Generalized finite difference method for a class of multidimensional space-fractional diffusion equations

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.