scholarly journals ALGORITHMS FOR NUMERICAL SOLVING OF 2D ANOMALOUS DIFFUSION PROBLEMS

2012 ◽  
Vol 17 (3) ◽  
pp. 447-455 ◽  
Author(s):  
Natalia Abrashina-Zhadaeva ◽  
Natalie Romanova
Keyword(s):  
Diffusion Equation ◽  
Finite Difference Methods ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Reaction Diffusion Equation ◽  
Stability And Convergence ◽  
Initial Boundary Value Problems ◽  
Fractional Order Differential Equations ◽  
Initial Boundary Value ◽  
Diffusion Problems

Fractional analog of the reaction diffusion equation is used to model the subdiffusion process. Diffusion equation with fractional Riemann–Liouville operator is analyzed in this paper. We offer finite-difference methods that can be used to solve the initial-boundary value problems for some time-fractional order differential equations. Stability and convergence theorems are proved.

Numerical solutions of the reaction-diffusion equation

2015 ◽  
Vol 25 (2) ◽  
pp. 265-271 ◽  
Author(s):  
G Wu ◽  
Eric Wai Ming Lee ◽  
Gao Li
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Diffusion Equation ◽  
Nonlinear Partial Differential Equations ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Reaction Diffusion Equation ◽  
Initial Boundary Value Problems ◽  
Content Type ◽  
Initial Boundary Value

Purpose – The purpose of this paper is to introduce variational iteration method (VIM) to construct equivalent integral equations for initial-boundary value problems of nonlinear partial differential equations. The Lagrange multipliers become the integral kernels. Design/methodology/approach – Using the discrete numerical integral formula, the general way is given to solve the famous reaction-diffusion equation numerically. Findings – With the given explicit solution, the results show the conveniences of the general numerical schemes and numerical simulation of the reaction-diffusion is finally presented in the cases of various coefficients. Originality/value – The method avoids the treatment of the time derivative as that in the classical finite difference method and the VIM is introduced to construct equivalent integral equations for initial-boundary value problems of nonlinear partial differential equations.

On a singular initial-boundary-value problem for a reaction-diffusion equation arising from a simple model of isothermal chemical autocatalysis

1992 ◽  
Vol 437 (1901) ◽  
pp. 657-671 ◽  
Keyword(s):  
Simple Model ◽  
Diffusion Equation ◽  
Boundary Value Problem ◽  
Rate Constant ◽  
Boundary Value ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Reaction Diffusion Equation ◽  
Initial Boundary Value

In this paper we examine the evolution that occurs when a localized input of an autocatalyst B is introduced into an expanse of a reactant A. The reaction is autocatalytic of order p,so A -> B at rate k [A] [B] p with rate constant k . We examine the case when 0 < p < 1, with p>/ 1 having been examined by Needham & Merkin (Phil. Trans. R. Soc. Lond. A 337, 261—274 (1991)). In particular, we show that the fully reacted state is not achieved (as t-> oo) via the propagation of a travelling wavefront (as for p>/ 1) but is approached uniformly in space as t-00.

scholarly journals Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation

2012 ◽  
Vol 15 (1) ◽  
Author(s):  
Yuri Luchko
Keyword(s):  
Diffusion Equation ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Initial Boundary Value Problems ◽  
The Maximum Principle ◽  
Time Fractional Diffusion Equation ◽  
Initial Boundary Value

AbstractIn this paper, some initial-boundary-value problems for the time-fractional diffusion equation are first considered in open bounded n-dimensional domains. In particular, the maximum principle well-known for the PDEs of elliptic and parabolic types is extended for the time-fractional diffusion equation. In its turn, the maximum principle is used to show the uniqueness of solution to the initial-boundary-value problems for the time-fractional diffusion equation. The generalized solution in the sense of Vladimirov is then constructed in form of a Fourier series with respect to the eigenfunctions of a certain Sturm-Liouville eigenvalue problem. For the onedimensional time-fractional diffusion equation $$(D_t^\alpha u)(t) = \frac{\partial } {{\partial x}}\left( {p(x)\frac{{\partial u}} {{\partial x}}} \right) - q(x)u + F(x,t), x \in (0,l), t \in (0,T)$$ the generalized solution to the initial-boundary-value problem with Dirichlet boundary conditions is shown to be a solution in the classical sense. Properties of this solution are investigated including its smoothness and asymptotics for some special cases of the source function.

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