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The method of fundamental solution (MFS) is an efficient meshless method forsolving a boundary value problem in an exterior unbounded domain. The numericalsolution obtained by the MFS is accurate, while the corresponding matrixequation is ill-conditioned. A modified MFS (MMFS) with the proper basisfunctions is proposed by the introduction of the modified Trefftz method (MTM).The concrete expressions of the corresponding condition numbers and thesolvability by these methods are mathematically proven. Thereby, the optimalparameter minimizing the condition number is also mathematically given.Numerical experiments show that the condition numbers of the matricescorresponding to the MTM and the MMFS are reduced and that the numericalsolution by the MMFS is more accurate than the one by the conventional method.

Mathematical and Numerical Studies on Meshless Methods for Exterior Unbounded Domain Problems