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THE FUNDAMENTAL AND NUMERICAL SOLUTIONS OF THE RIESZ SPACE-FRACTIONAL REACTION–DISPERSION EQUATION

Part of: Markov processes Partial differential equations, initial value and time-dependent initial-boundary value problems Real functions Stochastic processes

Published online by Cambridge University Press:  01 July 2008

J. CHEN ,
F. LIU ,
I. TURNER  and
V. ANH
J. CHEN
Affiliation:
School of Science, Jimei University, Xiamen 361021, China
F. LIU*
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, Queensland 4001, Australia (email: f.liu@qut.edu.au) School of Mathematical Sciences, South China University of Technology, Guangzhou 510640, China
I. TURNER
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, Queensland 4001, Australia (email: f.liu@qut.edu.au)
V. ANH
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, Queensland 4001, Australia (email: f.liu@qut.edu.au)
*
✉For correspondence; e-mail: f.liu@qut.edu.au
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Abstract

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A Riesz space-fractional reaction–dispersion equation (RSFRDE) is obtained from the classical reaction–dispersion equation (RDE) by replacing the second-order space derivative with a Riesz derivative of order β∈(1,2]. In this paper, using Laplace and Fourier transforms, we obtain the fundamental solution for a RSFRDE. We propose an explicit finite-difference approximation for a RSFRDE in a bounded spatial domain, and analyse its stability and convergence. Some numerical examples are presented.

Keywords

random walkfractional derivativesprobability densitystabilityconvergence

MSC classification

Secondary: 60J60: Diffusion processes 60G52: Stable processes 26A33: Fractional derivatives and integrals 65M12: Stability and convergence of numerical methods 65M06: Finite difference methods
Type
Research Article
Information
The ANZIAM Journal , Volume 50 , Issue 1 , July 2008 , pp. 45 - 57
Copyright
Copyright © Australian Mathematical Society 2009

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