Non-Singular Generalized RBF Solution and Weaker Singularity MFS: Laplace Equation and Anisotropic Laplace Equation

Abstract

This paper introduces a singular distance function r(s) in terms of a symmetric non-negative metric tensor S. If S satisfies a quadratic matrix equation involving a parameter beta, then for the Laplace equation r(s)(beta) is a non-singular generalized radial basis function solution if 2 > beta > 0, and a weaker singularity fundamental solution if -1 < beta < 0. With a unit vector as a medium to express S, we can derive the metric tensor in closed form and prove that S is a singular projection operator. For the anisotropic Laplace equation, the corresponding closed-form representation of S is also derived. The concept of non-singular generalized radial basis function solution for the Laplace-type equations is novel and useful, which has not yet appeared in the literature. In addition, a logarithmic type method of fundamental solutions is developed for the anisotropic Laplace equation. Owing to non-singularity and weaker singularity of the bases of solutions, numerical experiments verify the accuracy and efficiency of the proposed methods.