A Unified Meshfree Pseudospectral Method for Solving Both Classical and Fractional PDEs

Abstract

In this paper, we propose a meshfree method based on the Gaussian radial\nbasis function (RBF) to solve both classical and fractional PDEs. The proposed\nmethod takes advantage of the analytical Laplacian of Gaussian functions so as\nto accommodate the discretization of the classical and fractional Laplacian in\na single framework and avoid the large computational cost for numerical\nevaluation of the fractional derivatives. These important merits distinguish it\nfrom other numerical methods for fractional PDEs. Moreover, our method is\nsimple and easy to handle complex geometry and local refinement, and its\ncomputer program implementation remains the same for any dimension $d \\ge 1$.\nExtensive numerical experiments are provided to study the performance of our\nmethod in both approximating the Dirichlet Laplace operators and solving PDE\nproblems. Compared to the recently proposed Wendland RBF method, our method\nexactly incorporates the Dirichlet boundary conditions into the scheme and is\nfree of the Gibbs phenomenon as observed in the literature. Our studies suggest\nthat to obtain good accuracy the shape parameter cannot be too small or too\nbig, and the optimal shape parameter might depend on the RBF center points and\nthe solution properties.\n