臺灣博碩士論文加值系統

本文主要應用微分轉換法求解非線性振動系統問題，包括具有超越函數外力系統、耦合聯立系統及Duffing系統問題，並依據所得數值結果探討系統運動之週期與混沌現象；另外亦應用微分轉換法配合有限差分法之混合法求解二維Sine-Gordon系統之孤立子解，並討論孤立波運動情形及兩孤立波碰撞後之運動情形；最後嘗試應用混合法求解時變型邊界之線性對流擴散問題。
微分轉換法係運用微分轉換將系統統御方程式轉換成代數方程式，經由一個或多個顯性之迭代方程式計算出有限個 頻譜離散函數值，以逆微分轉換求得原微分方程式之數值解。文中所用之混合法應用於求解偏微分方程式時，係以微分轉換法處理時間域問題，而以有限差分法處理空間域問題，求解時在滿足邊界與起始條件下，代入經轉換後之代數迭代方程式求得轉換域之解，最後經由逆微分轉換求得其數值解。
本文所使用之數值解法與其它數值方法比較結果顯示，微分轉換法或配合有限差分之混合法在非線性動態系統研究方面，係為一有效且精確求解的方法之一。

This paper presents the differential transformation method to investigate the behaviors of various nonlinear vibration problems including the transcendental function system, coupling system and Duffing’s oscillator equations, and discuss the periodic and chaotic motion according to its numerical results. And the use of a hybrid method which combines differential transformation and finite difference methods in the solution of the two-dimensional Sine- Gordon equation, and discuss the collision of two expanding circular ring solitons. Also, the hybrid method is employed to solve the two-dimensional linear advection-diffusion problem.
Initially, the basic rules of differential transformation are applied to the nonlinear governing equations and its initial conditions. The initial values are then used to obtain the solution of the next time step. Applying an iterative procedure in which the solutions of one time step are used as the initial values of the next step, the differential equation is solved over the entire time domain. The solution of differential equation is then obtained by applying inverse differential transformation.
It is shown that the results obtained are in good agreement with the analytical solutions, and that the results are more accurate than those provided by other approximate numerical methods. Hence, the hybrid method is one of the most accurate and effective techniques for this kind of nonlinear problems.

中文摘要 I
英文摘要 II
目錄 IV
表目錄 VII
圖目錄 VIII
符號說明 XV
第一章 緒論 1
　　1-1 研究背景及目的 1
　　1-2 文獻回顧 2
　　1-3 本文架構 4
第二章 微分轉換法 6
　　2-1 前言 6
　　2-2 微分轉換法之基本定義 6
　　2-3 微分轉換之基本運算法則 10
　　2-3-1 線性運算 10
　　2-3-2 乘法運算 11
　　2-3-3 除法運算 11
　　2-3-4 微分運算 12
　　2-4 譜儲存法 13
第三章 非線性振動問題 16
　　3-1 前言 16
　　3-2 單自由度系統 16
　　3-3 Duffing系統 26
　　3-4 結果與討論 30
第四章 Duffing方程式及其混沌運動型態 43
　　4-1 前言 43
　　4-2 數值模擬 43
　　4-3 結果與討論 45
第五章 二維Sine-Gordon方程式之孤立子解 59
　　5-1 前言 59
　　5-2 數值模擬 60
5-3 結果與討論 69
第六章 應用混合法求解時變型邊界之線性對流擴散問題 86
　　6-1 前言 86
　　6-2 數值模擬 86
6-3 結果與討論 90
第七章 結論與建議 93
　　7-1 結論 93
　　7-2 建議及未來研究方向 94
參考文獻 95

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