This thesis consists of two parts. In the first part, to solve the boundary value problems of homogeneous equations, the fundamental solutions (FS) satisfying the homogeneous equations are chosen, and their linear combination is forced to satisfy the exterior and the interior boundary conditions. To avoid the logarithmic singularity, the source points of FS are located outside of the solution domain S. This method is called the method of fundamental solutions (MFS). The MFS was first used in Kupradze in 1963. Since then, there have appeared numerous reports of MFS for computation, but only a few for analysis. The part one of this thesis is to derive the eigenvalues for the Neumann and the Robin boundary conditions in the simple case, and to estimate the bounds of condition number for the mixed boundary conditions in some non-disk domains. The same exponential rates of Cond are obtained. And to report numerical results for two kinds of cases. (I) MFS for Motz's problem by adding singular functions. (II) MFS for Motz's problem by local refinements of collocation nodes. The values of traditional condition number are huge, and those of effective condition number are moderately large. However, the expansion coefficients obtained by MFS are scillatingly large, to cause another kind of instability: subtraction cancellation errors in the final harmonic solutions. Hence, for practical applications, the errors and the ill-conditioning must be balanced each other. To mitigate the ill-conditioning, it is suggested that the number of FS should not be large, and the distance between the source circle and the partial S should not be far, either. In the second part, to reduce the severe instability of MFS, the truncated singular value decomposition(TSVD) and Tikhonov regularization(TR) are employed. The computational formulas of the condition number and the effective condition number are derived, and their analysis is explored in detail. Besides, the error analysis of TSVD and TR is also made. Moreover, the combination of TSVD and TR is proposed and called the truncated Tikhonov regularization in this thesis, to better remove some effects of infinitesimal sigma_{min} and high frequency eigenvectors.