在本研究中利用MQ 半徑基底函數法（Multiquadrics Radial Basis Functions Method）解微分方程式，並探討其計算之準確性和適用性。MQ 半徑基底函數中，包含一個形狀參數，此形狀參數之值對整體計算準確性有相當大之影響。當形狀參數增加時，通常可以得到較好之結果，但卻會造成ill-condition 之問題；另一方面，增加佈點點數亦可增加計算之準確性，但有限於目前一般個人電腦，其所能處理矩陣大小之上限約為1500×1500，故在點數增加上有所侷限。
MQ半徑基底函數法，為所謂免切割(mesh-free or meshless )的數值方法，故其可在非規則區間中應用隨機佈點之方式求解。本研究亦探討佈點隨機程度會對整個計算之影響，得到隨機佈點的分散程度越均勻，計算結果越佳；同時在隨機佈點之情況下，其形狀參數如果使用各個參考點跟參考點之間最近之距離，會有較大之誤差產生，而此時如果改用所有最參考點和參考點最近距離之平均值，可以得到較好之結果。
當區間切割應用在此半徑基底函數法時，在形狀參數較小時，可以得到不錯之結果，但因區間與區間之多點重合緣故，使所需計算矩陣之條件數大幅上升，進而導致計算結果發散或無法完全收斂。雖然應用區間切割法，可以突破目前個人電腦所能處理之矩陣大小上限，讓總佈點數再增加，不過依然會有ill-condition 之問題產生。

Several influential factors in solving differential equations by the MQ RBFs method (Multiquardics Radial Basis Functions Method) are investigated in this study.It is found that the accuracy of solutions generally increases with increasing shape parameters contained in the MQ RBFs and the number of node points in the domain.
However, large values of the shape parameter usually lead the method ill-conditioned.In addition, the resultant coefficient matrix becomes unsolvable when the total number of node points exceeds 1500 on a typical personal computer.
MQ RBFs method is a truly meshless algorithm, which has shown to be effective in solving complicated physical problems with irregular domains. A mechanism to allocate the node points in irregular domains is introduced in this study. The ‘randomness’ of the randomly-distributed node points can be adjusted by a b-factor. It
is found that accuracy of solutions increases with decreasing the ‘randomness’ of the distribution of the node points. In addition, the solution accuracy increases further by utilizing improved shape parameters which are estimated by the average distance of the neighboring node points.
Method of overlapping domain decomposition is also incorporated into the MQ RBFs method for solving partial differential equations. Solutions by overlapping domain decomposition can have the same accuracy as in the situation without domain
decomposition for small values of shape parameters. However, for large shape parameters, increasing the overlapping points will cause the matrix of sub-domain ill-conditioned. The merit of overlapping domain decomposition is that problems with large numbers of node points, which are originally unsolvable without domain decomposition, can now be solved by this approach.

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