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

摘要

In this paper, we consider the numerical solution of a fractional partial differential equa-tion 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 α ∈ (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 β∈(0,1) and of order α ∈(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 Griinwald 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.