Moving pseudo-boundary method of fundamental solutions for nonlinear potential problems

Abstract

In the method of fundamental solutions (MFS), the solution of a boundary value problem (BVP) is approximated by a linear combination of fundamental solutions expressed in terms of sources which are located on a pseudo–boundary outside the domain of the problem. A major issue in such applications is the optimal placement of these sources. In this work we try to address this issue by letting some parameters which control the location of the sources free, to be determined as well as the coefficients of the fundamental solutions in the MFS approximations. When this approach is applied to nonlinear BVPs, this results in a system of nonlinear equations which can be solved by means of the MATLAB© routines lsqnonlin and fsolve. We investigate the effect of providing the exact Jacobian of the system to these routines to the overall efficacy of the approach. The results of several numerical experiments are presented and analyzed.