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博碩士論文 etd-0722105-213025 詳細資訊
Title page for etd-0722105-213025
論文名稱
Title
半解析法求解常微分邊值問題
Semi-Analytic Method for Boundary Value Problems of ODEs
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
62
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2005-05-26
繳交日期
Date of Submission
2005-07-22
關鍵字
Keywords
半解析法.、Sturm-Liouville問題、冪級數、邊值問題、奇異擾動問題
singularly perturbed problem, power series, boundary value problem, Sturm-Liouville problem, semi-analytic method.
統計
Statistics
本論文已被瀏覽 5732 次,被下載 2574
The thesis/dissertation has been browsed 5732 times, has been downloaded 2574 times.
中文摘要
在這篇論文中,我們展示了冪級數的效能,結合數值方法來求解邊值問題及常微方程的Sturm-Liouville特徵值問題。這一類方法通常稱做數值符號、數值解析、或半解析法。
在第一章中,我們發展了一個適合的演算法,用來自動決定冪級數的項數以達到要求的準確性。冪級數的展開點可以被自由的選取,也可使用
Abstract
In this thesis, we demonstrate the capability of power series, combined with numerical methods, to solve boundary value problems and Sturm-Liouville eigenvalue problems of ordinary differential equations. This kind of schemes is usually called the numerical-symbolic, numerical-analytic or semi-analytic method.
In the first chapter, we develop an adaptive algorithm, which automatically decides the terms of power series to reach desired accuracy. The expansion point of power series can be chosen freely. It is also possible to combine several power series piecewisely. We test it on several models, including the second and higher order linear or nonlinear differential equations. For nonlinear problems, the same procedure works similarly to linear problems. The only differences are the nonlinear recurrence of the coefficients and a nonlinear equation, instead of linear, to be solved.
In the second chapter, we use our semi-analytic method to solve singularly perturbed problems. These problems arise frequently in fluid mechanics and other branches of applied mathematics. Due to the existence of boundary or interior layers, its solution is very steep at certain point. So the terms of series need to be large in order to reach the desired accuracy. To improve its efficiency, we have a strategy to select only a few required basis from the whole polynomial family. Our method is shown to be a parameter diminishing method.
A specific type of boundary value problem, called the Sturm-Liouville eigenvalue problem, is very important in science and engineering. They can also be solved by our semi-analytic method. This is our focus in the third chapter. Our adaptive method works very well to compute its eigenvalues and eigenfunctions with desired accuracy. The numerical results are very satisfactory.
目次 Table of Contents
1. Boundary Value Problems of ODEs....................2
1.1 Introduction....................................2
1.2 Semi-Analytic Method............................3
1.3 Coefficients of Power Series....................5
1.4 Second Order Linear Equations...................9
1.5 Second Order Nonlinear Equations...............13
1.6 Higher Order Equations.........................17
2. Singularly Perturbed Differential Equations.......20
2.1 Introduction...................................20
2.2 Semi-Analytic Method...........................21
2.3 Spectral Methods...............................27
2.4 Parameter Diminishing..........................32
2.5 Convergence Analysis...........................35
3. Sturm-Liouville Eigenvalue Problems...............42
3.1 Introduction...................................42
3.2 Semi-Analytic Method...........................43
3.3 Numerical Experiments..........................45
3.4 Conclusion.....................................55
參考文獻 References
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17.G. Papakaliatakis and T. E. Simos, A New Method for the Numerical Solution of Fourth-Order BVP's with Oscillating Solutions, Comput. Math. Appl. 32(1996):1-10.
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