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Numerical methods for fractional partial differential equations with Riesz space fractional derivatives

Yang, Qianqian, Liu, Fawang, & Turner, I. (2010) Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Applied Mathematical Modelling, 34(1), pp. 200-218.

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Abstract

In this paper, we consider the numerical solution of a fractional partial differential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finite domain. Two types of FPDE-RSFD are considered: the Riesz fractional diffusion equation (RFDE) and the Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second-order space derivative with the Riesz fractional derivative of order αset membership, variant(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order βset membership, variant(0,1) and of order αset membership, variant(1,2], respectively. Firstly, analytic solutions of both the RFDE and RFADE are derived. Secondly, three numerical methods are provided to deal with the Riesz space fractional derivatives, namely, the L1/L2-approximation method, the standard/shifted Grünwald method, and the matrix transform method (MTM). Thirdly, the RFDE and RFADE are transformed into a system of ordinary differential equations, which is then solved by the method of lines. Finally, numerical results are given, which demonstrate the effectiveness and convergence of the three numerical methods.

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ID Code: 29844
Item Type: Journal Article
Keywords: Fractional Advection-Dispersion Equation, Riesz Space Fractional Derivative, L1/L2-approximation Method, Standard/shifted Grunwald Method, Matrix Transform Method, Method of Lines
DOI: 10.1016/j.apm.2009.04.006
ISSN: 0307-904X
Subjects: Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > APPLIED MATHEMATICS (010200)
Divisions: Past > QUT Faculties & Divisions > Faculty of Science and Technology
Past > Schools > Mathematical Sciences
Deposited On: 20 Jan 2010 16:30
Last Modified: 01 Mar 2012 06:28

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