  The method of fundamental solution (MFS) is an efficient meshless method for
solving a boundary value problem in an exterior unbounded domain. The numerical
solution obtained by the MFS is accurate, while the corresponding matrix
equation is ill-conditioned. A modified MFS (MMFS) with the proper basis
functions is proposed by the introduction of the modified Trefftz method (MTM).
The concrete expressions of the corresponding condition numbers and the
solvability by these methods are mathematically proven. Thereby, the optimal
parameter minimizing the condition number is also mathematically given.
Numerical experiments show that the condition numbers of the matrices
corresponding to the MTM and the MMFS are reduced and that the numerical
solution by the MMFS is more accurate than the one by the conventional method.
