A Unified Meshfree Pseudospectral Method For Solving Both Classical And Fractional Pdes

In this paper, we propose a meshfree method based on the Gaussian radial basis function (RBF) to solve both classical and fractional PDEs. The proposed method takes advantage of the analytical Laplacian of Gaussian functions so as to accommodate the discretization of the classical and fractional Laplacians in a single framework and avoid the large computational cost for numerical evaluation of the fractional derivatives. These important merits distinguish our method from other existing methods for fractional PDEs. Moreover, our method is simple and easy when handling complex geometries and local refinements, and its computer program implementation remains the same for any dimension d >= 1. Extensive numerical experiments are provided to study the performance of our method in both approximating the Dirichlet Laplace operators and solving classical and fractional PDE problems. We show that our method has spectral accuracy and can achieve good approximation even with a small number of points. Compared to the recently proposed Wendland RBF method, our method exactly incorporates the Dirichlet boundary conditions into the scheme and is free of the Gibbs phenomenon as observed in the wendland method.