求解二階耦合邊界值問題的李群SL(2,R)打靶法之研究

The Study of the Lie-group SL(2,R) Shooting Method to Solve Coupled Second-order Boundary Value Problems

劉明鑫(Ming-Xin Liu)

指導教授 ： 劉進賢

國立臺灣大學/工學院/土木工程學研究所/碩士(2013年)

https://doi.org/10.6342/NTU.2013.11210

摘要

科學家或工程師在研究某些工程或是理化問題時，會產生許多的數學問題，而這類的問題會產生一些非線性方程式、偏微分方程式或是方程組像是薛丁格方程式(Schrodinger equation)、史特母-劉維爾方程式(Sturm-Liouville equation)、哈密頓-亞可比方程式(Hamilton–Jacobi equation)等等數學問題，而在這些數學問題當中所隱含的物理問題正是我們必須要解決的，因為這當中的問題是與我們的生活息息相關的。只有為數不多的微分方程式或方程組可以得到解析解(Analytic solution)，而在較為複雜的非線性微分方程式當中，我們必需利用數值分析的方式去得到其數值解(numerical solution) 。在本篇論文當中我們採用了許多前人的理論作為基礎，我們採用了學者劉發展的保群算法，藉由李群的封閉性質，將複雜的二階非線性邊界值問題、二階耦合非線性邊界值問題，轉換成初始值問題(Initial-value problem)，再配合四階龍格－庫塔法(Runge-Kutta methods)求出問題之數值解(Numerical solution)。本文當中使用了程式語言MATLAB進行數值分析，並進行繪圖將其模擬狀況具現化。

關鍵字

保群算法 ； 李群打靶法 ； 耦合邊界值問題 ； 四階龍格－庫塔法

並列摘要

When scientist and engineer are doing research concerning physics and construction, it may come out lots of mathematical problems, and such kind of problems may produce non-linear ordinary differential equation(ODE)、partial differential equation(PDE) or equations like Schrodinger equation、Sturm-Liouville equation、Hamilton–Jacobi equation, and so on. However, the most important thing is that we have to find the solution to the problem and understand the meaning in the field of physics, it’s because it closely related with our life. Besides, there is not much ODE or equation we can get the analytical solution, so we have to reach the numerical solution by using the techniques of modern numerical methods. In this thesis, we consult many articles as our theoretical basis, and use the group preserving schemes (GPS) developed by Liu. We can transform the complicated second-order nonlinear boundary value problem (BVP) and the coupled second-order nonlinear boundary value problem (BVP) into initial value problem (IVP), with the closure property of the Lie-group, then we use the Runge-Kutta methods to get the numerical solution. Finally, we use programming language MATLAB for the numerical analysis and plot the numerical results.

並列關鍵字

group preserving schemes (GPS) ； Lie-group shooting method (LGSM) ； Coupled boundary value problem (BVP) ； Runge-Kutta methods(RK4)

參考文獻

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[4] Liu, C.-S., Cone of non-linear dynamical system and group preserving schemes, International Journal of Non-linear Mechanics Vol. 36,pp. 1047-1068 (2001).

[9] Liu, C.-S, Efficient shooting methods for the second order ordinary differential equations CMES: Computer Modeling in Engineering & Sciences Vol. 15 , pp. 69-86 (2006).