A method of fundamental solutions without fictitious boundary

TLDR: The singular boundary method (SBM) as mentioned in this paper employs the singular fundamental solution of the governing equation of interest as the interpolation basis function to avoid the singularity at the origin.

Abstract: This paper proposes a novel meshless boundary method called the singular boundary method (SBM). This method is mathematically simple, easy-to-program, and truly meshless. Like the method of fundamental solutions (MFS), the SBM employs the singular fundamental solution of the governing equation of interest as the interpolation basis function. However, unlike the MFS, the source and collocation points of the SBM coincide on the physical boundary without the requirement of introducing fictitious boundary. In order to avoid the singularity at the origin, this method proposes an inverse interpolation technique to evaluate the singular diagonal elements of the MFS coefficient matrix. The SBM is successfully tested on a benchmark problems, which shows that the method has a rapid convergence rate and is numerically stable.