A modified method of fundamental solutions with source on the boundary for solving laplace equations with circular and arbitrary domains

Abstract

A boundary-type method for solving the Laplace problems using the modified method of fundamental solutions (MMFS) is proposed. The present method (MMFS) implements the singular fundamental solutions to evaluate the solutions, and it can locate the source points on the real boundary as contrasted to the conventional MFS, where a fictitious boundary is needed to avoid the singularity of diagonal term of influence matrices. The diagonal term of influence matrices for arbitrary domain can be novelly determined by relating the MFS with the indirect BEM and are also solved for circular domain analytically by using separable kernels and circulants. The major difficulty of the coincidence of the source and collocation points in the conventional MFS is thereby overcome. The off-diagonal coefficients of influence matrices can be easily determined by using the two-point function. The ill-posed nature of the conventional MFS then disappears. Finally, we provide numerical evidences that the present method improves the accuracy of the solution after comparing with the conventional MFS, in particular for complicated boundaries in which the conventional MFS may encounter difficulties. Good agreements are observed as comparing with analytic or other numerical solutions.