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


Autoria(s): Yang, Qianqian; Liu, Fawang; Turner, I.
Data(s)

2010

Resumo

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.

Formato

application/pdf

Identificador

http://eprints.qut.edu.au/29844/

Publicador

Elsevier

Relação

http://eprints.qut.edu.au/29844/1/Numerical_methods_for_fractional_partial_differntial_equations_with_Riesz_space_fractional_derivatives_.pdf

DOI:10.1016/j.apm.2009.04.006

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.

Fonte

Faculty of Science and Technology; Mathematical Sciences

Palavras-Chave #010200 APPLIED MATHEMATICS #Fractional Advection-Dispersion Equation #Riesz Space Fractional Derivative #L1/L2-approximation Method #Standard/shifted Grunwald Method #Matrix Transform Method #Method of Lines
Tipo

Journal Article