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IMA Journal of Applied Mathematics Advance Access originally published online on October 3, 2008
IMA Journal of Applied Mathematics 2008 73(6):850-872; doi:10.1093/imamat/hxn033
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© The Author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

The fundamental solution and numerical solution of the Riesz fractional advection–dispersion equation

S. Shen

School of Mathematical Sciences, HuaQiao University, Quanzhou, Fujian, China

F. Liu{dagger}

School of Mathematical Sciences, Queensland University of Technology, Queensland 4001, Australia and School of Mathematical Sciences, South China University of Technology, Guangzhou 510640, China

V. Anh and I. Turner

School of Mathematical Sciences, Queensland University of Technology, Queensland 4001, Australia

{dagger} Email: fwliu{at}xmu.edu.cn

Received on November 29, 2006; Revision received October 4, 2007. Accepted on August 28, 2008

In this paper, we consider a Riesz fractional advection–dispersion equation (RFADE), which is derived from the kinetics of chaotic dynamics. The RFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives by the Riesz fractional derivatives of order {alpha} isin (0, 1) and β isin (1, 2], respectively. We derive the fundamental solution for the Riesz fractional advection–dispersion equation with an initial condition (RFADE-IC). We investigate a discrete random walk model based on an explicit finite-difference approximation for the RFADE-IC and prove that the random walk model belongs to the domain of attraction of the corresponding stable distribution. We also present explicit and implicit difference approximations for the Riesz fractional advection–dispersion equation with initial and boundary conditions (RFADE-IBC) in a finite domain. Stability and convergence of these numerical methods for the RFADE-IBC are discussed. Some numerical examples are given to show that the numerical results are in good agreement with our theoretical analysis.

Keywords: Riesz fractional derivative; advection–dispersion equation; discrete random walk; stability; convergence.


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