. A preliminary chapter serves the reader both as a basic reference source and as an ongoing thread that runs through the subsequent chapters. From representation theory and Gerstenhaber algebras to control theory, from differential equations to Finsler geometry and Lepage manifolds, the book introduces young researchers in Mathematics to a wealth of different topics, encouraging a multidisciplinary approach to research. As such, it is suitable for students in doctoral courses, and will also benefit researchers who want to expand their field of interest.
Applicability of the Fokker-Planck equation to the description of diffusion effects on nucleation
Sorokin, M. V.; Dubinko, V. I.; Borodin, V. A.
2017-01-01
The nucleation of islands in a supersaturated solution of surface adatoms is considered taking into account the possibility of diffusion profile formation in the island vicinity. It is shown that the treatment of diffusion-controlled cluster growth in terms of the Fokker-Planck equation is justified only provided certain restrictions are satisfied. First of all, the standard requirement that diffusion profiles of adatoms quickly adjust themselves to the actual island sizes (adiabatic principle) can be realized only for sufficiently high island concentration. The adiabatic principle is essential for the probabilities of adatom attachment to and detachment from island edges to be independent of the adatom diffusion profile establishment kinetics, justifying the island nucleation treatment as the Markovian stochastic process. Second, it is shown that the commonly used definition of the "diffusion" coefficient in the Fokker-Planck equation in terms of adatom attachment and detachment rates is justified only provided the attachment and detachment are statistically independent, which is generally not the case for the diffusion-limited growth of islands. We suggest a particular way to define the attachment and detachment rates that allows us to satisfy this requirement as well. When applied to the problem of surface island nucleation, our treatment predicts the steady-state nucleation barrier, which coincides with the conventional thermodynamic expression, even though no thermodynamic equilibrium is assumed and the adatom diffusion is treated explicitly. The effect of adatom diffusional profiles on the nucleation rate preexponential factor is also discussed. Monte Carlo simulation is employed to analyze the applicability domain of the Fokker-Planck equation and the diffusion effect beyond it. It is demonstrated that a diffusional cloud is slowing down the nucleation process for a given monomer interaction with the nucleus edge.
The Analysis of the Two-dimensional Diffusion Equation With a Source
Directory of Open Access Journals (Sweden)
Sunday Augustus REJU
2006-07-01
Full Text Available This study presents a new variant analysis and simulations of the two-dimensional energized wave equation remarkably different from the diffusion equations studied earlier studied. The objective functional and the dynamical energized wave are penalized to form a function called the Hamiltonian function. From this function, we obtained the necessary conditions for the optimal solutions using the maximum principle. By applying the Fourier solution to the first order differential equation, the analytical solutions for the state and control are obtained. The solutions are simulated to give visual physical interpretation of the waves and the numerical values.
Walker, Christoph
2010-01-01
The paper focuses on positive solutions to a coupled system of parabolic equations with nonlocal initial conditions. Such equations arise as steady-state equations in an age-structured predator-prey model with diffusion. By using global bifurcation techniques, we describe the structure of the set of positive solutions with respect to two parameters measuring the intensities of the fertility of the species. In particular, we establish co-existence steady-states, i.e. solutions which are nonnegative and nontrivial in both components.
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.
Energy Technology Data Exchange (ETDEWEB)
Schunert, Sebastian; Hammer, Hans; Lou, Jijie; Wang, Yaqi; Ortensi, Javier; Gleicher, Frederick; Baker, Benjamin; DeHart, Mark; Martineau, Richard
2016-11-01
The common definition of the diffusion coeffcient as the inverse of three times the transport cross section is not compat- ible with voids. Morel introduced a non-local tensor diffusion coeffcient that remains finite in voids[1]. It can be obtained by solving an auxiliary transport problem without scattering or fission. Larsen and Trahan successfully applied this diffusion coeffcient for enhancing the accuracy of diffusion solutions of very high temperature reactor (VHTR) problems that feature large, optically thin channels in the z-direction [2]. It is demonstrated that a significant reduction of error can be achieved in particular in the optically thin region. Along the same line of thought, non-local diffusion tensors are applied modeling the TREAT reactor confirming the findings of Larsen and Trahan [3]. Previous work of the authors have introduced a flexible Nonlinear-Diffusion Acceleration (NDA) method for the first order S N equations discretized with the discontinuous finite element method (DFEM), [4], [5], [6]. This NDA method uses a scalar diffusion coeffcient in the low-order system that is obtained as the flux weighted average of the inverse transport cross section. Hence, it su?ers from very large and potentially unbounded diffusion coeffcients in the low order problem. However, it was noted that the choice of the diffusion coeffcient does not influence consistency of the method at convergence and hence the di?usion coeffcient is essentially a free parameter. The choice of the di?usion coeffcient does, however, affect the convergence behavior of the nonlinear di?usion iterations. Within this work we use Morel’s non-local di?usion coef- ficient in the aforementioned NDA formulation in lieu of the flux weighted inverse of three times the transport cross section. The goal of this paper is to demonstrate that significant en- hancement of the spectral properties of NDA can be achieved in near void regions. For testing the spectral properties of the NDA
Fokker-Planck Type Equations with Sobolev Diffusion Coefficients and BV Drift Coefficients
Institute of Scientific and Technical Information of China (English)
De Jun LUO
2013-01-01
Combining Le Bris and Lions' arguments with Ambrosio's commutator estimate for BV vector fields,we prove in this paper the existence and uniqueness of solutions to the Fokker-Planck type equations with Sobolev diffusion coefficients and BV drift coefficients.
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.
Rate of Convergence to Barenblatt Profiles for the Fast Diffusion Equation
Fila, Marek; Winkler, Michael; Yanagida, Eiji
2011-01-01
We study the asymptotic behaviour of positive solutions of the Cauchy problem for the fast diffusion equation near the extinction time. We find a continuum of rates of convergence to a self-similar profile. These rates depend explicitly on the spatial decay rates of initial data.
Regularity criteria for 3D Boussinesq equations with zero thermal diffusion
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Zhuan Ye
2015-04-01
Full Text Available In this article, we consider the three-dimensional (3D incompressible Boussinesq equations with zero thermal diffusion. We establish a regularity criterion for the local smooth solution in the framework of Besov spaces in terms of the velocity only.
Blow-up criterion for the zero-diffusive Boussinesq equations via the velocity components
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Weihua Wang
2015-03-01
Full Text Available This article concerns the blow up for the smooth solutions of the three-dimensional Boussinesq equations with zero diffusivity. It is shown that if any two components of the velocity field $u$ satisfy $$ \\int_0^T \\frac{ \\||u_1|+|u_2|\\|^q_{L^{p,\\infty}} } {1+\\ln ( e+\\|\
Directory of Open Access Journals (Sweden)
Yan Sheng-Ping
2015-01-01
Full Text Available In this article, we first propose the local fractional Laplace series expansion method, which is a coupling method of series expansion method and Laplace transform via local fractional differential operator. An illustrative example for handling the diffusion equation arising in fractal heat transfer is given.
On Asymptotic Behavior for Reaction Diffusion Equation with Small Time Delay
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Xunwu Yin
2011-01-01
Full Text Available We investigate the asymptotic behavior of scalar diffusion equation with small time delay ut-Δu=f(ut,u(t-τ. Roughly speaking, any bounded solution will enter and stay in the neighborhood of one equilibrium when the equilibria are discrete.
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.
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.
The precise time-dependent solution of the Fokker–Planck equation with anomalous diffusion
Energy Technology Data Exchange (ETDEWEB)
Guo, Ran; Du, Jiulin, E-mail: jiulindu@aliyun.com
2015-08-15
We study the time behavior of the Fokker–Planck equation in Zwanzig’s rule (the backward-Ito’s rule) based on the Langevin equation of Brownian motion with an anomalous diffusion in a complex medium. The diffusion coefficient is a function in momentum space and follows a generalized fluctuation–dissipation relation. We obtain the precise time-dependent analytical solution of the Fokker–Planck equation and at long time the solution approaches to a stationary power-law distribution in nonextensive statistics. As a test, numerically we have demonstrated the accuracy and validity of the time-dependent solution. - Highlights: • The precise time-dependent solution of the Fokker–Planck equation with anomalous diffusion is found. • The anomalous diffusion satisfies a generalized fluctuation–dissipation relation. • At long time the time-dependent solution approaches to a power-law distribution in nonextensive statistics. • Numerically we have demonstrated the accuracy and validity of the time-dependent solution.
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.
Directory of Open Access Journals (Sweden)
Jose Villa-Morales
2014-02-01
Full Text Available We give a sufficient condition for blow up of positive mild solutions to an initial value problem for a nonautonomous weakly coupled system with distinct fractional diffusions. The proof is based on the study of blow up of a particular system of ordinary differential equations.
Boundedness of a Derived Function of a Solution About a Class of Diffusion Variational Equations
Institute of Scientific and Technical Information of China (English)
Kun Hui LIU
2004-01-01
In this paper the boundedness of a derived function of a solution about a class of diffusion variational equations is discussed. The application of it to related stochastic analysis problems is also illustrated. What should be emphasized is that the problem discussed and the ways proved in this paper are fundamentally new and the conclusion of this paper is fairly profound.
Convective－diffusive Cahn－Hilliard Equation with Concentration Dependent Mobility
Institute of Scientific and Technical Information of China (English)
刘长春; 尹景学
2003-01-01
In this paer,we study the global existence of classical solutions for the convective-diffusive Cahn-Hilliard Equation with concentration dependent mobility.Based on the Schauder type estimates,we establish the global existence of classical solutions.
Renormalization group analysis of the gluon mass equation
Aguilar, A. C.; Binosi, D.; Papavassiliou, J.
2014-04-01
We carry out a systematic study of the renormalization properties of the integral equation that determines the momentum evolution of the effective gluon mass in pure Yang-Mills theory, without quark effects taken into account. A detailed, all-order analysis of the complete kernel appearing in this particular equation, derived in the Landau gauge, reveals that the renormalization procedure may be accomplished through the sole use of ingredients known from the standard perturbative treatment of the theory, with no additional assumptions. However, the subtle interplay of terms operating at the level of the exact equation gets distorted by the approximations usually employed when evaluating the aforementioned kernel. This fact is reflected in the form of the obtained solutions, for which the deviations from the correct behavior are best quantified by resorting to appropriately defined renormalization-group invariant quantities. This analysis, in turn, provides a solid guiding principle for improving the form of the kernel, and furnishes a well-defined criterion for discriminating between various possibilities. Certain renormalization-group inspired Ansätze for the kernel are then proposed, and their numerical implications are explored in detail. One of the solutions obtained fulfills the theoretical expectations to a high degree of accuracy, yielding a gluon mass that is positive definite throughout the entire range of physical momenta, and displays in the ultraviolet the so-called "power-law" running, in agreement with standard arguments based on the operator product expansion. Some of the technical difficulties thwarting a more rigorous determination of the kernel are discussed, and possible future directions are briefly mentioned.
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.
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.
Some aspects of fractional diffusion equations of single and distributed order
Mainardi, Francesco; Gorenflo, Rudolf
2007-01-01
The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order $\\beta \\in (0,1)$. The fundamental solution for the Cauchy problem is interpreted as a probability density of a self-similar non-Markovian stochastic process related to a phenomenon of sub-diffusion (the variance grows in time sub-linearly). A further generalization is obtained by considering a continuous or discrete distribution of fractional time derivatives of order less than one. Then the fundamental solution is still a probability density of a non-Markovian process that, however, is no longer self-similar but exhibits a corresponding distribution of time-scales.
Stokes, Peter W; Read, Wayne; White, Ronald D
2014-01-01
The solution of a Caputo time fractional diffusion equation of order $0<\\alpha<1$ is found in terms of the solution of a corresponding integer order diffusion equation. We demonstrate a linear time mapping between these solutions that allows for accelerated computation of the solution of the fractional order problem. In the context of an $N$-point finite difference time discretisation, the mapping allows for an improvement in time computational complexity from $O\\left(N^{2}\\right)$ to $O\\left(N^{\\alpha}\\right)$, given a precomputation of $O\\left(N^{1+\\alpha}\\ln N\\right)$. The mapping is applied successfully to the least-squares fitting of a fractional advection diffusion model for the current in a time-of-flight experiment, resulting in a computational speed up in the range of one to three orders of magnitude for realistic problem sizes.
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
Natural gas diffusion through the cap rock is mainly by means ofdissolving in water, so its concentration can be replaced by solubility, which varies with temperature, pressure and salinity in strata. Under certain geological conditions the maximal solubility is definite, so the diffusion com-putation can be handled approximately by stable state equation. Furthermore, on the basis of the restoration of the paleo-buried history, the diffusion is calculated with the dynamic method, and the result is very close to the real diffusion value in the geological history.
Energy Technology Data Exchange (ETDEWEB)
Shestakov, A I; Vignes, R M; Stolken, J S
2010-01-05
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 system is coupled to a diffusion equation for the matter temperature. We are interested in modeling annealing of silica (SiO{sub 2}). We derive boundary conditions at a planar air-silica interface taking account of reflectivities. 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 CO{sub 2} laser, {lambda} = 10.59 {micro}m. In 2D, we anneal a disk (radius = 0.4, thickness = 0.4 cm) with a laser, Gaussian profile (r{sub 0} = 0.5 mm for 1/e decay).
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.
Banik, S K; Ray, D S; Banik, Suman Kumar; Bag, Bidhan Chandra; Ray, Deb Shankar
2002-01-01
Traditionally, the quantum Brownian motion is described by Fokker-Planck or diffusion equations in terms of quasi-probability distribution functions, e.g., Wigner functions. These often become singular or negative in the full quantum regime. In this paper a simple approach to non-Markovian theory of quantum Brownian motion using {\\it true probability distribution functions} is presented. Based on an initial coherent state representation of the bath oscillators and an equilibrium canonical distribution of the quantum mechanical mean values of their co-ordinates and momenta we derive a generalized quantum Langevin equation in $c$-numbers and show that the latter is amenable to a theoretical analysis in terms of the classical theory of non-Markovian dynamics. The corresponding Fokker-Planck, diffusion and the Smoluchowski equations are the {\\it exact} quantum analogues of their classical counterparts. The present work is {\\it independent} of path integral techniques. The theory as developed here is a natural ext...
Prediction equations for diffusing capacity (transfer factor of lung for North Indians
Directory of Open Access Journals (Sweden)
Sunil Kumar Chhabra
2016-01-01
Full Text Available Background: Prediction equations for diffusing capacity of lung for carbon monoxide (DLCO, alveolar volume (VA, and DLCO/VA using the current standardization guidelines are not available for Indian population. The present study was carried out to develop equations for these parameters for North Indian adults and examine the ethnic diversity in predictions. Materials and Methods: DLCO was measured by single-breath technique and VA by single-breath helium dilution using standardized methodology in 357 (258 males, 99 females normal nonsmoker adult North Indians and DLCO/VA was computed. The subjects were randomized into training and test datasets for development of prediction equations by multiple linear regressions and for validation, respectively. Results: For males, the following equations were developed: DLCO, −7.813 + 0.318 × ht −0.624 × age + 0.00552 × age 2 ; VA, −8.152 + 0.087 × ht −0.019 × wt; DLCO/VA, 7.315 − 0.037 × age. For females, the equations were: DLCO, −44.15 + 0.449 × ht −0.099 × age; VA, −6.893 + 0.068 × ht. A statistically acceptable prediction equation was not obtained for DLCO/VA in females. It was therefore computed from predicted DLCO and predicted VA. All equations were internally valid. Predictions of DLCO by Indian equations were lower than most Caucasian predictions in both males and females and greater than the Chinese predictions for males. Conclusion: This study has developed validated prediction equations for DLCO, VA, and DLCO/VA in North Indians. Substantial ethnic diversity exists in predictions for DLCO and VA with Caucasian equations generally yielding higher values than the Indian or Chinese equations. However, DLCO/VA predicted by the Indian equations is slightly higher than that by other equations.
Xi-Zhong, Liu
2012-01-01
In this paper, We derive the symmetry group theorem to the Lin-Tsien equation by using the modified CK's direct method, from which we obtain the corresponding symmetry group. More importantly, conservation laws corresponding to the Kac-Moody-Virasoro symmetry algebra of Lin-Tsien equation is obtained up to second order group invariants.
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...
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.
Multiple Group Analysis in Multilevel Structural Equation Model Across Level 1 Groups.
Ryu, Ehri
2015-01-01
This article introduces and evaluates a procedure for conducting multiple group analysis in multilevel structural equation model across Level 1 groups (MG1-MSEM; Ryu, 2014). When group membership is at Level 1, multiple group analysis raises two issues that cannot be solved by a simple extension of the standard multiple group analysis in single-level structural equation model. First, the Level 2 data are not independent between Level 1 groups. Second, the standard procedure fails to take into account the dependency between members of different Level 1 groups within the same cluster. The MG1-MSEM approach provides solutions to these problems. In MG1-MSEM, the Level 1 mean structure is necessary to represent the differences between Level 1 groups within clusters. The Level 2 model is the same regardless of Level 1 group membership. A simulation study examined the performance of MUML (Muthén's maximum likelihood) estimation in MG1-MSEM. The MG1-MSEM approach is illustrated for both a multilevel path model and a multilevel factor model using empirical data sets.
