In this paper, we propose a high-order method for numerical solution of space fractional diffusion equation (SFDE) in one and two dimensions. The space fractional derivative of order \(1<\alpha < 2\) is described in Caputo’s sense. The spline approximation of Caputo is considered which has second-order accuracy in space. To improve the spatial accuracy, Richardson extrapolation method is presented. As a result, the high-order method can be viewed as the modification of the existing jobs (Sousa, Comput Math Appl 62:938–944, 2011; Salehi et al., Appl Math Comput 336:465–480, 2018). For the two-dimensional case, an alternating direction implicit (ADI) scheme is considered to split the equation into two separate one-dimensional equations. Moreover, the proposed scheme is extended to two-sided SFDE case. Numerical results confirm the theoretical supports and the effectiveness of the proposed scheme.

中文翻译：

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样条法结合Richardson外推数值解空间分数扩散方程

本文针对一维和二维空间分数扩散方程（SFDE）的数值解提出了一种高阶方法。阶\（1 <\ alpha <2 \）的空间分数导数是Caputo所描述的。卡普托的样条近似被认为在空间上具有二阶精度。为了提高空间精度，提出了理查森外推法。结果，高阶方法可以看作是对现有工作的修改（Sousa，Comput Math Appl 62：938-944，2011； Salehi等人，Appl Math Comput 336：465-480，2018）。对于二维情况，考虑使用交替方向隐式（ADI）方案将方程拆分为两个单独的一维方程。此外，该方案扩展到双面SFDE的情况。数值结果证实了所提出方案的理论支持和有效性。