臺灣博碩士論文加值系統

本研究中，我們考慮關於兩點邊界值問題的數值方法。首先，我們介紹一些常微分方程的基本概念。其次，研究初始值問題的數值方法。逼近一階初始值問題的方法可以被分成兩類：一類逐點逼近原始問題；另一類稱為皮卡德迭代的方法，作積分去逼近原始問題中所包含的函數。二階初始值問題也被列入考慮，我們介紹尤拉法去給出逼近值。再者，我們將注意力放在兩點邊界值問題的數值方法。除了傳統方法，我們也研究在2017年被提出的打靶投射法[3]。第四部份，我們應用牛頓法，割線法以及打靶投射法，獲得邊界值問題的數值結果。最後，我們做出結論並提出一些待解問題。

In this thesis, we consider numerical methods of two-point boundary-value problems. First, we introduce some basic concepts of ordinary differential equations. Second, numerical methods of initial-value problems are studied. For approximating first-order initial-value problems, the methods can be divided into two types：one approximates the original problem point by point；the other one, called Picard iteration, performs integration to approximate the function contained in the original problem. Second-order initial-value problems are also taken into consideration, and we introduce Euler’s method to give approximations. Third, we focus our attention on numerical methods of two-point boundary-value problems. In addition to traditional methods, we also study the shooting-projection method [3], proposed in 2017. Fourth, we give numerical results, to which are obtained by applying Newton’s method, secant method, and shooting-projection method, of boundary-value problems. Finally, we arrive at a conclusion and indicate some problems unsolved.

Contents
Acknowledgements...........................................................I
摘要......................................................................II
Abstract.................................................................III
Contents..................................................................IV
List of Figures............................................................V

1. Introduction of Ordinary Differential Equations.........................1

2. Numerical Solutions of Initial-Value Problems...........................3
2-1 The Elementary Theory of Initial -Value Problems.......................3
2-2 Euler’s Method and Higher-Order Taylor Methods.........................4
2-3 Runge-Kutta Methods....................................................8
2-4 Picard Iteration......................................................17
2-5 Euler’s Method for Second-Order Initial-Value Problems................18

3. Numerical Solutions of Boundary-Value Problems.........................22
3-1 The Elementary Theory of Boundary -Value Problems.....................22
3-2 The Linear Shooting Method............................................22
3-3 Shooting Methods for Two-Point Boundary-Value Problems................25
3-4 Finite-Difference Methods.............................................31

4. Numerical Results of Some Boundary-Value Problems......................40

5. Conclusion.............................................................49

References................................................................50
List of Figures
Fig. 4.1..................................................................40
Fig. 4.2..................................................................41
Fig. 4.3..................................................................42
Fig. 4.4..................................................................43
Fig. 4.5..................................................................44
Fig. 4.6..................................................................45
Fig. 4.7..................................................................46
Fig. 4.8..................................................................47
Fig. 4.9..................................................................48

[1] Ivan Dimov, Stefka Fidanova, & Ivan Lirkov (Eds.) (2014). Numerical Methods and Applications. Borovets, Bulgaria：Springer.
[2] Richard L. Burden, & J. Douglas Faires (2010). Numerical Analysis, 9th edition. Boston, MA：Brooks/Cole.
[3] Stefan M. Filipov, Ivan D. Gospodinov, & Istvan Farago (2017). Shooting-projection method for two-point boundary value problems. Applied Mathematics Letters, 72, 10-15.
[4] Thomas, L.H. (1949). Elliptic Problems in Linear Differential Equations over a Network. Watson Sci. Comput. Lab Report, Columbia University, New York.