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.
Asymptotic Limit of a Singularly Perturbed Stationary Diffusion Equation: The Case of a Limit Cycle
Ge, Hao
2010-01-01
A limit cycle for a nonlinear ordinary differential equation has a sustained, stationary oscillation in time; Any non-trivial stationary stochastic process also exhibits stationary oscillations in time, though with randomness and a stationary probability density. A reconciliation of these two views of oscillatory dynamics has been elusive, although it becomes increasingly important in the biochemical modeling of cellular dynamics, where stochatic models based on the chemical master equation and the deterministic model based on the Law of Mass Action are routinely compared. Using a singularly perturbed stationary diffusion equation as a model for the chemical master equation with sufficiently large volume, $\\epsilon \\leftrightarrow 1/V$, we show that its stationary solution $u(\\vx)$ exhibits a clear separation of the exponentially and algebraic small contributions: $u(\\vx)=C_{\\epsilon}(\\vx) e^{-\\phi(\\vx)/\\epsilon}$, in which $\\phi(x)\\ge 0$ and $=0$ on the entire stable limit cycle. On the limit cycle, $C_0(\\vx...
MAXIMAL ATTRACTORS OF CLASSICAL SOLUTIONS FOR REACTION DIFFUSION EQUATIONS WITH DISPERSION
Institute of Scientific and Technical Information of China (English)
Li Yanling; Ma Yicheng
2005-01-01
The paper first deals with the existence of the maximal attractor of classical solution for reaction diffusion equation with dispersion, and gives the sup-norm estimate for the attractor. This estimate is optimal for the attractor under Neumann boundary condition. Next, the same problem is discussed for reaction diffusion system with uniformly contracting rectangle, and it reveals that the maximal attractor of classical solution for such system in the whole space is only necessary to be established in some invariant region.Finally, a few examples of application are given.
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.
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.
Analysis and numerical simulation of the diffusive wave approximation of the shallow water equations
Santillana, Mauricio
In this dissertation, the quantitative and qualitative aspects of modeling shallow water flow driven mainly by gravitational forces and dominated by shear stress, using an effective equation often referred to in the literature as the diffusive wave approximation of the shallow water equations (DSW) are presented. These flow conditions arise for example in overland flow and water flow in vegetated areas such as wetlands. The DSW equation arises in shallow water flow models when special assumptions are used to simplify the shallow water equations and contains as particular cases: the Porous Medium equation and the time evolution of the p-Laplacian. It has been successfully applied as a suitable model to simulate overland flow and water flow in vegetated areas such as wetlands; yet, no formal mathematical analysis has been carried out addressing, for example, conditions for which weak solutions may exist, and conditions for which a numerical scheme can be successful in approximating them. This thesis represents a first step in that direction. The outline of the thesis is as follows. First, a survey of relevant results coming from the studies of doubly nonlinear diffusion equations that can be applied to the DSW equation when topographic effects are ignored, is presented. Furthermore, an original proof of existence of weak solutions using constructive techniques that directly lead to the implementation of numerical algorithms to obtain approximate solutions is shown. Some regularity results about weak solutions are presented as well. Second, a numerical approach is proposed as a means to understand some properties of solutions to the DSW equation, when topographic effects are considered, and conditions for which the continuous and discontinuous Galerkin methods will succeed in approximating these weak solutions are established.
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
Maassen, Jesse; Lundstrom, Mark
2016-03-01
Understanding ballistic phonon transport effects in transient thermoreflectance experiments and explaining the observed deviations from classical theory remains a challenge. Diffusion equations are simple and computationally efficient but are widely believed to break down when the characteristic length scale is similar or less than the phonon mean-free-path. Building on our prior work, we demonstrate how well-known diffusion equations, namely, the hyperbolic heat equation and the Cattaneo equation, can be used to model ballistic phonon effects in frequency-dependent periodic steady-state thermal transport. Our analytical solutions are found to compare excellently to rigorous numerical results of the phonon Boltzmann transport equation. The correct physical boundary conditions can be different from those traditionally used and are paramount for accurately capturing ballistic effects. To illustrate the technique, we consider a simple model problem using two different, commonly used heating conditions. We demonstrate how this framework can easily handle detailed material properties, by considering the case of bulk silicon using a full phonon dispersion and mean-free-path distribution. This physically transparent approach provides clear insights into the nonequilibrium physics of quasi-ballistic phonon transport and its impact on thermal transport properties.
From quantum stochastic differential equations to Gisin-Percival state diffusion
Parthasarathy, K. R.; Usha Devi, A. R.
2017-08-01
Starting from the quantum stochastic differential equations of Hudson and Parthasarathy [Commun. Math. Phys. 93, 301 (1984)] and exploiting the Wiener-Itô-Segal isomorphism between the boson Fock reservoir space Γ (L2(R+ ) ⊗(Cn⊕Cn ) ) and the Hilbert space L2(μ ) , where μ is the Wiener probability measure of a complex n-dimensional vector-valued standard Brownian motion {B (t ) ,t ≥0 } , we derive a non-linear stochastic Schrödinger equation describing a classical diffusion of states of a quantum system, driven by the Brownian motion B. Changing this Brownian motion by an appropriate Girsanov transformation, we arrive at the Gisin-Percival state diffusion equation [N. Gisin and J. Percival, J. Phys. A 167, 315 (1992)]. This approach also yields an explicit solution of the Gisin-Percival equation, in terms of the Hudson-Parthasarathy unitary process and a randomized Weyl displacement process. Irreversible dynamics of system density operators described by the well-known Gorini-Kossakowski-Sudarshan-Lindblad master equation is unraveled by coarse-graining over the Gisin-Percival quantum state trajectories.
Institute of Scientific and Technical Information of China (English)
LIANG Hui; ZHAO Wei; DAI Dejun; ZHANG Jun
2014-01-01
Diapycnal mixing is important in oceanic circulation. An inverse method in which a semi-explicit scheme is applied to discretize the one-dimensional temperature diffusion equation is established to estimate the vertical temperature diffusion coefficient based on the observed temperature profiles. The sensitivity of the inverse model in the idealized and actual conditions is tested in detail. It can be found that this inverse model has high feasibility under multiple situations ensuring the stability of the inverse model, and can be considered as an efficient way to estimate the temperature diffusion coefficient in the weak current regions of the ocean. Here, the hydrographic profiles from Argo floats are used to estimate the temporal and spatial distribution of the vertical mixing in the north central Pacific based on this inverse method. It is further found that the vertical mixing in the upper ocean displays a distinct seasonal variation with the amplitude decreasing with depth, and the vertical mixing over rough topography is stronger than that over smooth topography. It is suggested that the high-resolution profiles from Argo floats and a more reasonable design of the inverse scheme will serve to understand mixing processes.
Lamb, George L
1995-01-01
INTRODUCTORY APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS. With Emphasis on Wave Propagation and Diffusion. This is the ideal text for students and professionals who have some familiarity with partial differential equations, and who now wish to consolidate and expand their knowledge. Unlike most other texts on this topic, it interweaves prior knowledge of mathematics and physics, especially heat conduction and wave motion, into a presentation that demonstrates their interdependence. The result is a superb teaching text that reinforces the reader's understanding of both mathematics and physic
The Transport Equation in Optically Thick Media: Discussion of IMC and its Diffusion Limit
Energy Technology Data Exchange (ETDEWEB)
Szoke, A. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Brooks, E. D. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
2016-07-12
We discuss the limits of validity of the Implicit Monte Carlo (IMC) method for the transport of thermally emitted radiation. The weakened coupling between the radiation and material energy of the IMC method causes defects in handling problems with strong transients. We introduce an approach to asymptotic analysis for the transport equation that emphasizes the fact that the radiation and material temperatures are always different in time-dependent problems, and we use it to show that IMC does not produce the correct diffusion limit. As this is a defect of IMC in the continuous equations, no improvement to its discretization can remedy it.
Existence of Young measure solutions of a class of singular diffusion equations
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
The first initial-boundary value problem of a class of singular diffusion equations with the flux sublinear growth and the potential without convexity is investigated. Such equations may be strongly degenerate, singular and forward-backward. Inspired by the idea in a recent work of Demoulini, we first discuss the regular case by introducing the Young measure solutions and prove the existence of such solutions. Consequently, we approximate the extreme case by the method of regularization. By means of some uniform estimates and some techniques, the existence of Young measure solutions with bounded variation is established.
Directory of Open Access Journals (Sweden)
Yanli Zhou
2013-01-01
Full Text Available Stochastic delay differential equations with jumps have a wide range of applications, particularly, in mathematical finance. Solution of the underlying initial value problems is important for the understanding and control of many phenomena and systems in the real world. In this paper, we construct a robust Taylor approximation scheme and then examine the convergence of the method in a weak sense. A convergence theorem for the scheme is established and proved. Our analysis and numerical examples show that the proposed scheme of high order is effective and efficient for Monte Carlo simulations for jump-diffusion stochastic delay differential equations.
Makarenkov, A. M.; Seregina, E. V.; Stepovich, M. A.
2017-05-01
Using the diffusion equation as an example, results of applying the projection Galerkin method for solving time-independent heat and mass transfer equations in a semi-infinite domain are presented. The convergence of the residual corresponding to the approximate solution of the timeindependent diffusion equation obtained by the projection method using the modified Laguerre functions is proved. Computational results for a two-dimensional toy problem are presented.
Improvement of nonlinear diffusion equation using relaxed geometric mean filter for low PSNR images
DEFF Research Database (Denmark)
Nadernejad, Ehsan
2013-01-01
A new method to improve the performance of low PSNR image denoising is presented. The proposed scheme estimates edge gradient from an image that is regularised with a relaxed geometric mean filter. The proposed method consists of two stages; the first stage consists of a second order nonlinear...... anisotropic diffusion equation with new neighboring structure and the second is a relaxed geometric mean filter, which processes the output of nonlinear anisotropic diffusion equation. The proposed algorithm enjoys the benefit of both nonlinear PDE and relaxed geometric mean filter. In addition, the algorithm...... will not introduce any artefacts, and preserves image details, sharp corners, curved structures and thin lines. Comparison of the results obtained by the proposed method, with those of other methods, shows that a noticeable improvement in the quality of the denoised images, that were evaluated subjectively...
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)
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.
Using anisotropic diffusion equations in pixon domain for image de-noising
DEFF Research Database (Denmark)
Nadernejad, Ehsan; Forchhammer, Søren; Sharifzadeh, Sara
2013-01-01
Image enhancement is an essential phase in many image processing algorithms. In any image de-noising algorithm, it is a major concern to keep the interesting structures of the image. Such interesting structures in an image often correspond to the discontinuities in the image (edges). In this paper......, we propose a new algorithm for image de-noising using anisotropic diffusion equations in pixon domain. In this approach, diffusion equations are applied on the pixonal model of the image. The algorithm has been examined on a variety of standard images and the performance has been compared...... with algorithms known from the literature. The experimental results show that in comparison with the other existing methods, the proposed algorithm has a better performance in de-noising and preserving image edges....
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)
Zanette, Rodrigo; Petersen, Caudio Zen [Univ. Federal de Pelotas, Capao do Leao (Brazil). Programa de Pos Graduacao em Modelagem Matematica; Schramm, Marcello [Univ. Federal de Pelotas (Brazil). Centro de Engenharias; Zabadal, Jorge Rodolfo [Univ. Federal do Rio Grande do Sul, Tramandai (Brazil)
2017-05-15
In this paper a solution for the one-dimensional steady state Multilayer Multigroup Neutron Diffusion Equation in cartesian geometry by Fictitious Borders Power Method and a perturbative analysis of this solution is presented. For each new iteration of the power method, the neutron flux is reconstructed by polynomial interpolation, so that it always remains in a standard form. However when the domain is long, an almost singular matrix arises in the interpolation process. To eliminate this singularity the domain segmented in R regions, called fictitious regions. The last step is to solve the neutron diffusion equation for each fictitious region in analytical form locally. The results are compared with results present in the literature. In order to analyze the sensitivity of the solution, a perturbation in the nuclear parameters is inserted to determine how a perturbation interferes in numerical results of the solution.
Application of diffusion-reaction equations to model carious lesion progress
Lewandowska, Katarzyna D.; Kosztołowicz, Tadeusz
2012-04-01
Nonlinear equations that describe the diffusion-reaction process with one static and one mobile substance are used to model a carious lesion process. The system under consideration consists of two initially separated substances A (an acid causing caries) and C (a static enamel mineral) which react chemically according to the formula A+C→0̸(inert). The so-called surface layer, which is formed in this process and in which chemical reactions can be neglected, is also included in this model. Changes in the substance concentrations are calculated approximately using the perturbation method. We show that the experimental data on the enamel mineral concentrations are well described by the analytical solutions of the diffusion-reaction equations.
Cubic B-Spline Collocation Method for One-Dimensional Heat and Advection-Diffusion Equations
Joan Goh; Ahmad Abd. Majid; Ahmad Izani Md. Ismail
2012-01-01
Numerical solutions of one-dimensional heat and advection-diffusion equations are obtained by collocation method based on cubic B-spline. Usual finite difference scheme is used for time and space integrations. Cubic B-spline is applied as interpolation function. The stability analysis of the scheme is examined by the Von Neumann approach. The efficiency of the method is illustrated by some test problems. The numerical results are found to be in good agreement with the exact solution.
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.
Blowup Analysis for a Nonlocal Diffusion Equation with Reaction and Absorption
Directory of Open Access Journals (Sweden)
Yulan Wang
2012-01-01
Full Text Available We investigate a nonlocal reaction diffusion equation with absorption under Neumann boundary. We obtain optimal conditions on the exponents of the reaction and absorption terms for the existence of solutions blowing up in finite time, or for the global existence and boundedness of all solutions. For the blowup solutions, we also study the blowup rate estimates and the localization of blowup set. Moreover, we show some numerical experiments which illustrate our results.
Institute of Scientific and Technical Information of China (English)
王文洽
2003-01-01
A new discrete approximation to the convection term of the covection-diffusionequation was constructed in Saul' yev type difference scheme, then the alternating segmentCrank-Nicolson( ASC-N) method for solving the convection-diffusion equation with variablecoefficient was developed. The ASC-N method is unconditionally stable. Numericalexperiment shows that this method has the obvious property of parallelism and accuracy. Themethod can be used directly on parallel computers.
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.
Moving-boundary problems for the time-fractional diffusion equation
Directory of Open Access Journals (Sweden)
Sabrina D. Roscani
2017-02-01
Full Text Available We consider a one-dimensional moving-boundary problem for the time-fractional diffusion equation. The time-fractional derivative of order $\\alpha\\in (0,1$ is taken in the sense of Caputo. We study the asymptotic behaivor, as t tends to infinity, of a general solution by using a fractional weak maximum principle. Also, we give some particular exact solutions in terms of Wright functions.
Institute of Scientific and Technical Information of China (English)
Zhongping LI; Wanjuan DU; Chunlai MU
2013-01-01
In this paper,we first find finite travelling-wave solutions,and then investigate the short time development of interfaces for non-Newtonian diffusion equations with strong absorption.We show that the initial behavior of the interface depends on the concentration of the mass of u(x,0) near x =0.More precisely,we find a critical value of the concentration,which separates the heating front of interfaces from the cooling front of them.
Institute of Scientific and Technical Information of China (English)
Ningning YAN; Zhaojie ZHOU
2008-01-01
In this paper,we study a posteriori error estimates of the edge stabilization Galerkin method for the constrained optimal control problem governed by convection-dominated diffusion equations.The residual-type a posteriori error estimators yield both upper and lower bounds for control u measured in L2-norm and for state y and costate p measured in energy norm.Two numerical examples are presented to illustrate the effectiveness of the error estimators provided in this paper.
Energy Technology Data Exchange (ETDEWEB)
Ceolin, C., E-mail: celina.ceolin@gmail.com [Universidade Federal de Santa Maria (UFSM), Frederico Westphalen, RS (Brazil). Centro de Educacao Superior Norte; Schramm, M.; Bodmann, B.E.J.; Vilhena, M.T., E-mail: celina.ceolin@gmail.com [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica
2015-07-01
Recently the stationary neutron diffusion equation in heterogeneous rectangular geometry was solved by the expansion of the scalar fluxes in polynomials in terms of the spatial variables (x; y), considering the two-group energy model. The focus of the present discussion consists in the study of an error analysis of the aforementioned solution. More specifically we show how the spatial subdomain segmentation is related to the degree of the polynomial and the Lipschitz constant. This relation allows to solve the 2-D neutron diffusion problem for second degree polynomials in each subdomain. This solution is exact at the knots where the Lipschitz cone is centered. Moreover, the solution has an analytical representation in each subdomain with supremum and infimum functions that shows the convergence of the solution. We illustrate the analysis with a selection of numerical case studies. (author)
Nonperturbative renormalization group study of the stochastic Navier-Stokes equation.
