Reproducing kernel particle method for two-dimensional time-space fractional diffusion equations in irregular domains

In recent years, the fractional differential equations have attracted a lot of attention due to their interested characteristics. Meshfree methods are highly accurate and have been extensively explored in engineering and mechanics fields. However, there is few research to develop the reproducing kernel particle method (RKPM), one of the widely used meshfree approach, for fractional partial differential equations. In this work, we solve time-space fractional diffusion equations in 2D regular and irregular domains. The temporal Caputo fractional derivatives are discretized by the L1 finite difference scheme and the spatial Laplacian fractional derivatives are discretized by RKPM based on the matrix transfer method. Especially, the corrected weighted shifted Griinwald-Letnikov scheme is utilized for temporally non-smooth solutions. Numerical examples in rectangular, circular, sector and human brain-like irregular domains are given to assess the efficiency and accuracy of the proposed numerical scheme. The spatial Laplacian fractional derivatives discretized by conventional finite difference method in the rectangular domain are also presented for comparison. The results indicate that RKPM is very effective for analyzing the considered fractional equations in various domains, which lays a concrete foundation for our further research of real application of human brain modeling.