臺灣博碩士論文加值系統

本文主要在探討Lazer-McKenna吊橋模型的多重週期解。欲求並延拓非線性偏微分方程組之週期解是一種很困難的工作，本文提供一種演算法來求整個解路徑，並延拓隨著一個參數變動的非線性偏微分方程組之週期解，我們稱之為虛擬弧長延拓法(Pesudoarclength continuation method)。我們將虛擬弧長延拓法應用在Lazer-McKenna吊橋模型上，而虛擬弧長延拓法是基於打靶法、牛頓法、Crank-Nicolson法、猜測及解法(predictor-solver)及隱函數定理等數值方法，且我們將利用其來探討Lazer-McKenna吊橋模型週期解的解路徑。本文中，我們將藉由係數的改變、分歧圖的討論及週期解圖形的探討，並說明產生多重週期解的參數區間與初始條件的相互關係。

The main purpose of this paper is to investigate the multiple periodic solution of the Lezer-McKenna suspension bridge model.To obtain and continue the path of periodic solution of the systems of nonlinear partial differential equation is a more difficult task. An algorithm for continuation of the path of periodic solutions in partial differential equations dependent on a parameter is pre-sented in this article and is successfully applied to the Lezer-McKenna suspension bridge model. The pseudoarclength continuation method for the continuation of path of periodic solutions is based on the shooting method, Newton method, Crank-Nicolson method, predictor-solver method and implicit function theorem. We will use pseudoarclength continuation method to investigate the path of the Lazer-McKenna suspension bridge periodic solutions. In this paper, we discuss bifur-cation diagrams and graphs of periodic solution of our model dependent on several coefficients. We also discuss the relationship between the parameter interval of multi-ple periodic solutions and initial conditions of the model.

第一章 緒論
第二章 分歧理論與虛擬弧長延拓法
2.1 分歧問題
2.2 分歧理論
2.3 局部延拓法
2.3.1 預測法
2.3.2 解法
2.4 虛擬弧長延拓法
第三章 數值解法
3.1 u_xxxx項的離散
3.2 初始值問題解法
3.3 週期解之求法
3.4 虛擬弧長延拓法求解路徑
3.5 Lazer-McKenna吊橋模型週期解之主要演算法
第四章 數值實驗
4.1 演算法的測試
4.2 分歧圖的定性分析
4.3 係數改變的影響
4.4 多重週期解區間
4.5 多重週期解的立體圖
第五章 結論
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