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研究生： 張中維
Chang, Chung-Wei
論文名稱： 有界對稱區間上基於 Chebyshev 譜方法解四階非線性 Helmholtz 方程
Using Chebyshev spectral collocation method to solve the fourth-order nonlinear Helmholtz equation in the bounded symmetric domain
指導教授： 李金龍
Li, Chin-Lung
口試委員: 張延彰
Chang, Yen-Chang
李俊憲
Li, Chun-Hsien
學位類別： 碩士
Master
系所名稱： 理學院 - 計算與建模科學研究所
Institute of Computational and Modeling Science
論文出版年： 2021
畢業學年度： 109
語文別： 英文
論文頁數： 62
中文關鍵詞： 有界對稱區間 、Chebyshev 譜方法 、四階非線性 Helmholtz 方程
外文關鍵詞： bounded symmetric domain, Chebyshev spectral collocation method, fourth-order nonlinear Helmholtz equation
相關次數： 點閱：2  下載：0 
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    • 四階非線性亥姆霍茲方程是一個在研究四階非線性薛丁格方程性質中非常重要的結果。在物理中，四階非線性亥姆霍茲方程與四階非線性薛丁格方程，光學，水波理論與橋墩的數學模型有關。我們提出切比雪夫譜方法使其在滿足對稱邊界上解出四階亥姆霍茲方程數值解。基於 Matlab 軟體，數值結果將被用來描述我們在一維與二維配合一些邊界條件與對稱區間的四階亥姆霍茲方程的理論結果。最後，我們模擬圖形並且總結一維四階非線性亥姆霍茲方程與二維四階非線性亥姆霍茲方程在任意對稱長度定義域中的解。

    The fourth-order NLH equation is an important result to investigate the behavior of the fourth-order nonlinear Schrödinger equation. In physics, the fourth-order NLH equation is related to the fourth-order nonlinear Schrödinger equation, the mathematical models of optics, wave theory, and suspension bridge. We propose the Chebyshev spectral collocation differential method such that the method can satisfy the symmetric domain to numerically solve the fourth-order NLH equation. Based on the Matlab software, the numerical results are presented to illustrate our theoretical results in the one-dimensional and two-dimensional fourth-order NLH equation with some boundary conditions and symmetric domain. Finally, we simulate the graphs and conclude our results which contain various symmetric lengths, one-dimensional, and two-dimensional fourth-order NLH equations.

    1 Introduction........................................................ 1 2 The Chebyshev Spectral Collocation Method in Symmetric Domain........5 2.1 The Chebyshev Differential Matrix..................................5 2.2 The Chebyshev Differential Matrix for Higher-Order Derivatives ... 8 3 The Fourth-Order ODE with the Dirichlet and Neumann Conditions......10 3.1 Nonlinear ODE with the Boundary Conditions are Zeros..............10 3.2 Linear ODE with the Boundary Conditions are Real Numbers..........15 4 The Fourth-Order PDE with the Dirichlet and Neumann Conditions......24 4.1 Nonlinear PDE with the Boundary Conditions are Zeros .............25 4.2 Nonlinear PDE with the Boundary Conditions are Real Numbers.......29 5 Conclusions.........................................................37 References............................................................38 6 Appendix............................................................42

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