應用再生核配置法於非線性疊代分析

Abstract

在強形式配置法之非線性相關研究中，本研究為首先以再生核函數配置法(RKCM)求解半線性橢圓偏微分方程問題之分析。在疊代方法中，使用擬牛頓疊代法與牛頓疊代法分別求解三個例題，包含三角函數、指數函數與多項式組合三角函數之問題，根據數值分析之結果，兩疊代法呈現之收斂行為非常相似，然傳統的牛頓疊代法較擬牛頓疊代法收斂速度快，且數值結果較穩定，但擬牛頓疊代法每一疊代步所花費的運算時間較少，隨著配置法中之點數增加時，節省的時間將大幅增加。

In the nonlinear related research of the strong form collocation methods, this is the first work using the reproducing kernel collocation method (RKCM) to solve the semilinear elliptic partial differential equations. As for the iteration schemes, we adopt both the quasi-Newton iteration method and Newton iteration method to solve three examples with the following types of solutions: a trigonometric function, an exponential function, and a trigonometric function combined with a polynomial. Based on our numerical results, the two iteration methods show similar convergence behavior. The Newton iteration method converges faster and is more stable than the quasi-Newton iteration method. But the quasi-Newton iteration method requires less CPU time in each iterative step. Therefore, as the number of collocation points increases, the quasi-Newton iteration method will save more time.