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. 200218.

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Abstract
In this paper, we consider the numerical solution of a fractional partial differential equation with Riesz space fractional derivatives (FPDERSFD) on a finite domain. Two types of FPDERSFD 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 secondorder 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 firstorder and secondorder 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/L2approximation 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 
Refereed:  Yes 
Keywords:  Fractional AdvectionDispersion Equation, Riesz Space Fractional Derivative, L1/L2approximation Method, Standard/shifted Grunwald Method, Matrix Transform Method, Method of Lines 
DOI:  10.1016/j.apm.2009.04.006 
ISSN:  0307904X 
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 06:30 
Last Modified:  29 Feb 2012 20:28 
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