Fractional diffusion equations have been widely used to accurately describe anomalous solute transport in complex media. This paper proposes a local meshless method named the generalized finite difference method (GFDM), to solve a class of multidimensional space fractional diffusion equations (SFDEs) in a finite domain. In the GFDM, the spatial derivative terms are expressed as linear combinations of neighboring-node values with different weighting coefficients using the moving least-square approximation. An explicit formula for the SFDE is then obtained. The numerical solution is achieved by solving a sparse linear system. Four numerical examples are provided to verify the effectiveness of the proposed method. Numerical analysis indicates that the relative errors of prediction results are stable and less than 1% (0.001–1%). The method can also be applied for irregular grids with acceptable accuracy.

中文翻译：

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一类多维空间分数扩散方程的广义有限差分法

分数扩散方程已被广泛用于准确描述复杂介质中的异常溶质输运。本文提出了一种称为广义有限差分法（GFDM）的局部无网格方法，在有限域中求解一类多维空间分数扩散方程（SFDE）。在 GFDM 中，空间导数项表示为使用移动最小二乘法近似的具有不同加权系数的相邻节点值的线性组合。然后获得 SFDE 的显式公式。数值解是通过求解稀疏线性系统来实现的。提供了四个数值例子来验证所提出方法的有效性。数值分析表明，预测结果的相对误差稳定，小于1%（0.001-1%）。