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Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. (English) Zbl 1185.65200
Summary: 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 $\alpha \in \left(1,2\right]$. 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 $\beta \in \left(0,1\right)$ and of order $\alpha \in \left(1,2\right]$, 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.
##### MSC:
 65M99 Numerical methods for IVP of PDE 26A33 Fractional derivatives and integrals (real functions) 35R11 Fractional partial differential equations