Numerical methods for the fractional partial

In this paper, we consider the numerical solutions of a fractional partial differential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finite domain. Two kinds of FPDE-RSFD are considered: the Riesz fractional diffusion equation (RFDE) and the Riesz fractional advection-dispersion equation (RFADE). RFDE is obtained from the standard diffusion equation by replacing the secondorder space derivative with the Riesz fractional derivative of order α ∈ (1, 2]; RFADE is obtained from the standard advection-dispersion equation by replacing the 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 RFDE and RFADE are derived. Secondly, three numerical methods are provided to deal with the Riesz space fractional derivatives, the L1/L2approximation method, the standard/shifted Grünwald method, and a new matrix transform method (MTM). Thirdly, the RFDE and RFADE are transformed into a system of ordinary differential equations (ODE), which is then solved by the method of lines (MOL). Finally, numerical results are given, which are in good agreement with the analytic solutions.