Mejía-Monasterio, Carlos; Muratore-Ginanneschi, Paolo
2012-07-01
We study the renormalization group flow of the average action of the stochastic Navier-Stokes equation with power-law forcing. Using Galilean invariance, we introduce a nonperturbative approximation adapted to the zero-frequency sector of the theory in the parametric range of the Hölder exponent 4-2ε of the forcing where real-space local interactions are relevant. In any spatial dimension d, we observe the convergence of the resulting renormalization group flow to a unique fixed point which yields a kinetic energy spectrum scaling in agreement with canonical dimension analysis. Kolmogorov's -5/3 law is, thus, recovered for ε = 2 as also predicted by perturbative renormalization. At variance with the perturbative prediction, the -5/3 law emerges in the presence of a saturation in the ε dependence of the scaling dimension of the eddy diffusivity at ε = 3/2 when, according to perturbative renormalization, the velocity field becomes infrared relevant.
Equation of Diffusion of a Composite Mixture into a Composite Medium
Kravchuk, A. S.; Kravchuk, A. I.; Popova, T. S.
2016-07-01
The equation of diffusion of a composite mixture into a composite medium has been obtained for the first time. The assumption used is that the macropoint of the medium, i.e., an elementary macrovolume, in which the statistical parameters of distribution of inhomogeneities coincide with the corresponding values assigned for the medium as a whole, is small compared to the geometric dimensions of the volume considered. The ″Reuss-Voigt fork″ has been obtained for determining the limits of the change in the diffusion coefficient. Thereafter the fork is narrowed to the ″Kravchuk-Tarasyuk fork.″ Effective diffusion coefficients are obtained as an arithmetic mean value of the Kravchuk-Tarasyuk fork. The found averaged physical parameters can be used in solving specific physical problems for inhomogeneous media.
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.
Saxena, R. K.; Mathai, A. M.; Haubold, H. J.
2015-10-01
This paper deals with the investigation of the computational solutions of an unified fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the generalized fractional time-derivative defined by Hilfer (2000), the space derivative of second order by the Riesz-Feller fractional derivative and adding the function ϕ (x, t) which is a nonlinear function governing reaction. The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of the H-function. The main result obtained in this paper provides an elegant extension of the fundamental solution for the space-time fractional diffusion equation obtained earlier by Mainardi et al. (2001, 2005) and a result very recently given by Tomovski et al. (2011). Computational representation of the fundamental solution is also obtained explicitly. Fractional order moments of the distribution are deduced. At the end, mild extensions of the derived results associated with a finite number of Riesz-Feller space fractional derivatives are also discussed.
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.
Sierakowski, Adam J.; Lukassen, Laura J.
2016-11-01
In the shear flow of non-Brownian particles, we describe the long-time diffusive processes stochastically using a Fokker-Planck equation. Previous work has indicated that a Fokker-Planck equation coupling the probability densities of position and velocity spaces may be appropriate for describing this phenomenon. The stochastic description, integrated over velocity space to obtain a reduced position-space Fokker-Planck equation, contains unknown space diffusion coefficients. In this work, we use the Physalis method for simulating disperse particle flows to verify the colored-noise velocity space model (an Ornstein-Uhlenbeck process) by comparing the simulated long-time diffusion rate with the diffusion rate proposed by the theory. We then use the simulated data to calculate the unknown space diffusion coefficients that appear in the reduced position-space Fokker-Planck equation and summarize the results. This study was partially supported by US NSF Grant CBET1335965.
Functional renormalisation group equations for supersymmetric field theories
Energy Technology Data Exchange (ETDEWEB)
Synatschke-Czerwonka, Franziska
2011-01-11
This work is organised as follows: In chapter 2 the basic facts of quantum field theory are collected and the functional renormalisation group equations are derived. Chapter 3 gives a short introduction to the main concepts of supersymmetry that are used in the subsequent chapters. In chapter 4 the functional RG is employed for a study of supersymmetric quantum mechanics, a supersymmetric model which are studied intensively in the literature. A lot of results have previously been obtained with different methods and we compare these to the ones from the FRG. We investigate the N=1 Wess-Zumino model in two dimensions in chapter 5. This model shows spontaneous supersymmetry breaking and an interesting fixed-point structure. Chapter 6 deals with the three dimensional N=1 Wess-Zumino model. Here we discuss the zero temperature case as well as the behaviour at finite temperature. Moreover, this model shows spontaneous supersymmetry breaking, too. In chapter 7 the two-dimensional N=(2,2) Wess-Zumino model is investigated. For the superpotential a non-renormalisation theorem holds and thus guarantees that the model is finite. This allows for a direct comparison with results from lattice simulations. (orig.)
Directory of Open Access Journals (Sweden)
Kanittha Yimnak
2014-01-01
Full Text Available The meshless local Pretrov-Galerkin method (MLPG with the test function in view of the Heaviside step function is introduced to solve the system of coupled nonlinear reaction-diffusion equations in two-dimensional spaces subjected to Dirichlet and Neumann boundary conditions on a square domain. Two-field velocities are approximated by moving Kriging (MK interpolation method for constructing nodal shape function which holds the Kronecker delta property, thereby enhancing the arrangement nodal shape construction accuracy, while the Crank-Nicolson method is chosen for temporal discretization. The nonlinear terms are treated iteratively within each time step. The developed formulation is verified in two numerical examples with investigating the convergence and the accuracy of numerical results. The numerical experiments revealing the solutions by the developed formulation are stable and more precise.
Luchko, Yuri; Mainardi, Francesco
2013-06-01
In this paper, the one-dimensional time-fractional diffusion-wave equation with the Caputo fractional derivative of order α, 1 ≤ α ≤ 2 and with constant coefficients is revisited. It is known that the diffusion and the wave equations 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. We show that the time-fractional diffusion-wave equation interpolates between these two different responses and investigate the behavior of its fundamental solution for the signalling problem in detail. In particular, the maximum location, the maximum value, and the propagation velocity of the maximum point of the fundamental solution for the signalling problem are described analytically and calculated numerically.
Energy Technology Data Exchange (ETDEWEB)
Potemki, Valeri G. [Moscow State Engineering Physics Institute (Technical University), Moscow (Russian Federation). Dept. of Automatics and Electronics; Borisevich, Valentine D.; Yupatov, Sergei V. [Moscow State Enineering Physics Institute (Technical University), Moscow (Russian Federation). Dept. of Technical Physics
1996-12-31
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) 7 refs., 10 figs.
On the master equation approach to diffusive grain-surface chemistry: the H, O, CO system
Stantcheva, T; Herbst, E
2002-01-01
We have used the master equation approach to study a moderately complex network of diffusive reactions occurring on the surfaces of interstellar dust particles. This network is meant to apply to dense clouds in which a large portion of the gas-phase carbon has already been converted to carbon monoxide. Hydrogen atoms, oxygen atoms, and CO molecules are allowed to accrete onto dust particles and their chemistry is followed. The stable molecules produced are oxygen, hydrogen, water, carbon dioxide (CO2), formaldehyde (H2CO), and methanol (CH3OH). The surface abundances calculated via the master equation approach are in good agreement with those obtained via a Monte Carlo method but can differ considerably from those obtained with standard rate equations.
A Two-grid Method with Expanded Mixed Element for Nonlinear Reaction-diffusion Equations
Institute of Scientific and Technical Information of China (English)
Wei Liu; Hong-xing Rui; Hui Guo
2011-01-01
Expanded mixed finite element approximation of nonlinear reaction-diffusion equations is discussed. The equations considered here are used to model the hydrologic and bio-geochemical phenomena. To linearize the mixed-method equations, we use a two-grid method involving a small nonlinear system on a coarse gird of size H and a linear system on a fine grid of size h. Error estimates are derived which demonstrate that the error is O(△t + hk+1 + H2k+2-d/2) (k ≥ 1), where k is the degree of the approximating space for the primary variable and d is the spatial dimension. The above estimates are useful for determining an appropriate H for the coarse grid problems.
Saxena, R K; Haubold, H J
2011-01-01
The object of this paper is to present a computable solution of a fractional partial differential equation associated with a Riemann-Liouville derivative of fractional order as the time-derivative and Riesz-Feller fractional derivative as the space derivative. The method followed in deriving the solution is that of joint Laplace and Fourier transforms. The solution is derived in a closed and computable form in terms of the H-function. It provides an elegant extension of the results given earlier by Debnath, Chen et al., Haubold et al., Mainardi et al., Saxena et al., and Pagnini et al. The results obtained are presented in the form of four theorems. Some results associated with fractional Schroeodinger equation and fractional diffusion-wave equation are also derived as special cases of the findings.
A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains
Bonforte, Matteo; Vázquez, Juan Luis
2015-10-01
We investigate quantitative properties of the nonnegative solutions to the nonlinear fractional diffusion equation, , posed in a bounded domain, , with m > 1 for t > 0. As we use one of the most common definitions of the fractional Laplacian , 0 zero Dirichlet boundary conditions. We consider a general class of very weak solutions of the equation, and obtain a priori estimates in the form of smoothing effects, absolute upper bounds, lower bounds, and Harnack inequalities. We also investigate the boundary behaviour and we obtain sharp estimates from above and below. In addition, we obtain similar estimates for fractional semilinear elliptic equations. Either the standard Laplacian case s = 1 or the linear case m = 1 are recovered as limits. The method is quite general, suitable to be applied to a number of similar problems.
Higher-order Solution of Stochastic Diffusion equation with Nonlinear Losses Using WHEP technique
El-Beltagy, Mohamed A.
2014-01-06
Using Wiener-Hermite expansion with perturbation (WHEP) technique in the solution of the stochastic partial differential equations (SPDEs) has the advantage of converting the problem to a system of deterministic equations that can be solved efficiently using the standard deterministic numerical methods [1]. The Wiener-Hermite expansion is the only known expansion that handles the white/colored noise exactly. The main statistics, such as the mean, covariance, and higher order statistical moments, can be calculated by simple formulae involving only the deterministic Wiener-Hermite coefficients. In this poster, the WHEP technique is used to solve the 2D diffusion equation with nonlinear losses and excited with white noise. The solution will be obtained numerically and will be validated and compared with the analytical solution that can be obtained from any symbolic mathematics package such as Mathematica.
Lie Group Classifications and Non-differentiable Solutions for Time-Fractional Burgers Equation
Institute of Scientific and Technical Information of China (English)
WU Guo-Cheng
2011-01-01
Lie group method provides an efficient tool to solve nonlinear partial differential equations.This paper suggests Lie group method for fractional partial differential equations.A time-fractional Burgers equation is used as an example to illustrate the effectiveness of the Lie group method and some classes of exact solutions are obtained.
Use versus Nonuse of Repeater Examinees in Common Item Linear Equating with Nonrandom Groups.
Cope, Ronald T.
This study considers the use of repeaters when test equating. The subjects consist of five groups of applicants to a professional certification program. Each group comprises first time examinees and repeaters. The procedures include a common item linear equating with nonrandom groups, use of equating chains, and the use of total examinee group…
Hellander, Andreas; Lawson, Michael J.; Drawert, Brian; Petzold, Linda
2014-06-01
The efficiency of exact simulation methods for the reaction-diffusion master equation (RDME) is severely limited by the large number of diffusion events if the mesh is fine or if diffusion constants are large. Furthermore, inherent properties of exact kinetic-Monte Carlo simulation methods limit the efficiency of parallel implementations. Several approximate and hybrid methods have appeared that enable more efficient simulation of the RDME. A common feature to most of them is that they rely on splitting the system into its reaction and diffusion parts and updating them sequentially over a discrete timestep. This use of operator splitting enables more efficient simulation but it comes at the price of a temporal discretization error that depends on the size of the timestep. So far, existing methods have not attempted to estimate or control this error in a systematic manner. This makes the solvers hard to use for practitioners since they must guess an appropriate timestep. It also makes the solvers potentially less efficient than if the timesteps were adapted to control the error. Here, we derive estimates of the local error and propose a strategy to adaptively select the timestep when the RDME is simulated via a first order operator splitting. While the strategy is general and applicable to a wide range of approximate and hybrid methods, we exemplify it here by extending a previously published approximate method, the diffusive finite-state projection (DFSP) method, to incorporate temporal adaptivity.
Hellander, Andreas; Lawson, Michael J; Drawert, Brian; Petzold, Linda
2015-01-01
The efficiency of exact simulation methods for the reaction-diffusion master equation (RDME) is severely limited by the large number of diffusion events if the mesh is fine or if diffusion constants are large. Furthermore, inherent properties of exact kinetic-Monte Carlo simulation methods limit the efficiency of parallel implementations. Several approximate and hybrid methods have appeared that enable more efficient simulation of the RDME. A common feature to most of them is that they rely on splitting the system into its reaction and diffusion parts and updating them sequentially over a discrete timestep. This use of operator splitting enables more efficient simulation but it comes at the price of a temporal discretization error that depends on the size of the timestep. So far, existing methods have not attempted to estimate or control this error in a systematic manner. This makes the solvers hard to use for practitioners since they must guess an appropriate timestep. It also makes the solvers potentially less efficient than if the timesteps are adapted to control the error. Here, we derive estimates of the local error and propose a strategy to adaptively select the timestep when the RDME is simulated via a first order operator splitting. While the strategy is general and applicable to a wide range of approximate and hybrid methods, we exemplify it here by extending a previously published approximate method, the Diffusive Finite-State Projection (DFSP) method, to incorporate temporal adaptivity. PMID:26865735
Hybrid simplified spherical harmonics with diffusion equation for light propagation in tissues
Chen, Xueli; Sun, Fangfang; Yang, Defu; Ren, Shenghan; Zhang, Qian; Liang, Jimin
2015-08-01
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.
Simulation of a fast diffuse optical tomography system based on radiative transfer equation
Motevalli, S. M.; Payani, A.
2016-12-01
Studies show that near-infrared (NIR) light (light with wavelength between 700nm and 1300nm) undergoes two interactions, absorption and scattering, when it penetrates a tissue. Since scattering is the predominant interaction, the calculation of light distribution in the tissue and the image reconstruction of absorption and scattering coefficients are very complicated. Some analytical and numerical methods, such as radiative transport equation and Monte Carlo method, have been used for the simulation of light penetration in tissue. Recently, some investigators in the world have tried to develop a diffuse optical tomography system. In these systems, NIR light penetrates the tissue and passes through the tissue. Then, light exiting the tissue is measured by NIR detectors placed around the tissue. These data are collected from all the detectors and transferred to the computational parts (including hardware and software), which make a cross-sectional image of the tissue after performing some computational processes. In this paper, the results of the simulation of an optical diffuse tomography system are presented. This simulation involves two stages: a) Simulation of the forward problem (or light penetration in the tissue), which is performed by solving the diffusion approximation equation in the stationary state using FEM. b) Simulation of the inverse problem (or image reconstruction), which is performed by the optimization algorithm called Broyden quasi-Newton. This method of image reconstruction is faster compared to the other Newton-based optimization algorithms, such as the Levenberg-Marquardt one.
Collins, Kimberlee C.; Maznev, Alexei A.; Tian, Zhiting; Esfarjani, Keivan; Nelson, Keith A.; Chen, Gang
2013-09-01
The relaxation of an one-dimensional transient thermal grating (TTG) in a medium with phonon-mediated thermal transport is analyzed within the framework of the Boltzmann transport equation (BTE), with the goal of extracting phonon mean free path (MFP) information from TTG measurements of non-diffusive phonon transport. Both gray-medium (constant MFP) and spectrally dependent MFP models are considered. In the gray-medium approximation, an analytical solution is derived. For large TTG periods compared to the MFP, the model yields an exponential decay of grating amplitude with time in agreement with Fourier's heat diffusion equation, and at shorter periods, phonon transport transitions to the ballistic regime, with the decay becoming strongly non-exponential. Spectral solutions are obtained for Si and PbSe at 300 K using phonon dispersion and lifetime data from density functional theory calculations. The spectral decay behaviors are compared to several approximate models: a single MFP solution, a frequency-integrated gray-medium model, and a "two-fluid" BTE solution. We investigate the utility of using the approximate models for the reconstruction of phonon MFP distributions from non-diffusive TTG measurements.
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.
Space-Time Fractional Diffusion-Advection Equation with Caputo Derivative
Directory of Open Access Journals (Sweden)
José Francisco Gómez Aguilar
2014-01-01
Full Text Available An alternative construction for the space-time fractional diffusion-advection equation for the sedimentation phenomena is presented. The order of the derivative is considered as 0<β, γ≤1 for the space and time domain, respectively. The fractional derivative of Caputo type is considered. In the spatial case we obtain the fractional solution for the underdamped, undamped, and overdamped case. In the temporal case we show that the concentration has amplitude which exhibits an algebraic decay at asymptotically large times and also shows numerical simulations where both derivatives are taken in simultaneous form. In order that the equation preserves the physical units of the system two auxiliary parameters σx and σt are introduced characterizing the existence of fractional space and time components, respectively. A physical relation between these parameters is reported and the solutions in space-time are given in terms of the Mittag-Leffler function depending on the parameters β and γ. The generalization of the fractional diffusion-advection equation in space-time exhibits anomalous behavior.
Korneev, V. G.
