A mesh-free pseudospectral approach to estimating the fractional Laplacian via radial basis functions

This paper investigates the use of radial basis function (RBF) interpolants to estimate a function's fractional Laplacian of a given order through a mesh-free pseudospectral method. The mesh-free approach yields an algorithm that can be implemented in high dimensional settings without adjustment. Moreover, the fractional Laplacian is defined in terms of the Fourier transform, and the symmetry of RBFs can be exploited to simplify the estimation problem. Convergence rates are established for RBFs when the function whose fractional Laplacian to be estimated is compactly supported. Further results demonstrate convergence when a function is in the native space for a Wendland RBF (i.e. a Sobolev space) and satisfies a certain L-1 condition. Numerical experiments demonstrate the developed method by estimating the fractional Laplacian of several functions and by solving a fractional Poisson equation with extended Dirichlet condition in one and two dimensions. (C) 2019 Elsevier Inc. All rights reserved.