"二階與三階耦合邊值問題的李群 SL(n,R) 打靶法之研究"

Abstract

邊值問題是許多工程及數學領域上很常見的問題，也有許多其他形式的問題經轉換而成邊值問題，如何有效精確地求解邊值問題是工程師很重要的課題。邊值問題當中的耦合邊值問題具有兩條或以上的方程式，方程式彼此之間交互影響，因為方程式的相依性，增加求解的難度。本文所使用的求解兩點邊值問題的方法為李群打靶法，其中運用的一步保群算法具有李群的封閉性、快速計算…等等的優點，已經精確地解決了許多二階或是三階的邊值問題。本論文利用李群打靶法的優點，將求解的問題推廣至三階與二階耦合邊值問題，推導一個新形式的李群打靶法，並結合不同李群來做求解，驗證李群打靶法在耦合邊值問題上，仍具有準確性。結合工程數學、廣義中值定理等等觀念，推導產生李群打靶法的步驟後，由常用的市售程式語言之一MATLAB來實行，以期推廣至更複雜的邊值問題，如高階耦合、多重耦合邊值問題…等等。

In enginerring and mathematics, boundary value problems are common problems. There are many other form problems that are transformed to boundary value problems. How to accurately solve the boundary value problems is a very important subject for engineers. The coupled boundary value problem is one kind of boundary value problems with two or more equations, and there are interactions between equations. Because of the cross dependencies of equatioms, the difficulty of solving boundary value problems increases. In this study, the numerical solution for two-point boundary value problems is the Lie-group shooting method(LGSM). By using the advandtages of Lie-group`s closure property and quick calculation ,…etc, the one-step group preserving schemes has been used to accurately solve many second-order or third-order boundary value problems. In this paper ,the advantages of using the LGSM for solving the prombles will be extended to two and three-order coupled boundary value prombles. Developing a new form of LGSM, and combining different Lie-groups for solving the coupled boundary value problems. We will prove that the new form LGSM for solving coupled boundary value problem is accurate. After deducing the steps of using LGSM with engineering mathematics and Generalized mid-point rule…etc, we will use one of the commonly used commercial programming language MATLAB to implement steps. We expect LGSM to be extended to more complex boundary value problems, such as high-order coupled, multi-coupled boundary value problems…and so on.