2016-11-01
Efficiency of the error control of numerical solutions of partial differential equations entirely depends on the two factors: accuracy of an a posteriori error majorant and the computational cost of its evaluation for some test function/vector-function plus the cost of the latter. In the paper consistency of an a posteriori bound implies that it is the same in the order with the respective unimprovable a priori bound. Therefore, it is the basic characteristic related to the first factor. The paper is dedicated to the elliptic diffusion-reaction equations. We present a guaranteed robust a posteriori error majorant effective at any nonnegative constant reaction coefficient (r.c.). For a wide range of finite element solutions on a quasiuniform meshes the majorant is consistent. For big values of r.c. the majorant coincides with the majorant of Aubin (1972), which, as it is known, for relatively small r.c. (< ch -2 ) is inconsistent and looses its sense at r.c. approaching zero. Our majorant improves also some other majorants derived for the Poisson and reaction-diffusion equations.
Huang, Ding-jiang; Ivanova, Nataliya M.
2016-02-01
In this paper, we explain in more details the modern treatment of the problem of group classification of (systems of) partial differential equations (PDEs) from the algorithmic point of view. More precisely, we revise the classical Lie algorithm of construction of symmetries of differential equations, describe the group classification algorithm and discuss the process of reduction of (systems of) PDEs to (systems of) equations with smaller number of independent variables in order to construct invariant solutions. The group classification algorithm and reduction process are illustrated by the example of the generalized Zakharov-Kuznetsov (GZK) equations of form ut +(F (u)) xxx +(G (u)) xyy +(H (u)) x = 0. As a result, a complete group classification of the GZK equations is performed and a number of new interesting nonlinear invariant models which have non-trivial invariance algebras are obtained. Lie symmetry reductions and exact solutions for two important invariant models, i.e., the classical and modified Zakharov-Kuznetsov equations, are constructed. The algorithmic framework for group analysis of differential equations presented in this paper can also be applied to other nonlinear PDEs.
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.
Analytical solutions of a fractional diffusion-advection equation for solar cosmic-ray transport
Energy Technology Data Exchange (ETDEWEB)
Litvinenko, Yuri E.; Effenberger, Frederic, E-mail: yuril@waikato.ac.nz [Department of Mathematics, University of Waikato, P.B. 3105 Hamilton (New Zealand)
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.
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.
Regularity and mass conservation for discrete coagulation-fragmentation equations with diffusion
Cañizo, José A; Fellner, Klemens
2009-01-01
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 $L^2$ 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.
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.
Yang-Baxter Maps, Discrete Integrable Equations and Quantum Groups
Bazhanov, Vladimir V
2015-01-01
For every quantized Lie algebra there exists a map from the tensor square of the algebra to itself, which by construction satisfies the set-theoretic Yang-Baxter equation. This map allows one to define an integrable discrete quantum evolution system on quadrilateral lattices, where local degrees of freedom (dynamical variables) take values in a tensor power of the quantized Lie algebra. The corresponding equations of motion admit the zero curvature representation. The commuting Integrals of Motion are defined in the standard way via the Quantum Inverse Problem Method, utilizing Baxter's famous commuting transfer matrix approach. All elements of the above construction have a meaningful quasi-classical limit. As a result one obtains an integrable discrete Hamiltonian evolution system, where the local equation of motion are determined by a classical Yang-Baxter map and the action functional is determined by the quasi-classical asymptotics of the universal R-matrix of the underlying quantum algebra. In this paper...
Kordilla, Jannes; Pan, Wenxiao; Tartakovsky, Alexandre
2014-12-14
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.
Raghib, M; Levin, S A; Kevrekidis, I G
2010-06-01
We propose a (time) multiscale method for the coarse-grained analysis of collective motion and decision-making in self-propelled particle models of swarms comprising a mixture of 'naïve' and 'informed' individuals. The method is based on projecting the particle configuration onto a single 'meta-particle' that consists of the elongation of the flock together with the mean group velocity and position. We find that the collective states can be associated with the transient and asymptotic transport properties of the random walk followed by the meta-particle, which we assume follows a continuous time random walk (CTRW). These properties can be accurately predicted at the macroscopic level by an advection-diffusion equation with memory (ADEM) whose parameters are obtained from a mean group velocity time series obtained from a single simulation run of the individual-based model.
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.
Asymptotic Analysis to a Diffusion Equation with a Weighted Nonlo cal Source
Institute of Scientific and Technical Information of China (English)
JIANG Liang-jun
2015-01-01
In this paper, we deal with the blow-up property of the solution to the diffusion equation ut=∆u+a(x)f(u) RΩh(u)dx, x∈Ω, t>0 subject to the null Dirichlet boundary condition. We will show that under certain conditions, the solution blows up in finite time and prove that the set of all blow-up points is the whole region. Especially, in case of f(s)=sp, h(s)=sq, 0≤p≤1, p+q>1, we obtain the asymptotic behavior of the blow up solution.
Threshold phenomena for symmetric decreasing solutions of reaction-diffusion equations
Muratov, C B
2012-01-01
We study the long time behavior of solutions of the Cauchy problem for nonlinear reaction-diffusion equations in one space dimension with the nonlinearity of bistable, ignition or monostable type. We prove a one-to-one relation between the long time behavior of the solution and the limit value of its energy for symmetric decreasing initial data in $L^2$ under minimal assumptions on the nonlinearities. The obtained relation allows to establish sharp threshold results between propagation and extinction for monotone families of initial data in the considered general setting.
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.
Cubic B-Spline Collocation Method for One-Dimensional Heat and Advection-Diffusion Equations
Directory of Open Access Journals (Sweden)
Joan Goh
2012-01-01
Full Text Available Numerical solutions of one-dimensional heat and advection-diffusion equations are obtained by collocation method based on cubic B-spline. Usual finite difference scheme is used for time and space integrations. Cubic B-spline is applied as interpolation function. The stability analysis of the scheme is examined by the Von Neumann approach. The efficiency of the method is illustrated by some test problems. The numerical results are found to be in good agreement with the exact solution.
Relation between the complex Ginzburg-Landau equation and reaction-diffusion System
Institute of Scientific and Technical Information of China (English)
Shao Xin; Ren Yi; Ouyang Qi
2006-01-01
The complex Ginzburg-Landau equation(CGLE)has been used to describe the travelling wave behaviour in reaction-diffusion (RD) systems. We argue that this description is valid only when the RD system is close to the Hopf bifurcation,and is not valid when a RD system is away from the onset.To test this,we study spirals and anti-spirals in the chlorite-iodide-malonic acid (CIMA) reaction and the corresponding CGLE.Numerical simulations confirm that the CGLE can only be applied to the CIMA reaction when it is very near the Hopf onset.
A Monte Carlo synthetic-acceleration method for solving the thermal radiation diffusion equation
Energy Technology Data Exchange (ETDEWEB)
Evans, Thomas M., E-mail: evanstm@ornl.gov [Oak Ridge National Laboratory, 1 Bethel Valley Rd., Oak Ridge, TN 37831 (United States); Mosher, Scott W., E-mail: moshersw@ornl.gov [Oak Ridge National Laboratory, 1 Bethel Valley Rd., Oak Ridge, TN 37831 (United States); Slattery, Stuart R., E-mail: sslattery@wisc.edu [University of Wisconsin–Madison, 1500 Engineering Dr., Madison, WI 53716 (United States); Hamilton, Steven P., E-mail: hamiltonsp@ornl.gov [Oak Ridge National Laboratory, 1 Bethel Valley Rd., Oak Ridge, TN 37831 (United States)
2014-02-01
We present a novel synthetic-acceleration-based Monte Carlo method for solving the equilibrium thermal radiation diffusion equation in three spatial dimensions. The algorithm performance is compared against traditional solution techniques using a Marshak benchmark problem and a more complex multiple material problem. Our results show that our Monte Carlo method is an effective solver for sparse matrix systems. For solutions converged to the same tolerance, it performs competitively with deterministic methods including preconditioned conjugate gradient and GMRES. We also discuss various aspects of preconditioning the method and its general applicability to broader classes of problems.
Free boundary value problems for a class of generalized diffusion equation
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
The transport behavior of free boundary value problems for a class of generalized diffusion equations was studied. Suitable similarity transformations were used to convert the problems into a class of singular nonlinear two-point boundary value problems and similarity solutions were numerical presented for different representations of heat conduction function, convection function, heat flux function, and power law parameters by utilizing the shooting technique. The results revealed the flux transfer mechanism and the character as well as the effects of parameters on the solutions.
Prozorov, A. A.; Trifonov, A. Yu.; Shapovalov, A. V.
2015-07-01
Asymptotic solutions of the nonlocal, one-dimensional Fisher-Kolmogorov-Petrovskii-Piskunov equation with fractional derivatives in the diffusion operator are constructed. The fractional derivative is defined in accordance with the approaches of Weyl, Grünwald-Letnilkov, and Liouville. Asymptotic solutions are constructed in a class of functions that are a perturbation of the found exact quasistationary solution and tend at large times to this quasistationary solution. It is shown that the presence of fractional derivatives leads to drift of the center of mass of the initial distribution and breaks its symmetry.
Theoretical study of some nodal methods for the solution of the diffusion equation. Numerical tests
Energy Technology Data Exchange (ETDEWEB)
Fedon-Magnaud, C.
1983-08-01
The nodal methods used in the solution of the neutron multigroup diffusion equation are described. A new formulation of this methods is obtained in order to have a comparison with the finite element methods. After a brief review of nonconforming finite element theory, we use a Radau formula to establish the equivalence with nodal schemes. Convergence theorems and error estimations are then obtained. In the last part, numerical calculations are performed for two reactor test configurations. Comparisons are done between nodal or nonconforming schemes and more classical methods (F.D., conforming F.E.) wich are used in reactor analysis.
Directory of Open Access Journals (Sweden)
Hassan Kamil Jassim
2016-01-01
Full Text Available We used the local fractional variational iteration transform method (LFVITM coupled by the local fractional Laplace transform and variational iteration method to solve three-dimensional diffusion and wave equations with local fractional derivative operator. This method has Lagrange multiplier equal to minus one, which makes the calculations more easily. The obtained results show that the presented method is efficient and yields a solution in a closed form. Illustrative examples are included to demonstrate the high accuracy and fast convergence of this new method.
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.
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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.
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.
Hamilton-Jacobi equation, heteroclinic chains and Arnol'd diffusion in three time scales systems
Gallavotti, G; Mastropietro, V; Gallavotti, Giovanni; Gentile, Guido; Mastropietro, Vieri
1998-01-01
Interacting systems consisting of two rotators and a point mass near a hyperbolic fixed point are considered, in a case in which the uncoupled systems have three very different characteristic time scales. The abundance of quasi periodic motions in phase space is studied via the Hamilton-Jacobi equation. The main result, a high density theorem of invariant tori, is derived by the classical canonical transformation method extending previous results. As an application the existence of long heteroclinic chains (and of Arnol'd diffusion) is proved for systems interacting through a trigonometric polynomial in the angle variables.
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.
THE EXTINCTION BEHAVIOR OF THE SOLUTIONS FOR A CLASS OF REACTION-DIFFUSION EQUATIONS
Institute of Scientific and Technical Information of China (English)
陈松林
2001-01-01
The methods of Lp estimation are used to discuss the extinction phenomena of the solutions to the following reaction-diffusion equations with initial-boudnary values u/ t = Au-λ |u|γ-1u-βu ((x,t) ∈Ω×(0,+∞)),u(x,t) | Ω×(0.+∞) = 0,u(x,0) = uo(x) ∈ H1 0(Ω) ∩ L1+γ(Ω) (x ∈Ω).Sufficient and necessary conditions about the extinction of the solutions is given Here λ＞0, γ＞ 0, β＞ 0 are constants, Ω∈ RN is bounded with smooth boundary Ω At last,it is simulated with a higher order equation by using the present methods.
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.
Dagdug, Leonardo; Vazquez, Marco-Vinicio; Berezhkovskii, Alexander M.; Zitserman, Vladimir Yu.; Bezrukov, Sergey M.
2012-05-01
The generalized Fick-Jacobs equation is widely used to study diffusion of Brownian particles in three-dimensional tubes and quasi-two-dimensional channels of varying constraint geometry. We show how this equation can be applied to study the slowdown of unconstrained diffusion in the presence of obstacles. Specifically, we study diffusion of a point Brownian particle in the presence of identical cylindrical obstacles arranged in a square lattice. The focus is on the effective diffusion coefficient of the particle in the plane perpendicular to the cylinder axes, as a function of the cylinder radii. As radii vary from zero to one half of the lattice period, the effective diffusion coefficient decreases from its value in the obstacle free space to zero. Using different versions of the generalized Fick-Jacobs equation, we derive simple approximate formulas, which give the effective diffusion coefficient as a function of the cylinder radii, and compare their predictions with the values of the effective diffusion coefficient obtained from Brownian dynamics simulations. We find that both Reguera-Rubi and Kalinay-Percus versions of the generalized Fick-Jacobs equation lead to quite accurate predictions of the effective diffusion coefficient (with maximum relative errors below 4% and 7%, respectively) over the entire range of the cylinder radii from zero to one half of the lattice period.
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.
Multiple Imputation Strategies for Multiple Group Structural Equation Models
Enders, Craig K.; Gottschall, Amanda C.
2011-01-01
Although structural equation modeling software packages use maximum likelihood estimation by default, there are situations where one might prefer to use multiple imputation to handle missing data rather than maximum likelihood estimation (e.g., when incorporating auxiliary variables). The selection of variables is one of the nuances associated…
Group Classification and Exact Solutions of a Class of Variable Coefficient Nonlinear Wave Equations
Institute of Scientific and Technical Information of China (English)
HUANG Ding-Jiang; MEI Jian-Qin; ZHANG Hong-Qing
2009-01-01
Complete group classification of a class of variable coefficient (1 + 1)-dimensional wave equations is performed.The possible additional equivalence transformations between equations from the class under consideration and the conditional equivalence groups are also investigated. These allow simplification of the results of the classification and further applications of them. The derived Lie symmetries are used to construct exact solutions of special forms of these equations via the classical Lie method. Nonclassical symmetries of the wave equations are discussed.
Directory of Open Access Journals (Sweden)
Yong Huang
2012-01-01
Full Text Available The Bäcklund transformations and abundant exact explicit solutions for a class of nonlinear wave equation are obtained by the extended homogeneous balance method. These solutions include the solitary wave solution of rational function, the solitary wave solutions, singular solutions, and the periodic wave solutions of triangle function type. In addition to rederiving some known solutions, some entirely new exact solutions are also established. Explicit and exact particular solutions of many well-known nonlinear evolution equations which are of important physical significance, such as Kolmogorov-Petrovskii-Piskunov equation, FitzHugh-Nagumo equation, Burgers-Huxley equation, Chaffee-Infante reaction diffusion equation, Newell-Whitehead equation, Fisher equation, Fisher-Burgers equation, and an isothermal autocatalytic system, are obtained as special cases.
On the numerical solution of the diffusion equation with a nonlocal boundary condition
Directory of Open Access Journals (Sweden)
Dehghan Mehdi
2003-01-01
Full Text Available Parabolic partial differential equations with nonlocal boundary specifications feature in the mathematical modeling of many phenomena. In this paper, numerical schemes are developed for obtaining approximate solutions to the initial boundary value problem for one-dimensional diffusion equation with a nonlocal constraint in place of one of the standard boundary conditions. The method of lines (MOL semidiscretization approach is used to transform the model partial differential equation into a system of first-order linear ordinary differential equations (ODEs. The partial derivative with respect to the space variable is approximated by a second-order finite-difference approximation. The solution of the resulting system of first-order ODEs satisfies a recurrence relation which involves a matrix exponential function. Numerical techniques are developed by approximating the exponential matrix function in this recurrence relation. We use a partial fraction expansion to compute the matrix exponential function via Pade approximations, which is particularly useful in parallel processing. The algorithm is tested on a model problem from the literature.
Quantum Master Equation for QED in Exact Renormalization Group
Igarashi, Yuji; Sonoda, Hidenori
2007-01-01
Recently, one of us (H.S.) gave an explicit form of the Ward-Takahashi identity for the Wilson action of QED. We first rederive the identity using a functional method. The identity makes it possible to realize the gauge symmetry even in the presence of a momentum cutoff. In the cutoff dependent realization, the abelian nature of the gauge symmetry is lost, breaking the nilpotency of the BRS transformation. Using the Batalin-Vilkovisky formalism, we extend the Wilson action by including the antifield contributions. Then, the Ward-Takahashi identity for the Wilson action is lifted to a quantum master equation, and the modified BRS transformation regains nilpotency. We also obtain a flow equation for the extended Wilson action.
Renormalization group equation for f (R ) gravity on hyperbolic spaces
Falls, Kevin; Ohta, Nobuyoshi
2016-10-01
We derive the flow equation for the gravitational effective average action in an f (R ) truncation on hyperbolic spacetimes using the exponential parametrization of the metric. In contrast to previous works on compact spaces, we are able to evaluate traces exactly using the optimized cutoff. This reveals in particular that all modes can be integrated out for a finite value of the cutoff due to a gap in the spectrum of the Laplacian, leading to the effective action. Studying polynomial solutions, we find poorer convergence than has been found on compact spacetimes even though at small curvature the equations only differ in the treatment of certain modes. In the vicinity of an asymptotically free fixed point, we find the universal beta function for the R2 coupling and compute the corresponding effective action which involves an R2log (R2) quantum correction.
