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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 \(\alpha\in (1,2]\). 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 \(\beta \in (0,1)\) and of order \(\alpha \in (1,2]\), 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.

MSC:
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
26A33 Fractional derivatives and integrals
35R11 Fractional partial differential equations
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