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

机译：具有Riesz空间分数阶导数的分数阶偏微分方程的数值方法

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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.
机译：在本文中，我们考虑了有限域上带有Riesz空间分数导数（FPDE-RSFD）的分数阶偏微分方程的数值解。考虑两种类型的FPDE-RSFD：Riesz分数对流扩散方程（RFDE）和Riesz分数对流扩散方程（RFADE）。通过将二阶空间导数替换为α∈（1,2]的Riesz分数导数，可以从标准扩散方程获得RFDE；通过替换一阶和导数，可以从标准对流扩散方程获得RFADE。分别具有阶数为β∈（0,1）和阶数为α∈（1,2）的Riesz分数阶导数的二阶空间导数，首先，导出RFDE和RFADE的解析解；其次，三种数值方法提供了处理Liesz空间分数导数的方法，即L1 / L2逼近法，标准/移位Griinwald方法和矩阵变换方法（MTM）；其三，将RFDE和RFADE变换为一个普通系统微分方程，然后用线法求解，最后给出数值结果，证明了这三种数值方法的有效性和收敛性。

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