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Ping Zhang
2016-01-01
Full Text Available Variational multiscale element free Galerkin (VMEFG method is applied to Burgers’ equation. It can be found that, for the very small diffusivity coefficients, VMEFG method still suffers from instability in the presence of boundary or interior layers. In order to overcome this problem, the high order low-pass filter is used to smooth the solution. Three test examples with very small diffusion are presented and the solutions obtained are compared with exact solutions and some other numerical methods. The numerical results are found in which the VMEFG coupled with low-pass filter works very well for Burgers’ equation with very small diffusivity coefficients.
Vass, József
2016-01-01
A discretization scheme is introduced for a set of convection-diffusion equations with a non-linear reaction term, where the convection velocity is constant for each reactant. This constancy allows a transformation to new spatial variables, which ensures the global stability of discretization. Convection-diffusion equations are notorious for their lack of stability, arising from the algebraic interaction of the convection and diffusion terms. Unexpectedly, our implemented numerical algorithm proves to be faster than computing exact solutions derived for a special case, while remaining reasonably accurate, as demonstrated in our runtime and error analysis.
Energy Technology Data Exchange (ETDEWEB)
Kwok, Sau Fa, E-mail: kwok@dfi.uem.br
2012-08-15
A Langevin equation with multiplicative white noise and its corresponding Fokker-Planck equation are considered in this work. From the Fokker-Planck equation a transformation into the Wiener process is provided for different orders of prescription in discretization rule for the stochastic integrals. A few applications are also discussed. - Highlights: Black-Right-Pointing-Pointer Fokker-Planck equation corresponding to the Langevin equation with mul- tiplicative white noise is presented. Black-Right-Pointing-Pointer Transformation of diffusion processes into the Wiener process in different prescriptions is provided. Black-Right-Pointing-Pointer The prescription parameter is associated with the growth rate for a Gompertz-type model.
A non-scale-invariant form for coarse-grained diffusion-reaction equations
Ostvar, Sassan; Wood, Brian D.
2016-09-01
The process of mixing and reaction is a challenging problem to understand mathematically. Although there have been successes in describing the effective properties of mixing and reaction under a number of regimes, process descriptions for early times have been challenging for cases where the structure of the initial conditions is highly segregated. In this paper, we use the method of volume averaging to develop a rigorous theory for diffusive mixing with reactions from initial to asymptotic times under highly segregated initial conditions in a bounded domain. One key feature that arises in this development is that the functional form of the averaged differential mass balance equations is not, in general, scale invariant. Upon upscaling, an additional source term arises that helps to account for the initial configuration of the reacting chemical species. In this development, we derive the macroscopic parameters (a macroscale source term and an effectiveness factor modifying the reaction rate) defined in the macroscale diffusion-reaction equation and provide example applications for several initial configurations.
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.
Yuste, S. B.; Abad, E.; Escudero, C.
2016-09-01
We present a classical, mesoscopic derivation of the Fokker-Planck equation for diffusion in an expanding medium. To this end, we take a conveniently generalized Chapman-Kolmogorov equation as the starting point. We obtain an analytical expression for the Green's function (propagator) and investigate both analytically and numerically how this function and the associated moments behave. We also study first-passage properties in expanding hyperspherical geometries. We show that in all cases the behavior is determined to a great extent by the so-called Brownian conformal time τ (t ) , which we define via the relation τ ˙=1 /a2 , where a (t ) is the expansion scale factor. If the medium expansion is driven by a power law [a (t ) ∝tγ with γ >0 ] , then we find interesting crossover effects in the mixing effectiveness of the diffusion process when the characteristic exponent γ is varied. Crossover effects are also found at the level of the survival probability and of the moments of the first passage-time distribution with two different regimes separated by the critical value γ =1 /2 . The case of an exponential scale factor is analyzed separately both for expanding and contracting media. In the latter situation, a stationary probability distribution arises in the long-time limit.
Planar diffusions with rank-based characteristics and perturbed Tanaka equations
Fernholz, E Robert; Karatzas, Ioannis; Prokaj, Vilmos
2011-01-01
For given non negative constants $g$, $h$, $\\rho$, $\\sigma$ with $\\rho^2+\\sigma^2 =1$ and $g+h>0$, we construct a diffusion process $(X_1, X_2)$ with values in the plane and infinitesimal generator $L = 1_{x_1>x_2} p(\\partial_{x_1},\\partial_{x_2}) + 1_{x_2>x_1} p(\\partial_{x_2},\\partial_{x_1}), $ where $ p(x,y)= \\frac{\\rho^2}2 x^2 + \\frac{\\sigma^2}2 y^2 - h x + g y. $ We compute the transition probabilities of this process, discuss its realization in terms of appropriate systems of stochastic differential equations, study its dynamics under a time reversal, and note that these involve singularly continuous components governed by local time. Crucial in our analysis are properties of Brownian and semimartingale local time; properties of the generalized perturbed Tanaka equation $dZ(t) = f(Z(t)) dM(t) + dN(t)$, $Z(0)=\\xi $$ driven by suitable continuous, orthogonal semimartingales $M$ and $N$ and with $f$ of bounded variation, which we study here in detail; and those of a one-dimensional diffusion $Y$ with bang-...
Calculating the renormalisation group equations of a SUSY model with Susyno
Fonseca, Renato M
2011-01-01
Susyno is a Mathematica package dedicated to the computation of the 2-loop renormalisation group equations of a supersymmetric model based on any gauge group (the only exception being multiple U(1) groups) and for any field content.
Singh, Brajesh K; Srivastava, Vineet K
2015-04-01
The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multi-dimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineering and natural sciences, modelled in terms of differential equations.
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.
Studies of the accuracy of time integration methods for reaction-diffusion equations
Ropp, David L.; Shadid, John N.; Ober, Curtis C.
2004-03-01
In this study we present numerical experiments of time integration methods applied to systems of reaction-diffusion equations. Our main interest is in evaluating the relative accuracy and asymptotic order of accuracy of the methods on problems which exhibit an approximate balance between the competing component time scales. Nearly balanced systems can produce a significant coupling of the physical mechanisms and introduce a slow dynamical time scale of interest. These problems provide a challenging test for this evaluation and tend to reveal subtle differences between the various methods. The methods we consider include first- and second-order semi-implicit, fully implicit, and operator-splitting techniques. The test problems include a prototype propagating nonlinear reaction-diffusion wave, a non-equilibrium radiation-diffusion system, a Brusselator chemical dynamics system and a blow-up example. In this evaluation we demonstrate a "split personality" for the operator-splitting methods that we consider. While operator-splitting methods often obtain very good accuracy, they can also manifest a serious degradation in accuracy due to stability problems.
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.
Analyses of the Instabilities in the Discretized Diffusion Equations via Information Theory
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Maxence Bigerelle
2016-04-01
Full Text Available In a previous investigation (Bigerelle and Iost, 2004, the authors have proposed a physical interpretation of the instability λ = Δt/Δx2 > 1/2 of the parabolic partial differential equations when solved by finite differences. However, our results were obtained using integration techniques based on erf functions meaning that no statistical fluctuation was introduced in the mathematical background. In this paper, we showed that the diffusive system can be divided into sub-systems onto which a Brownian motion is applied. Monte Carlo simulations are carried out to reproduce the macroscopic diffusive system. It is shown that the amount of information characterized by the compression ratio of information of the system is pertinent to quantify the entropy of the system according to some concepts introduced by the authors (Bigerelle and Iost, 2007. Thanks to this mesoscopic discretization, it is proved that information on each sub-cell of the diffusion map decreases with time before the unstable equality λ = 1/2 and increases after this threshold involving an increase in negentropy, i.e., a decrease in entropy contrarily to the second principle of thermodynamics.
On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coefficient
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John D. Towers
2002-10-01
Full Text Available We study the Cauchy problem for the nonlinear (possibly strongly degenerate parabolic transport-diffusion equation $$ partial_t u + partial_x (gamma(xf(u=partial_x^2 A(u, quad A'(cdotge 0, $$ where the coefficient $gamma(x$ is possibly discontinuous and $f(u$ is genuinely nonlinear, but not necessarily convex or concave. Existence of a weak solution is proved by passing to the limit as $varepsilondownarrow 0$ in a suitable sequence ${u_{varepsilon}}_{varepsilon>0}$ of smooth approximations solving the problem above with the transport flux $gamma(xf(cdot$ replaced by $gamma_{varepsilon}(xf(cdot$ and the diffusion function $A(cdot$ replaced by $A_{varepsilon}(cdot$, where $gamma_{varepsilon}(cdot$ is smooth and $A_{varepsilon}'(cdot>0$. The main technical challenge is to deal with the fact that the total variation $|u_{varepsilon}|_{BV}$ cannot be bounded uniformly in $varepsilon$, and hence one cannot derive directly strong convergence of ${u_{varepsilon}}_{varepsilon>0}$. In the purely hyperbolic case ($A'equiv 0$, where existence has already been established by a number of authors, all existence results to date have used a singular maolinebreak{}pping to overcome the lack of a variation bound. Here we derive instead strong convergence via a series of a priori (energy estimates that allow us to deduce convergence of the diffusion function and use the compensated compactness method to deal with the transport term. Submitted April 29, 2002. Published October 27, 2002. Math Subject Classifications: 35K65, 35D05, 35R05, 35L80 Key Words: Degenerate parabolic equation; nonconvex flux; weak solution; discontinuous coefficient; viscosity method; a priori estimates; compensated compactness
Flow and Diffusion Equations for Fluid Flow in Porous Rocks for the Multiphase Flow Phenomena
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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.
A Unified Approach to Linear Equating for the Nonequivalent Groups Design
von Davier, Alina A.; Kong, Nan
2005-01-01
This article describes a new, unified framework for linear equating in a non-equivalent groups anchor test (NEAT) design. The authors focus on three methods for linear equating in the NEAT design--Tucker, Levine observed-score, and chain--and develop a common parameterization that shows that each particular equating method is a special case of the…
Boundary Layer Equations and Lie Group Analysis of a Sisko Fluid
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Gözde Sarı
2012-01-01
Full Text Available Boundary layer equations are derived for the Sisko fluid. Using Lie group theory, a symmetry analysis of the equations is performed. A partial differential system is transferred to an ordinary differential system via symmetries. Resulting equations are numerically solved. Effects of non-Newtonian parameters on the solutions are discussed.
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
Kunze, Markus
2011-01-01
We prove convergence of the solutions X_n of semilinear stochastic evolution equations dX_n(t) = (A_nX(t) + F_n(t,X_n(t)))dt + G_n(t,X_n(t))dW_H(t), X_n(0) = x_n, on a Banach space B, driven by a cylindrical Brownian motion W_H in a Hilbert space H. We assume that the operators A_n converge to A and the locally Lipschitz functions F_n and G_n converge to the locally Lipschitz functions F and G in an appropriate sense. Moreover, we obtain estimates for the lifetime of the solution X of the limiting problem in terms of the lifetimes of the approximating solutions X_n. We apply the results to prove global existence for reaction diffusion equations with multiplicative noise and a polynomially bounded reaction term satisfying suitable dissipativity conditions. The operator governing the linear part of the equation can be an arbitrary uniformly elliptic second order elliptic operator.
Energy Technology Data Exchange (ETDEWEB)
Ibragimov, Nail H [Department of Mathematics and Science, Blekinge Institute of Technology, SE-371 79 Karlskrona (Sweden); Meleshko, Sergey V [School of Mathematics, Institute of Science, Suranaree University of Technology, Nakhon Ratchasima 30000 (Thailand); Rudenko, Oleg V, E-mail: nib@bth.se, E-mail: sergey@math.sut.ac.th, E-mail: rudenko@acs366.phys.msu.ru [Department of Physics, Moscow State University, 119991 Moscow (Russian Federation)
2011-08-05
The paper deals with an evolutionary integro-differential equation describing nonlinear waves. A particular choice of the kernel in the integral leads to well-known equations such as the Khokhlov-Zabolotskaya equation, the Kadomtsev-Petviashvili equation and others. Since the solutions of these equations describe many physical phenomena, the analysis of the general model studied in this paper is important. One of the methods for obtaining solutions of differential equations is provided by the Lie group analysis. However, this method is not applicable to integro-differential equations. Therefore, we discuss new approaches developed in modern group analysis and apply them to the general model considered in this paper. Reduced equations and exact solutions are also presented.
Directory of Open Access Journals (Sweden)
Md. Nur Alam
2016-06-01
Full Text Available In this article, we apply the exp(-Φ(ξ-expansion method to construct many families of exact solutions of nonlinear evolution equations (NLEEs via the nonlinear diffusive predator–prey system and the Bogoyavlenskii equations. These equations can be transformed to nonlinear ordinary differential equations. As a result, some new exact solutions are obtained through the hyperbolic function, the trigonometric function, the exponential functions and the rational forms. If the parameters take specific values, then the solitary waves are derived from the traveling waves. Also, we draw 2D and 3D graphics of exact solutions for the special diffusive predator–prey system and the Bogoyavlenskii equations by the help of programming language Maple.
Sun, HongGuang; Liu, Xiaoting; Zhang, Yong; Pang, Guofei; Garrard, Rhiannon
2017-09-01
Fractional-order diffusion equations (FDEs) extend classical diffusion equations by quantifying anomalous diffusion frequently observed in heterogeneous media. Real-world diffusion can be multi-dimensional, requiring efficient numerical solvers that can handle long-term memory embedded in mass transport. To address this challenge, a semi-discrete Kansa method is developed to approximate the two-dimensional spatiotemporal FDE, where the Kansa approach first discretizes the FDE, then the Gauss-Jacobi quadrature rule solves the corresponding matrix, and finally the Mittag-Leffler function provides an analytical solution for the resultant time-fractional ordinary differential equation. Numerical experiments are then conducted to check how the accuracy and convergence rate of the numerical solution are affected by the distribution mode and number of spatial discretization nodes. Applications further show that the numerical method can efficiently solve two-dimensional spatiotemporal FDE models with either a continuous or discrete mixing measure. Hence this study provides an efficient and fast computational method for modeling super-diffusive, sub-diffusive, and mixed diffusive processes in large, two-dimensional domains with irregular shapes.
Guner, Ozkan; Bekir, Ahmet; Unsal, Omer; Cevikel, Adem C.
2017-01-01
In this paper, we pay attention to the analytical method named, ansatz method for finding the exact solutions of the variable-coefficient modified KdV equation and variable coefficient diffusion-reaction equation. As a result the singular 1-soliton solution is obtained. These solutions are important for the explanation of some practical physical problems. The obtained results show that these methods provides a powerful mathematical tool for solving nonlinear equations with variable coefficients. This method can be extended to solve other variable coefficient nonlinear partial differential equations.
Non-rigid registration of breast surfaces using the laplace and diffusion equations
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Ou Jao J
2010-02-01
Full Text Available Abstract A semi-automated, non-rigid breast surface registration method is presented that involves solving the Laplace or diffusion equations over undeformed and deformed breast surfaces. The resulting potential energy fields and isocontours are used to establish surface correspondence. This novel surface-based method, which does not require intensity images, anatomical landmarks, or fiducials, is compared to a gold standard of thin-plate spline (TPS interpolation. Realistic finite element simulations of breast compression and further testing against a tissue-mimicking phantom demonstrate that this method is capable of registering surfaces experiencing 6 - 36 mm compression to within a mean error of 0.5 - 5.7 mm.
Numerical methods for one-dimensional reaction-diffusion equations arising in combustion theory
Ramos, J. I.
1987-01-01
A review of numerical methods for one-dimensional reaction-diffusion equations arising in combustion theory is presented. The methods reviewed include explicit, implicit, quasi-linearization, time linearization, operator-splitting, random walk and finite-element techniques and methods of lines. Adaptive and nonadaptive procedures are also reviewed. These techniques are applied first to solve two model problems which have exact traveling wave solutions with which the numerical results can be compared. This comparison is performed in terms of both the wave profile and computed wave speed. It is shown that the computed wave speed is not a good indicator of the accuracy of a particular method. A fourth-order time-linearized, Hermitian compact operator technique is found to be the most accurate method for a variety of time and space sizes.
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.
Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations
Ducrot, Arnaud; Matano, Hiroshi
2012-01-01
We consider one-dimensional reaction-diffusion equations for a large class of spatially periodic nonlinearities (including multistable ones) and study the asymptotic behavior of solutions with Heaviside type initial data. Our analysis reveals some new dynamics where the profile of the propagation is not characterized by a single front, but by a layer of several fronts which we call a terrace. Existence and convergence to such a terrace is proven by using an intersection number argument, without much relying on standard linear analysis. Hence, on top of the peculiar phenomenon of propagation that our work highlights, several corollaries will follow on the existence and convergence to pulsating traveling fronts even for highly degenerate nonlinearities that have not been treated before.
