此論文運用Bessel和Neumann函數為基底之基本解法(MFS)去分析2D Helmholtz方程之誤差和穩定性。我們導出在有界的單連通區域之誤差範圍，然而條件數範圍的推導僅限制在圓盤區域上。由實驗中可看出，運用Bessel函數為基底之基本解法比起Neumann函數為基底更有效率。在Bessel函數為基底之基本解法中，有趣的是，源點(source points)的半徑未必大於方程解區域裡的最大半徑r_max 。這違反了在基本解(MFS)裡的一般條件: r_max < R 。最後，利用實驗數據去驗證理論分析和結論。

In the thesis, the error and stability analysis is made for the 2D Helmholtz equation by the method of fundamental solutions (MFS) using both Bessel and Neumann functions. The bounds of errors in bounded simply-connected domains are derived, while the bounds of condition number are derived only for disk domains. The MFS using Bessel functions is more efficient than the MFS using Neumann functions. Interestingly, for the MFS using Bessel functions, the radius R of the source points is not necessarily larger than the maximal radius r_max of the solution domain. This is against the traditional condition: r_max < R for MFS. Numerical experiments are carried out to support the analysis and conclusions made.

1 Introduction 4
2 Algorithms 4
3 Preliminary Lemmas 11
4 Error Analysis of MFS using Bessel functions for Small k 14
5 Stability Analysis for Disk Domains 21
6 MFS Using Neumann Functions 25
6.1 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6.2 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6.3 Bounds of Condition Numbers . . . . . . . . . . . 29
7 Numerical Experiments 30
8 Concluding Remarks 32

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