以帶有幅狀基底函數的基本法解橢圓偏微分方程

Solving Elliptic Partial Differential Equations by Fundamental Solutions with Radial Basis Functions

朱建柏(Jian-Bo Jhu)

指導教授 ： 王復國

大同大學/電機資訊學院/應用數學學系所/碩士(2011年)

摘要

中文摘要
不像傳統以兩個步驟求出橢圓偏微分方程式的特殊解及齊次解，我們以帶有幅狀基底函數MQ的基本解法，這是一種新組合的無網格方法，將特殊解及齊次解以一個矩陣系統的新型式求出。在二維的Poisson 方程和Helmholtz 方程在Dirichlet邊界條件下，以此方法計算出幾個例子，並且也以三種不同的幅狀基底函數(RBF)的Kansa’s method來計算，我們以圖表比較這兩種數值方法的精確性和穩定性。

關鍵字

無網格方法 ； 幅狀基底函數 ； 基本法

並列摘要

ABSTRACT
Unlike traditional ways of two stages of searching for particular solution and homogeneous solution of elliptic partial differential equation, we use the method of fundamental solution (MFS) with Multiquadric (MQ), one of radial basis functions (RBF), which is a new combination of meshless method. The particular solution and homogeneous solution can be solved by a linear matrix system through this new formulation. The numerical results of several examples of two dimensional Poisson equation and Helmholtz equation with Dirichlet condition are computed. In addition to the above method, the Kansa’s method of three different kinds of radial basis functions (RBF) are also applied to these examples. We illustrate and compare the accuracy and stability of these two numerical methods.

並列關鍵字

meshless method ； radial basis functions ； particular solution. ； method of fundamental solution

參考文獻

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