Reverse Smoothing Effects, Fine Asymptotics, and Harnack Inequalities for Fast Diffusion Equations
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Bonforte Matteo
2007-01-01
Full Text Available We investigate local and global properties of positive solutions to the fast diffusion equation in the good exponent range , corresponding to general nonnegative initial data. For the Cauchy problem posed in the whole Euclidean space , we prove sharp local positivity estimates (weak Harnack inequalities and elliptic Harnack inequalities; also a slight improvement of the intrinsic Harnack inequality is given. We use them to derive sharp global positivity estimates and a global Harnack principle. Consequences of these latter estimates in terms of fine asymptotics are shown. For the mixed initial and boundary value problem posed in a bounded domain of with homogeneous Dirichlet condition, we prove weak, intrinsic, and elliptic Harnack inequalities for intermediate times. We also prove elliptic Harnack inequalities near the extinction time, as a consequence of the study of the fine asymptotic behavior near the finite extinction time.
Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions
Ren, Jincheng; Sun, Zhi-zhong; Zhao, Xuan
2013-01-01
An effective finite difference scheme is considered for solving the time fractional sub-diffusion equation with Neumann boundary conditions. A difference scheme combining the compact difference approach the spatial discretization and L1 approximation for the Caputo fractional derivative is proposed and analyzed. Although the spatial approximation order at the Neumann boundary is one order lower than that for interior mesh points, the unconditional stability and the global convergence order O(τ+h4) in discrete L2 norm of the compact difference scheme are proved rigorously, where τ is the temporal grid size and h is the spatial grid size. Numerical experiments are included to support the theoretical results, and comparison with the related works are presented to show the effectiveness of our method.
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.
KPP reaction-diffusion equations with a non-linear loss inside a cylinder
Giletti, Thomas
2010-01-01
We consider in this paper a reaction-diffusion system in presence of a flow and under a KPP hypothesis. While the case of a single-equation has been extensively studied since the pioneering Kolmogorov-Petrovski-Piskunov paper, the study of the corresponding system with a Lewis number not equal to 1 is still quite open. Here, we will prove some results about the existence of travelling fronts and generalized travelling fronts solutions of such a system with the presence of a non-linear spacedependent loss term inside the domain. In particular, we will point out the existence of a minimal speed, above which any real value is an admissible speed. We will also give some spreading results for initial conditions decaying exponentially at infinity.
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
Classical implicit travelling wave solutions for a quasilinear convection-diffusion equation
Hearns, Jessica; Van Gorder, Robert A.
2012-11-01
We discuss classical implicit solutions to the partial differential equation ut=(H(u))xx+(G(u))x, a general convection-diffusion PDE with particular subcases appearing in many areas of fluids and astrophysics. As an illustrative example, and to compare our results with those present in the literature, we frequently consider travelling wave solutions for the quasilinear PDE ut=(um)xx+(un)x, which has been used to describe the flow of viscous fluids on an inclined bed and as a model of convection-diffusion processes. When n ⩾ m > 1, this equation can be used to model the flow of a fluid under gravity through a homogeneous and isotropic porous medium. The travelling wave ODE for both the general and more specific cases have a first integral which is used to obtain an implicit solution for the travelling wave profiles. We should mention that, for some values of m, the implicit relation can be solved in closed form for explicit exact solutions. In the case of n = 2m - 1, solving the implicit relation gives a general way of obtaining the solutions found in Vanaja [Vanaja, V., 2009. Physica Scripta 80, p. 045402] where the travelling wave solutions for the cases (m, n) = (2, 3) and (m, n) = (3, 5) were explicitly constructed using a more complicated ansatz method. For other more complicated cases where inversion cannot be performed, we apply the method of series reversion to construct series solutions from the implicit relations. Furthermore, we deduce the dependence of travelling wave solutions on the wave speed, even in cases where the explicit exact solution cannot be found.
Group classification of a generalized Black-Scholes-Merton equation
Bozhkov, Y.; Dimas, S.
2014-07-01
The complete group classification of a generalization of the Black-Scholes-Merton model is carried out by making use of the underlying equivalence and additional equivalence transformations. For each nonlinear case obtained through this classification, invariant solutions are given. To that end, two boundary conditions of financial interest are considered, the terminal and the barrier option conditions.
Mesh locking effects in the finite volume solution of 2-D anisotropic diffusion equations
Manzini, Gianmarco; Putti, Mario
2007-01-01
Strongly anisotropic diffusion equations require special techniques to overcome or reduce the mesh locking phenomenon. We present a finite volume scheme that tries to approximate with the best possible accuracy the quantities that are of importance in discretizing anisotropic fluxes. In particular, we discuss the crucial role of accurate evaluations of the tangential components of the gradient acting tangentially to the control volume boundaries, that are called into play by anisotropic diffusion tensors. To obtain the sought characteristics from the proposed finite volume method, we employ a second-order accurate reconstruction scheme which is used to evaluate both normal and tangential cell-interface gradients. The experimental results on a number of different meshes show that the scheme maintains optimal convergence rates in both L2 and H1 norms except for the benchmark test considering full Neumann boundary conditions on non-uniform grids. In such a case, a severe locking effect is experienced and documented. However, within the range of practical values of the anisotropy ratio, the scheme is robust and efficient. We postulate and verify experimentally the existence of a quadratic relationship between the anisotropy ratio and the mesh size parameter that guarantees optimal and sub-optimal convergence rates.
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.
Tang, Min; Wang, Yihong
2017-02-01
In magnetized plasma, the magnetic field confines the particles around the field lines. The anisotropy intensity in the viscosity and heat conduction may reach the order of 1012. When the boundary conditions are periodic or Neumann, the strong diffusion leads to an ill-posed limiting problem. To remove the ill-conditionedness in the highly anisotropic diffusion equations, we introduce a simple but very efficient asymptotic preserving reformulation in this paper. The key idea is that, instead of discretizing the Neumann boundary conditions locally, we replace one of the Neumann boundary condition by the integration of the original problem along the field line, the singular 1 / ɛ terms can be replaced by O (1) terms after the integration, which yields a well-posed problem. Small modifications to the original code are required and no change of coordinates nor mesh adaptation are needed. Uniform convergence with respect to the anisotropy strength 1 / ɛ can be observed numerically and the condition number does not scale with the anisotropy.
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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.
Solving the Christoffel equation: Phase and group velocities
Jaeken, Jan W.; Cottenier, Stefaan
2016-10-01
We provide christoffel, a Python tool for calculating direction-dependent phase velocities, polarization vectors, group velocities, power flow angles and enhancement factors based on the stiffness tensor of a solid. It is built in a modular way to allow for efficient and flexible calculations, and the freedom to select and combine results as desired. All derivatives are calculated analytically, which circumvents possible numerical sampling problems. GNUPlot scripts are provided for convenient visualization.
Energy Technology Data Exchange (ETDEWEB)
Kekenes-Huskey, P. M., E-mail: pkekeneshuskey@ucsd.edu [Department of Pharmacology, University of California San Diego, La Jolla, California 92093-0636 (United States); Gillette, A. K. [Department of Mathematics, University of Arizona, Tucson, Arizona 85721-0089 (United States); McCammon, J. A. [Department of Pharmacology, University of California San Diego, La Jolla, California 92093-0636 (United States); Department of Chemistry, Howard Hughes Medical Institute, University of California San Diego, La Jolla, California 92093-0636 (United States)
2014-05-07
The macroscopic diffusion constant for a charged diffuser is in part dependent on (1) the volume excluded by solute “obstacles” and (2) long-range interactions between those obstacles and the diffuser. Increasing excluded volume reduces transport of the diffuser, while long-range interactions can either increase or decrease diffusivity, depending on the nature of the potential. We previously demonstrated [P. M. Kekenes-Huskey et al., Biophys. J. 105, 2130 (2013)] using homogenization theory that the configuration of molecular-scale obstacles can both hinder diffusion and induce diffusional anisotropy for small ions. As the density of molecular obstacles increases, van der Waals (vdW) and electrostatic interactions between obstacle and a diffuser become significant and can strongly influence the latter's diffusivity, which was neglected in our original model. Here, we extend this methodology to include a fixed (time-independent) potential of mean force, through homogenization of the Smoluchowski equation. We consider the diffusion of ions in crowded, hydrophilic environments at physiological ionic strengths and find that electrostatic and vdW interactions can enhance or depress effective diffusion rates for attractive or repulsive forces, respectively. Additionally, we show that the observed diffusion rate may be reduced independent of non-specific electrostatic and vdW interactions by treating obstacles that exhibit specific binding interactions as “buffers” that absorb free diffusers. Finally, we demonstrate that effective diffusion rates are sensitive to distribution of surface charge on a globular protein, Troponin C, suggesting that the use of molecular structures with atomistic-scale resolution can account for electrostatic influences on substrate transport. This approach offers new insight into the influence of molecular-scale, long-range interactions on transport of charged species, particularly for diffusion-influenced signaling events
Numerical Calculation and Exergy Equations of Spray Heat Exchanger Attached to a Main Fan Diffuser
Cui, H.; Wang, H.; Chen, S.
2015-04-01
In the present study, the energy depreciation rule of spray heat exchanger, which is attached to a main fan diffuser, is analyzed based on the second law of thermodynamics. Firstly, the exergy equations of the exchanger are deduced. The equations are numerically calculated by the fourth-order Runge-Kutta method, and the exergy destruction is quantitatively effected by the exchanger structure parameters, working fluid (polluted air, i.e., PA; sprayed water, i.e., SW) initial state parameters and the ambient reference parameters. The results are showed: (1) heat transfer is given priority to latent transfer at the bottom of the exchanger, and heat transfer of convection and is equivalent to that of condensation in the upper. (2) With the decrease of initial temperature of SW droplet, the decrease of PA velocity or the ambient reference temperature, and with the increase of a SW droplet size or initial PA temperature, exergy destruction both increase. (3) The exergy efficiency of the exchanger is 72.1 %. An approach to analyze the energy potential of the exchanger may be provided for engineering designs.
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.
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.
Institute of Scientific and Technical Information of China (English)
QIN Xinqiang; MA Yichen; GONG Chunqiong
2004-01-01
A two-grid method for solving nonlinear convection-dominated diffusion equations is presented. The method use discretizations based on a characteristic mixed finite-element method and give the linearization for nonlinear systems by two steps. The error analysis shows that the two-grid scheme combined with the characteristic mixed finite-element method can decrease numerical oscillation caused by dominated convections and solve nonlinear advection-dominated diffusion problems efficiently.
Directory of Open Access Journals (Sweden)
Mohammad Siddique
2010-08-01
Full Text Available Parabolic partial differential equations with nonlocal boundary conditions arise in modeling of a wide range of important application areas such as chemical diffusion, thermoelasticity, heat conduction process, control theory and medicine science. In this paper, we present the implementation of positivity- preserving Padé numerical schemes to the two-dimensional diffusion equation with nonlocal time dependent boundary condition. We successfully implemented these numerical schemes for both Homogeneous and Inhomogeneous cases. The numerical results show that these Padé approximation based numerical schemes are quite accurate and easily implemented.
Institute of Scientific and Technical Information of China (English)
XIONG Lei; LI haijiao; ZHANG Lewen
2008-01-01
The fourth-order B spline wavelet scaling functions are used to solve the two-dimensional unsteady diffusion equation. The calculations from a case history indicate that the method provides high accuracy and the computational efficiency is enhanced due to the small matrix derived from this method.The respective features of 3-spline wavelet scaling functions, 4-spline wavelet scaling functions and quasi-wavelet used to solve the two-dimensional unsteady diffusion equation are compared. The proposed method has potential applications in many fields including marine science.
Institute of Scientific and Technical Information of China (English)
Jie－MinZHAN; Yao－SongCHEN
1996-01-01
An operator splitting method combining finite difference method and finite element method is proposed in this paper by using boundary-fitted coordinate system.The governing equation is split into advection and diffusion equations and solved by finit difference method using boundary-fitted coordinate system and finite element method respectively.An example for which analytic solution is available is used to verified the proposed methods and the agreement is very good.Numerical results show that it is very efficient.
Institute of Scientific and Technical Information of China (English)
XU JingBo; ZOU XuBo; GAO XiaoChun; FU Jian
2002-01-01
By making use of the dynamical algebraic approach, we study the two-mode Raman coupled model governed by the Milburn equation and find the exact solution of the Milburn equation without diffusion approximation. The exact solution is then used to discuss the influence of intrinsic decoherence on the revivals of atomic inversion, oscillation of the photon number distribution and squeezing of radiation field in the whole ranges of the decoherence parameter γ.
Yu, Zhiyong
2016-01-01
In this paper, we investigate infinite horizon jump-diffusion forward-backward stochastic differential equations under some monotonicity conditions. We establish an existence and uniqueness theorem, two stability results and a comparison theorem for solutions to such kind of equations. Then the theoretical results are applied to study a kind of infinite horizon backward stochastic linear-quadratic optimal control problems, and then differential game problems. The unique optimal controls for t...
Group classification and conservation laws of anisotropic wave equations with a source
Ibragimov, N. H.; Gandarias, M. L.; Galiakberova, L. R.; Bruzon, M. S.; Avdonina, E. D.
2016-08-01
Linear and nonlinear waves in anisotropic media are useful in investigating complex materials in physics, biomechanics, biomedical acoustics, etc. The present paper is devoted to investigation of symmetries and conservation laws for nonlinear anisotropic wave equations with specific external sources when the equations in question are nonlinearly self-adjoint. These equations involve two arbitrary functions. Construction of conservation laws associated with symmetries is based on the generalized conservation theorem for nonlinearly self-adjoint partial differential equations. First we calculate the conservation laws for the basic equation without any restrictions on the arbitrary functions. Then we make the group classification of the basic equation in order to specify all possible values of the arbitrary functions when the equation has additional symmetries and construct the additional conservation laws.
Group classification of steady two-dimensional boundary-layer stagnation-point flow equations
Nadjafikhah, Mehdi; Hejazi, Seyed Reza
2010-01-01
Lie symmetry group method is applied to study the boundary-layer equations for two-dimensional steady flow of an incompressible, viscous fluid near a stagnation point at a heated stretching sheet placed in a porous medium equation. The symmetry group and its optimal system are given, and group invariant solutions associated to the symmetries are obtained. Finally the structure of the Lie algebra symmetries is determined.
The Wilson exact renormalization group equation and the anomalous dimension parameter
Bervillier, C
2013-01-01
The non-linear way the anomalous dimension parameter has been introduced in the historic first version of the exact renormalization group equation is compared to current practice. A simple expression for the exactly marginal redundant operator proceeds from this non-linearity, whereas in the linear case, first order differential equations must be solved to get it. The role of this operator in the construction of the flow equation is highlighted.
Group Diffusion of Cognitive Effort as a Determinant of Attribution.
Quintanar, Leo R.; Pryor, John B.
The tendency for individuals to reduce their own efforts when others are available to respond has been called "social loafing." Social loafing has been found also to characterize collective endeavors on tasks considered cognitively efffortful. To test the hypothesis that reduced cognitive effort related to the presence of a coacting group would…
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}.
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.
Watkins, N. W.; Credgington, D.; Sanchez, R.; Chapman, S. C.
2007-12-01
Since the 1960s Mandelbrot has advocated the use of fractals for the description of the non-Euclidean geometry of many aspects of nature. In particular he proposed two kinds of model to capture persistence in time (his Joseph effect, common in hydrology and with fractional Brownian motion as the prototpe) and/or prone to heavy tailed jumps (the Noah effect, typical of economic indices, for which he proposed Lévy flights as an exemplar). Both effects are now well demonstrated in space plasmas, notably in indices quantifying Earth's auroral currents and in the turbulent solar wind. Models have, however, typically emphasised one of the Noah and Joseph parameters (the Lévy exponent μ and the temporal exponent β) at the other's expense. I will describe recent work [1] in which we studied a simple self-affine stable model-linear fractional stable motion, LFSM, which unifies both effects. I will discuss how this resolves some contradictions seen in earlier work. Such Noah-Joseph hybrid ("ambivalent" [2]) behaviour is highly topical in physics but is typically studied in the paradigm of the continuous time random walk (CTRW) [2,3] rather than LFSM. I will clarify the physical differences between these two pictures and present a recently-derived diffusion equation for LFSM. This replaces the second order spatial derivative in the equation of fBm [4] with a fractional derivative of order μ, but retains a diffusion coefficient with a power law time dependence rather than a fractional derivative in time (c.f. [2,3]). Intriguingly the self-similarity exponent extracted from the CTRW differs from that seen in LFSM. In the CTRW it is the ratio of μ to a temporal exponent, in LFSM it is an additive function of them. I will also show work in progress using an LFSM model and simple analytic scaling arguments to study the problem of the area between an LFSM curve and a threshold-related to the burst size measure introduced by Takalo and Consolini into solar- terrestrial physics
Some Diophantine equations from finite group theory: $\\Phi_m(x)=2p^n-1$
Luca, Florian; de Weger, Benne
2009-01-01
We show that the equation in the title (with $\\Phi_m$ the $m$th cyclotomic polynomial) has no integer solution with $n\\ge 1$ in the cases $(m,p)=(15,41), (15,5581),(10,271)$. These equations arise in a recent group theoretical investigation by Z. Akhlaghi, M. Khatami and B. Khosravi.
Real-space renormalization-group approach to field evolution equations.
Degenhard, Andreas; Rodríguez-Laguna, Javier
2002-03-01
An operator formalism for the reduction of degrees of freedom in the evolution of discrete partial differential equations (PDE) via real-space renormalization group is introduced, in which cell overlapping is the key concept. Applications to (1+1)-dimensional PDEs are presented for linear and quadratic equations that are first order in time.
