Application of the Method of Fundamental Solutions to Moving Boundary Problems and Complex Analysis

Abstract

The method of fundamental solutions is known as a mesh-free numerical method for solving potential problems. It has been actively used in numerous fields, especially in engineering. In this paper, two problems are treated as part of an extension of the application of the method of fundamental solutions. The first is a moving boundary problem for viscous fluids, known as the Hele–Shaw problem. It is shown that the method of fundamental solutions can be used for a spatial discretization mechanism that satisfies the geometric variational structures in an asymptotic sense. The second is the numerical analysis of conformal mappings, an essential research subject in complex analysis. By combining the dipole simulation method, which is an improvement of the method of fundamental solutions from the viewpoint of potential theory, with the complex dipole simulation method, which is an extension of the method of fundamental solutions to approximation theory of holomorphic functions, we show that a simple and mathematically natural bidirectional numerical method can be obtained to analyze conformal mappings.