Chen [3]證明利用基本解法耦合徑向基底函數來求解一維帕松方程式的近似解會收斂到Driscoll 和 Fornberg [2]利用拉格朗日插值多項式計算出來的結果。對於橢圓算子的特徵模組問題，我們可以得到類似的結果，即利用徑向基底函數求得的近似解會收斂到利用拉格朗日插值多項式為基底所計算出來的結果。在這篇論文中，我們比較利用徑向基底函數與拉格朗日插值多項式求解一維橢圓算子特徵模組問題的近似解，得出徑向基底函數求得的近似解會收斂到拉格朗日插值多項式的近似解之結論。

　　Chen el al. [3] showed that for 1D Poisson's equation the approximate solution obtained by using method of fundamental solutions (MFS) coupled with the radial basis functions (RBFs) converges in the sense of Lagrange interpolating polynomial using the result of Driscoll and Fornberg [2]. For an elliptic operator eigenmode problem we show that similar results could be obtained. That is the solution obtained by radial basis collocation schemes converges to the solution with Lagrange interpolating polynomial being the underline approximate solution. In this thesis, we compare the approximate solutions of 1-D elliptic eigenmodes problem using radial basis function and Lagrange interpolating polynomial in a collocation scheme, and conclude that the solution obtained from RBF collocation method converges to the solu- tion using Lagrange interpolating polynomial as approximate solution in the similar sense of Driscoll and Fornberg [2].

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Radial Basis Function Collocation Method . . . . . . . . . . . . . . . . 2
2.1 Radial Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 Solving One-Dimensional Eigenmodes Problem of Poisson
Equation Using Lagrange interpolating polynomial . . . .. . . . . 5
2.4 Solving One-Dimensional Eigenmodes Problem of Poisson
Equation using RBF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

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