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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.
反应扩散方程的奇摄动%SINGULAR PERTURBATION FOR REACTION DIFFUSION EQUATIONS
Institute of Scientific and Technical Information of China (English)
莫嘉琪; 王辉; 朱江
2003-01-01
The singularly perturbed initial boundary value problems for reaction diffusion equations are considered. Under suitable conditions and by using the theory of differential inequality, the asymptotic behavior of solution for initial boundary value problems are studied, where the reduced problems possess two intersecting solutions.
A remark on the Beale-Kato-Majda criterion for the 3D MHD equations with zero magnetic diffusivity
Gala, Sadek; Ragusa, Maria Alessandra
2016-06-01
In this work, we show that a smooth solution of the 3D MHD equations with zero magnetic diffusivity in the whole space ℝ3 breaks down if and only if a certain norm of the magnetic field blows up at the same time.
Institute of Scientific and Technical Information of China (English)
秦玉明; 李海燕
2014-01-01
This article is devoted to the study of global existence and exponential stability of solutions to an initial-boundary value problem of the quasilinear thermo-diffusion equations with second sound by means of multiplicative techniques and energy method provided that the initial data are close to the equilibrium and the relaxation kernel is strongly positive definite and decays exponentially.
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.
Costa, C. P.; Vilhena, M. T.; Moreira, D. M.; Tirabassi, T.
We present a three-dimensional solution of the steady-state advection-diffusion equation considering a vertically inhomogeneous planetary boundary layer (PBL). We reach this goal applying the generalized integral transform technique (GITT), a hybrid method that had solved a wide class of direct and inverse problems mainly in the area of heat transfer and fluid mechanics. The transformed problem is solved by the advection-diffusion multilayer model (ADMM) method, a semi-analytical solution based on a discretization of the PBL in sub-layers where the advection-diffusion equation is solved by the Laplace transform technique. Numerical simulations are presented and the performances of the solution are compared against field experiments data.
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.
Gorpas, Dimitris; Andersson-Engels, Stefan
2012-03-01
The solution of the forward problem in fluorescence molecular imaging is among the most important premises for the successful confrontation of the inverse reconstruction problem. To date, the most typical approach has been the application of the diffusion approximation as the forward model. This model is basically a first order angular approximation for the radiative transfer equation, and thus it presents certain limitations. The scope of this manuscript is to present the dual coupled radiative transfer equation and diffusion approximation model for the solution of the forward problem in fluorescence molecular imaging. The integro-differential equations of its weak formalism were solved via the finite elements method. Algorithmic blocks with cubature rules and analytical solutions of the multiple integrals have been constructed for the solution. Furthermore, specialized mapping matrices have been developed to assembly the finite elements matrix. As a radiative transfer equation based model, the integration over the angular discretization was implemented analytically, while quadrature rules were applied whenever required. Finally, this model was evaluated on numerous virtual phantoms and its relative accuracy, with respect to the radiative transfer equation, was over 95%, when the widely applied diffusion approximation presented almost 85% corresponding relative accuracy for the fluorescence emission.
Han, Renji; Dai, Binxiang
2017-06-01
The spatiotemporal pattern induced by cross-diffusion of a toxic-phytoplankton-zooplankton model with nonmonotonic functional response is investigated in this paper. The linear stability analysis shows that cross-diffusion is the key mechanism for the formation of spatial patterns. By taking cross-diffusion rate as bifurcation parameter, we derive amplitude equations near the Turing bifurcation point for the excited modes in the framework of a weakly nonlinear theory, and the stability analysis of the amplitude equations interprets the structural transitions and stability of various forms of Turing patterns. Furthermore, we illustrate the theoretical results via numerical simulations. It is shown that the spatiotemporal distribution of the plankton is homogeneous in the absence of cross-diffusion. However, when the cross-diffusivity is greater than the critical value, the spatiotemporal distribution of all the plankton species becomes inhomogeneous in spaces and results in different kinds of patterns: spot, stripe, and the mixture of spot and stripe patterns depending on the cross-diffusivity. Simultaneously, the impact of toxin-producing rate of toxic-phytoplankton (TPP) species and natural death rate of zooplankton species on pattern selection is also explored.
Reverse Smoothing Effects, Fine Asymptotics, and Harnack Inequalities for Fast Diffusion Equations
Directory of Open Access Journals (Sweden)
Juan Luis Vazquez
2006-11-01
Full Text Available We investigate local and global properties of positive solutions to the fast diffusion equation ut=ÃŽÂ”um in the good exponent range (dÃ¢ÂˆÂ’2+/d
Behaviour near extinction for the Fast Diffusion Equation on bounded domains
Bonforte, Matteo; Vazquez, Juan Luis
2010-01-01
We consider the Fast Diffusion Equation $u_t=\\Delta u^m$ posed in a bounded smooth domain $\\Omega\\subset \\RR^d$ with homogeneous Dirichlet conditions; the exponent range is $m_s=(d-2)_+/(d+2)
Richter, Otto; Moenickes, Sylvia; Suhling, Frank
2012-02-01
The spatial dynamics of range expansion is studied in dependence of temperature. The main elements population dynamics, competition and dispersal are combined in a coherent approach based on a system of coupled partial differential equations of the reaction-diffusion type. The nonlinear reaction terms comprise population dynamic models with temperature dependent reproduction rates subject to an Allee effect and mutual competition. The effect of temperature on travelling wave solutions is investigated for a one dimensional model version. One main result is the importance of the Allee effect for the crossing of regions with unsuitable habitats. The nonlinearities of the interaction terms give rise to a richness of spatio-temporal dynamic patterns. In two dimensions, the resulting non-linear initial boundary value problems are solved over geometries of heterogeneous landscapes. Geo referenced model parameters such as mean temperature and elevation are imported into the finite element tool COMSOL Multiphysics from a geographical information system. The model is applied to the range expansion of species at the scale of middle Europe.
A hybrid multi-scale computational scheme for advection-diffusion-reaction equation
Karimi, S.; Nakshatrala, K. B.
2016-12-01
Simulation of transport and reaction processes in porous media and subsurface science has become more vital than ever. Over the past few decades, a variety of mathematical models and numerical methodologies for porous media simulations have been developed. As the demand for higher accuracy and validity of the models grows, the issue of disparate temporal and spatial scales becomes more problematic. The variety of reaction processes and complexity of pore geometry poses a huge computational burden in a real-world or reservoir scale simulation. Meanwhile, methods based on averaging or up- scaling techniques do not provide reliable estimates to pore-scale processes. To overcome this problem, development of hybrid and multi-scale computational techniques is considered a promising approach. In these methods, pore-scale and continuum-scale models are combined, hence, a more reliable estimate to pore-scale processes is obtained without having to deal with the tremendous computational overhead of pore-scale methods. In this presentation, we propose a computational framework that allows coupling of lattice Boltzmann method (for pore-scale simulation) and finite element method (for continuum-scale simulation) for advection-diffusion-reaction equations. To capture disparate in time and length events, non-matching grid and time-steps are allowed. Apart from application of this method to benchmark problems, multi-scale simulation of chemical reactions in porous media is also showcased.
Full Symmetry Groups and Exact Solutions to BKP and GKP Equations
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Bo Ren
2014-01-01
Full Text Available We investigate the (2+1-dimensional nonlinear BKP and GKP equations with the modified direct CK’s method. Then, we get its Lie point groups and the full symmetry group, and a relationship is constructed between the new solutions and the old one. Based on the relationship, the new solutions can be obtained by using a given solution of the equations.
Conditional Similarity Reductions of Jimbo-Miwa Equation via the Classical Lie Group Approach
Institute of Scientific and Technical Information of China (English)
TANG Xiao-Yan; LIN Ji
2003-01-01
Recently, the Clarkson and Kruskal direct method has been modified to find new similarity reductions (conditional similarity reductions) of nonlinear systems and the results obtained by the modified direct method cannot be obtained by the current classical and/or non-classical Lie group approach. In this paper, we show that the conditional similarity reductions of the Jimbo-Miwa equation can be reobtained by adding an additional constraint equation to the original model to form a conditional equation system first and then solving the model system by means of the classical Lie group approach.
Group-based strategy diffusion in multiplex networks with weighted values
Yu, Jianyong; Jiang, J. C.; Xiang, Leijun
2017-03-01
The information diffusion of multiplex social networks has received increasing interests in recent years. Actually, the multiplex networks are made of many communities, and it should be gotten more attention for the influences of community level diffusion, besides of individual level interactions. In view of this, this work explores strategy interactions and diffusion processes in multiplex networks with weighted values from a new perspective. Two different groups consisting of some agents with different influential strength are firstly built in each layer network, the authority and non-authority groups. The strategy interactions between different groups in intralayer and interlayer networks are performed to explore community level diffusion, by playing two classical strategy games, Prisoner's Dilemma and Snowdrift Game. The impact forces from the different groups and the reactive forces from individual agents are simultaneously taken into account in intralayer and interlayer interactions. This paper reveals and explains the evolutions of cooperation diffusion and the influences of interlayer interaction tight degrees in multiplex networks with weighted values. Some thresholds of critical parameters of interaction degrees and games parameters settings are also discussed in group-based strategy diffusion.
Hsuen, Hsiao-Kuo
The performance equations for cathodes of polymer electrolyte fuel cells (PEFCs) that describe the dependence of cathode potential on current density are developed. Formulation of the performance equations starts from the reduction of a one-dimensional model that considers, in detail, the potential losses pertinent to the limitations of electron conduction, oxygen diffusion, proton migration, and the oxygen reduction reaction. In particular, non-uniform accumulation of liquid water in the gas diffuser, which partially blocks the gas channels and imposes a greater resistance for oxygen transport, is taken into account. Reduction of the one-dimensional model is implemented by approximating the oxygen concentration profile in the catalyst layer with a parabolic polynomial or a piecewise parabolic one determined by the occurrence of oxygen depletion. The final forms of the equations are obtained by applying the method of weighted residuals over the catalyst layer. The weighting function is selected in such a way that the weighted residuals can be analytically integrated. Potential losses caused by the various limiting processes can be quantitatively estimated by the performance equations. Thus, they provide a convenient diagnostic tool for the cathode performance. Computational results reveal that the performance equations agree well with the original one-dimensional model over an extensive range of parameter values. This indicates that the present performance equations can be used as a substitute for the one-dimensional model to provide quantitatively correct predictions for the cathode performance of PEFCs.
Institute of Scientific and Technical Information of China (English)
Hua-zhong Tang; Gerald Warnecke
2006-01-01
This paper presents a class of high resolution local time step schemes for nonlinear hyperbolic conservation laws and the closely related convection-diffusion equations, by projecting the solution increments of the underlying partial differential equations (PDE)at each local time step. The main advantages are that they are of good consistency, and it is convenient to implement them. The schemes are L∞ stable, satisfy a cell entropy inequality, and may be extended to the initial boundary value problem of general unsteady PDEs with higher-order spatial derivatives. The high resolution schemes are given by combining the reconstruction technique with a second order TVD Runge-Kutta scheme or a Lax-Wendroff type method, respectively.The schemes are used to solve a linear convection-diffusion equation, the nonlinear inviscid Burgers' equation, the one- and two-dimensional compressible Euler equations, and the two-dimensional incompressible Navier-Stokes equations. The numerical results show that the schemes are of higher-order accuracy, and efficient in saving computational cost,especially, for the case of combining the present schemes with the adaptive mesh method [15]. The correct locations of the slow moving or stronger discontinuities are also obtained,although the schemes are slightly nonconservative.
Directory of Open Access Journals (Sweden)
Xinzhi Liu
1998-01-01
Full Text Available This paper studies a class of high order delay partial differential equations. Employing high order delay differential inequalities, several oscillation criteria are established for such equations subject to two different boundary conditions. Two examples are also given.
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Giane Gonçalves
2005-07-01
Full Text Available Neste trabalho investigaremos as equações de difusão, usualmente aplicadas na descrição da difusão anômala, que empregam derivadas fracionárias tanto na variável temporal quanto na variável espacial. Em particular, para essas equações obteremos soluções exatas levando em conta uma condição inicial genérica e formularemos uma teoria deperturbação para o estudo de situações mais complexas. Também verificaremos que as derivadas fracionárias, quando aplicadas na parte temporal, possibilitam-nos o estudo de um processo de difusão anômala com o segundo momento finito, i.e., x 2 µ t a ( 0 1 , correspondendo aos casos, sub e superdifusivo, respectivamente. Em contraste, com a derivada fracionária aplicada na variável espacial que resulta em uma difusão anômala cujo segundo momento não é finito. Complementando o cenário acima, empregaremos o formalismo de caminhantes aleatórios para explorar as implicações obtidas por usar derivadas fracionárias na equação de difusão.In this work we investigate the anomalous diffusion equations, usually applied to describe the anomalous diffusion, which employ fractional derivatives for the time or the spatial variables. Inparticular, we obtain exact solutions by taking a generic initial condition into account and developing a perturbation theory to investigate complex situations. We also verify that the fractional derivatives, when applied to the time variable, lead us to a anomalous diffusion with second moment finite, i.e., x 2 µ t a ( 0 1 , corresponding to sub and superdifusive behavior, respectively. By way of contrast, the fractional derivative applied to the spatial variable results in a anomalous diffusion where the second moment is not finite. These equations generalize the usual diffusion equation in order to incorporate several situations. We also employ the continuous time random walking formalism to investigate the implications obtained by using fractional
On the use of SERPENT Monte Carlo code to generate few group diffusion constants
Energy Technology Data Exchange (ETDEWEB)
Piovezan, Pamela, E-mail: pamela.piovezan@ctmsp.mar.mil.b [Centro Tecnologico da Marinha em Sao Paulo (CTMSP), Sao Paulo, SP (Brazil); Carluccio, Thiago; Domingos, Douglas Borges; Rossi, Pedro Russo; Mura, Luiz Felipe, E-mail: fermium@cietec.org.b, E-mail: thiagoc@ipen.b [Fermium Tecnologia Nuclear, Sao Paulo, SP (Brazil); Instituto de Pesquisas Energeticas e Nucleares (IPEN/CNEN-SP), Sao Paulo, SP (Brazil)
2011-07-01
The accuracy of diffusion reactor codes strongly depends on the quality of the groups constants processing. For many years, the generation of such constants was based on 1-D infinity cell transport calculations. Some developments using collision probability or the method of characteristics allow, nowadays, 2-D assembly group constants calculations. However, these 1-D and 2-D codes how some limitations as , for example, on complex geometries and in the neighborhood of heavy absorbers. On the other hand, since Monte Carlos (MC) codes provide accurate neutro flux distributions, the possibility of using these solutions to provide group constants to full-core reactor diffusion simulators has been recently investigated, especially for the cases in which the geometry and reactor types are beyond the capability of the conventional deterministic lattice codes. The two greatest difficulties on the use of MC codes to group constant generation are the computational costs and the methodological incompatibility between analog MC particle transport simulation and deterministic transport methods based in several approximations. The SERPENT code is a 3-D continuous energy MC transport code with built-in burnup capability that was specially optimized to generate these group constants. In this work, we present the preliminary results of using the SERPENT MC code to generate 3-D two-group diffusion constants for a PWR like assembly. These constants were used in the CITATION diffusion code to investigate the effects of the MC group constants determination on the neutron multiplication factor diffusion estimate. (author)
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)
Energy Technology Data Exchange (ETDEWEB)
Sano, Y.; Yamamoto, S. [Yamaguchi University, Yamaguchi (Japan). Faculty of Engineering
1997-01-10
The integral form of the diffusion equation for a slab with a constant surface concentration based on the dissolved solid coordinate is derived, which presents the relation between desorption rate and integral average diffusion coefficient by the ratio of the 0 and 1st moments of concentration distribution. The relation between the 0 and 1st moments is quite insensitive to the concentration distribution governed by the diffusion equation with the concentration dependent diffusion coefficients and is satisfactorily estimated by the ratio of average to center concentration at the starting point of the regular regime. From these equations, one desorption curve is quantitatively transformed to the integral average diffusion coefficients as a function of center concentrations. The diffusion coefficient is calculated as a function of concentration by differentiating the integral average diffusion coefficients. As an example, the isothermal drying data of skim-milk aqueous solution are analyzed. 11 refs., 11 figs., 2 tabs.
Wazwaz, Abdul-Majid
2017-07-01
In this work we address the Lane-Emden boundary value problems which appear in chemical applications, biochemical applications, and scientific disciplines. We apply the variational iteration method to solve two specific models. The first problem models reaction-diffusion equation in a spherical catalyst, while the second problem models the reaction-diffusion process in a spherical biocatalyst. We obtain reliable analytical expressions of the concentrations and the effectiveness factors. Proper graphs will be used to illustrate the obtained results. The proposed analysis demonstrates reliability and efficiency applicability of the employed method.
Chapter 19: The age of scarplike landforms from diffusion-equation analysis
Hanks, Thomas C.
2000-01-01
The purpose of this paper is to review developments in the quantitative modeling of fault-scarp geomorphology, principally those since 1980. These developments utilize diffusionequation mathematics, in several different forms, as the basic model of fault-scarp evolution. Because solutions to the general diffusion equation evolve with time, as we expect faultscarp morphology to evolve with time, the model solutions carry information about the age of the structure and thus its time of formation; hence the inclusion of this paper in this volume. The evolution of fault-scarp morphology holds a small but special place in the much larger class of problems in landform evolution. In general, landform evolution means the evolution of topography as a function of both space and time. It is the outcome of the competition among those tectonic processes that make topography, erosive processes that destroy topography, and depositional processes that redistribute topography. Deposition and erosion can always be coupled through conservation-of-mass relations, but in general deposition occurs at great distance from the source region of detritus. Moreover, erosion is an inherently rough process whereas deposition is inherently smooth, as is evident from even casual inspection of shaded-relief, digital-elevation maps (e.g., Thelin and Pike, 1990; Simpson and Anders, 1992) and the current fascination with fractal representations oferoding terrains (e.g., Huang and Turcotte, 1989; Newman and Turcotte, 1990). Nevertheless, large-scale landform-evolution modeling, now a computationally intensive, advanced numerical exercise, is generating ever more realistic landforms (e.g., Willgoose and others, 1991a,b; Kooi and Beaumont, 1994; Tucker and Slingerland, 1994), although many of the rate coefficients remain poorly prescribed
Energy Technology Data Exchange (ETDEWEB)
Barajas-Solano, David A.; Tartakovsky, A. M.
2016-10-13
We present a hybrid scheme for the coupling of macro and microscale continuum models for reactive contaminant transport in fractured and porous media. The transport model considered is the advection-dispersion equation, subject to linear heterogeneous reactive boundary conditions. The Multiscale Finite Volume method (MsFV) is employed to define an approximation to the microscale concentration field defined in terms of macroscopic or \\emph{global} degrees of freedom, together with local interpolator and corrector functions capturing microscopic spatial variability. The macroscopic mass balance relations for the MsFV global degrees of freedom are coupled with the macroscopic model, resulting in a global problem for the simultaneous time-stepping of all macroscopic degrees of freedom throughout the domain. In order to perform the hybrid coupling, the micro and macroscale models are applied over overlapping subdomains of the simulation domain, with the overlap denoted as the handshake subdomain $\\Omega^{hs}$, over which continuity of concentration and transport fluxes between models is enforced. Continuity of concentration is enforced by posing a restriction relation between models over $\\Omega^{hs}$. Continuity of fluxes is enforced by prolongating the macroscopic model fluxes across the boundary of $\\Omega^{hs}$ to microscopic resolution. The microscopic interpolator and corrector functions are solutions to local microscopic advection-diffusion problems decoupled from the global degrees of freedom and from each other by virtue of the MsFV decoupling ansatz. The error introduced by the decoupling ansatz is reduced iteratively by the preconditioned GMRES algorithm, with the hybrid MsFV operator serving as the preconditioner.
Computer difference scheme for a singularly perturbed convection-diffusion equation
Shishkin, G. I.
2014-08-01
The Dirichlet problem for a singularly perturbed ordinary differential convection-diffusion equation with a perturbation parameter ɛ (that takes arbitrary values from the half-open interval (0, 1]) is considered. For this problem, an approach to the construction of a numerical method based on a standard difference scheme on uniform meshes is developed in the case when the data of the grid problem include perturbations and additional perturbations are introduced in the course of the computations on a computer. In the absence of perturbations, the standard difference scheme converges at an (δ st ) rate, where δ st = (ɛ + N -1)-1 N -1 and N + 1 is the number of grid nodes; the scheme is not ɛ-uniformly well conditioned or stable to perturbations of the data. Even if the convergence of the standard scheme is theoretically proved, the actual accuracy of the computed solution in the presence of perturbations degrades with decreasing ɛ down to its complete loss for small ɛ (namely, for ɛ = (δ-2max i, j |δ a {/i j }| + δ-1 max i, j |δ b {/i j }|), where δ = δ st and δ a {/i j }, δ b {/i j } are the perturbations in the coefficients multiplying the second and first derivatives). For the boundary value problem, we construct a computer difference scheme, i.e., a computing system that consists of a standard scheme on a uniform mesh in the presence of controlled perturbations in the grid problem data and a hypothetical computer with controlled computer perturbations. The conditions on admissible perturbations in the grid problem data and on admissible computer perturbations are obtained under which the computer difference scheme converges in the maximum norm for ɛ ∈ (0, 1] at the same rate as the standard scheme in the absence of perturbations.
Institute of Scientific and Technical Information of China (English)
QingdongCAI
1999-01-01
A new technique in the formulation of numerical scheme for hyperbolic equation is developed.It is different from the classical FD and FE methods.We begin with the algebraic equations with some undefined parameters,and get the difference equation through Taylor-series expansion.When the parameters in the partial difference equation are defined,the equation is what the scheme will simulated.The numerical example of the viscous BUrgers equation shows the validity of the scheme.This method deals with the numerical viscosity and dispersion exactly ,giving a preliminary explanation to some problems that the CFD face now.
p-Laplace equation in the Heisenberg group regularity of solutions
Ricciotti, Diego
2015-01-01
This works focuses on regularity theory for solutions to the p-Laplace equation in the Heisenberg group. In particular, it presents detailed proofs of smoothness for solutions to the non-degenerate equation and of Lipschitz regularity for solutions to the degenerate one. An introductory chapter presents the basic properties of the Heisenberg group, making the coverage self-contained. The setting is the first Heisenberg group, helping to keep the notation simple and allow the reader to focus on the core of the theory and techniques in the field. Further, detailed proofs make the work accessible to students at the graduate level.
Group Classification of a General Bond-Option Pricing Equation of Mathematical Finance
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Tanki Motsepa
2014-01-01
Full Text Available We carry out group classification of a general bond-option pricing equation. We show that the equation admits a three-dimensional equivalence Lie algebra. We also show that some of the values of the constants which result from group classification give us well-known models in mathematics of finance such as Black-Scholes, Vasicek, and Cox-Ingersoll-Ross. For all such values of these arbitrary constants we obtain Lie point symmetries. Symmetry reductions are then obtained and group invariant solutions are constructed for some cases.
Uniqueness of Solutions to Schrödinger Equations on Complex Semi-Simple Lie Groups
Indian Academy of Sciences (India)
Sagun Chanillo
2007-08-01
In this note we study the time-dependent Schrödinger equation on complex semi-simple Lie groups. We show that if the initial data is a bi-invariant function that has sufficient decay and the solution has sufficient decay at another fixed value of time, then the solution has to be identically zero for all time. We also derive Strichartz and decay estimates for the Schrödinger equation. Our methods also extend to the wave equation. On the Heisenberg group we show that the failure to obtain a parametrix for our Schrödinger equation is related to the fact that geodesics project to circles on the contact plane at the identity.
Szymkiewicz, Romuald; Gasiorowski, Dariusz
2012-09-01
SummaryWe consider solution of 2D nonlinear diffusive wave equation in a domain temporarily covered by a layer of water. A modified finite element method with triangular elements and linear shape functions is used for spatial discretization. The proposed modification refers to the procedure of spatial integration and leads to a more general algorithm involving a weighting parameter. The standard finite element method and the finite difference method are its particular cases. Time integration is performed using a two-stage difference scheme with another weighting parameter. The resulting systems of nonlinear algebraic equations are solved using the Picard and Newton iterative methods. It is shown that the two weighting parameters determine the accuracy and stability of the numerical solution as well as the convergence of iterative process. Accuracy analysis using the modified equation approach carried out for linear version of the governing equation allowed to evaluate the numerical diffusion and dispersion generated by the method as well as to explain its properties. As the finite element method accounts for the Neumann type of boundary conditions in a natural way, no special treatment of the boundary is needed. Consequently the problem of moving grid point, which must follow the shoreline, in the proposed approach is overcome automatically. The current position of moving boundary is obtained as a result of solution of the governing equation at fixed grid point.
Symmetries and Group-Invariant Solutions for Transonic Pressure-Gradient Equations
Institute of Scientific and Technical Information of China (English)
王丽真; 黄晴
2011-01-01
Lie symmetry group method is applied to study the transonic pressure-gradient equations in two-dimensional space. Its symmetry groups and corresponding optimal systems are determined, and several classes of irrotational groupinvariant solutions associated to the symmetries are obtained and special case of one-dimensional rarefaction wave is found.
Intestinal and diffuse carcinoma of the stomach among the ethnic and dialect groups in Singapore.
Teh, M; Lee, Y S
1987-08-15
This study attempts to determine the relative prevalence of intestinal-type and diffuse-type carcinomas (using a modified Lauren classification of gastric carcinomas) and to evaluate its significance in relation to the difference in stomach cancer risks among the different ethnic and Chinese dialect groups in Singapore. Of the 648 cases of primary adenocarcinoma of the stomach studied, 405 (62.5%) were of the intestinal type, 206 (31.8%) of the diffuse type, and 37 (5.7%) of the mixed type. Men had higher proportions of intestinal-type carcinoma than women. The intestinal-diffuse carcinoma ratio increased progressively with age. Although the relative intestinal-diffuse carcinoma ratios in women appeared to reflect the relative incidence rates of stomach cancer of the different ethnic and dialect groups, the ratios in men were inconsistent. Indian men had a higher intestinal-diffuse carcinoma ratio than Chinese men despite a lower incidence of stomach cancer. Hokkien men had the highest incidence of stomach cancer and the lowest intestinal-diffuse carcinoma ratio among the Chinese dialect groups. The use of the ratio as an indicator of relative risks for gastric cancer between populations of different genetic makeup is inconsistent and unreliable.
Directory of Open Access Journals (Sweden)
Hasibun Naher
2012-01-01
Full Text Available We construct new exact traveling wave solutions involving free parameters of the nonlinear reaction diffusion equation by using the improved (G′/G-expansion method. The second-order linear ordinary differential equation with constant coefficients is used in this method. The obtained solutions are presented by the hyperbolic and the trigonometric functions. The solutions become in special functional form when the parameters take particular values. It is important to reveal that our solutions are in good agreement with the existing results.
Symmetry groups and spiral wave solution of a wave propagation equation
Institute of Scientific and Technical Information of China (English)
张全举; 屈长征
2002-01-01
We study a third-order nonlinear evolution equation, which can be transformed to the modified KdV equation,using the Lie symmetry method. The Lie point symmetries and the one-dimensional optimal system of the symmetryalgebras are determined. Those symmetries are some types of nonlocal symmetries or hidden symmetries of the modifiedKdV equation. The group-invariant solutions, particularly the travelling wave and spiral wave solutions, are discussedin detail, and a type of spiral wave solution which is smooth in the origin is obtained.
Energy Technology Data Exchange (ETDEWEB)
Aviles, B.N.; Sutton, T.M.; Kelly, D.J. III.
1991-09-01
A generalized Runge-Kutta method has been employed in the numerical integration of the stiff space-time diffusion equations. The method is fourth-order accurate, using an embedded third-order solution to arrive at an estimate of the truncation error for automatic timestep control. The efficiency of the Runge-Kutta method is enhanced by a block-factorization technique that exploits the sparse structure of the matrix system resulting from the space and energy discretized form of the time-dependent neutron diffusion equations. Preliminary numerical evaluation using a one-dimensional finite difference code shows the sparse matrix implementation of the generalized Runge-Kutta method to be highly accurate and efficient when compared to an optimized iterative theta method. 12 refs., 5 figs., 4 tabs.
Jia, Junxiong; Peng, Jigen; Yang, Jiaqing
2017-04-01
In this paper, we focus on a space-time fractional diffusion equation with the generalized Caputo's fractional derivative operator and a general space nonlocal operator (with the fractional Laplace operator as a special case). A weak Harnack's inequality has been established by using a special test function and some properties of the space nonlocal operator. Based on the weak Harnack's inequality, a strong maximum principle has been obtained which is an important characterization of fractional parabolic equations. With these tools, we establish a uniqueness result of an inverse source problem on the determination of the temporal component of the inhomogeneous term, which seems to be the first theoretical result of the inverse problem for such a general fractional diffusion model.
Directory of Open Access Journals (Sweden)
B. Godongwana
2010-01-01
Full Text Available 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 vertical or horizontal orientation. The first-order limit of the Michaelis-Menten equation for substrate consumption was used to test the developed model against available analytical results. The results obtained from the application of this model, along with a biofilm growth kinetic model, will be useful in the derivation of an efficiency expression for enzyme production in an MBR.
Application of Linearized Anisotropic Diffusion Equation to Edge-Preserving Image Smoothing
Institute of Scientific and Technical Information of China (English)
YUANZejian; ZHENGNanning; ZHANGYuanlin
2004-01-01
A stable anisotropic diffusion model for edge-preservlng image smoothing is presented. By comparison with existing diffusion models~ this model has two significant new features. The first is that the diffusion model has the ability to adjust the diffusion coefficient and to regulate diffusion orientation simultaneously. The second is that the diffusion model being nonlinear in essence is linearized along time coordinates which makes the model more stable and thus the complexity of computation is reduced greatly. We use an Additive operator splitting scheme (AOS) that is an efficient and stable numerical method to implement the edge-preservlng image smoothing. The experimental results show that the model proposed in this paper is not only stable and efficient but can preserve image edge.
Institute of Scientific and Technical Information of China (English)
陈松林
2002-01-01
@@ The following Lemma stated (in: Applied Mathematics and Mechanics (English Edition),2001,22( 11 ): 1352 - 1356) was applied to describe the asymptotic behaviors of solutions of aclass of reaction-diffusion equation. On account of the special property of the super solution(i.e., a nonzero solution and null solution have C1 smooth connection) constructed in theLemma, we'll give the Lemma a detailed proof as follows.
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.
A New Lattice Bhatnagar-Gross-Krook Model for the Convection-Diffusion Equation with a Source Term
Institute of Scientific and Technical Information of China (English)
DENG Bin; SHI Bao-Chang; WANG Guang-Chao
2005-01-01
@@ A new lattice Bhatnagar-Gross-Krook (LBGK) model for the convection-diffusion equation with a source term is proposed. Unlike the models proposed previously, the present model does not require any additional assumption on the source term. Numerical results are found to be in excellent agreement with the analytical solutions. It is also found that the numerical accuracy of the model is much better than that of the existing models.
Directory of Open Access Journals (Sweden)
M.H. Tiwana
2017-04-01
Full Text Available This work investigates the fractional non linear reaction diffusion (FNRD system of Lotka-Volterra type. The system of equations together with the boundary conditions are solved by Homotopy perturbation transform method (HPTM. The series solutions are obtained for the two cases (homogeneous and non-homogeneous of FNRD system. The effect of fractional parameter on the mass concentration of two species are shown and discussed with the help of 3D graphs.
Energy Technology Data Exchange (ETDEWEB)
Hsieh, Chih-Chun [Department of Materials Science and Engineering, National Chung Hsing University, 250 Kuo-Kuang Rd., Taichung 402, Taiwan (China); Wu, Weite, E-mail: wwu@dragon.nchu.edu.t [Department of Materials Science and Engineering, National Chung Hsing University, 250 Kuo-Kuang Rd., Taichung 402, Taiwan (China)
2010-09-17
Research highlights: This article concentrates the phase transformation in {delta} {yields} {sigma} in dissimilar stainless steels using the Vitek equation and thermodynamics simulation during the multi-pass welding. The phase transformation in {delta} {yields} {sigma} is very important to the properties of stainless steel composites. In this study, the diffusion behavior of Cr, Ni and Si in the {delta}, {sigma}, and {gamma} phases were discussed using the DSC analysis and diffusion equation calculation. This method has a novelty for discussing the phase transformation in {delta} {yields} {sigma} 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 {sigma} 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 (D{sub Cr}{sup {delta}} and D{sub Ni}{sup {delta}}) of Cr and Ni are higher in {delta}-ferrite than (D{sub Cr}{sup {gamma}} and D{sub Ni}{sup {gamma}}) in the {gamma} phase and that they facilitate the precipitation of {sigma} phase in the third pass fusion zone. When the diffusion activation energy of Cr in {delta}-ferrite is equal to that of Ni in {delta}-ferrite (Q{sub dCr}{sup {delta}}=Q{sub dNi}{sup {delta}}), phase transformation of the {delta} {yields} {sigma} can be occurred.
Energy Technology Data Exchange (ETDEWEB)
Jin, Shi, E-mail: sjin@wisc.edu [Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706 (United States); Institute of Natural Sciences, Department of Mathematics, MOE-LSEC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240 (China); Lu, Hanqing, E-mail: hanqing@math.wisc.edu [Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706 (United States)
2017-04-01
In this paper, we develop an Asymptotic-Preserving (AP) stochastic Galerkin scheme for the radiative heat transfer equations with random inputs and diffusive scalings. In this problem the random inputs arise due to uncertainties in cross section, initial data or boundary data. We use the generalized polynomial chaos based stochastic Galerkin (gPC-SG) method, which is combined with the micro–macro decomposition based deterministic AP framework in order to handle efficiently the diffusive regime. For linearized problem we prove the regularity of the solution in the random space and consequently the spectral accuracy of the gPC-SG method. We also prove the uniform (in the mean free path) linear stability for the space-time discretizations. Several numerical tests are presented to show the efficiency and accuracy of proposed scheme, especially in the diffusive regime.