| 姓名 |
呂東丞(Dong-Chen. Leu)
查詢紙本館藏 |
畢業系所 |
物理學系 |
| 論文名稱 |
齊次平衡解析方法在求解非線性偏微分方程式的適用性分析
(Applicability Analysis of Homogeneous Balance Method in Solving Nonlinear Partial Differential Equations)
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| 相關論文 | |
| 檔案 |
 [Endnote RIS 格式]
[Bibtex 格式]
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| 摘要(中) |
在近代科學研究中，非線性效應的影響越來越不可忽略，非線性偏微分方程式的重要性也隨之提高。但大多非線性系統不是絕對可解模型，因此研究學者發展了不少數學方法來解析非線性偏微分方程式。
在本文中我們主要使用齊次平衡解析法（Homogeneous Balance Method），希望找出除了數值模擬以外的方法求解非線性方程式。在此研究中我們利用解析解與數值模擬做比對，並以一常見的絕對可解模型－KdV方程式作為基準。
研究中發現，不同的初始條件會造成不同的精準度。而齊次平衡方法在除了孤波之外的計算並無法表現色散現象以及非線性現象的特徵。希望由此研究結果確認此方法的可信度或可使用的範圍，以加速往後對非線性系統的運算。 |
| 摘要(英) |
In nowadays scientific researches, the nonlinear effects become non ignorable. Therefore, the technique for solving nonlinear partial differential equations is indispensable. Most nonlinear partial differential equations are not exact solvable, therefore, many researchers are trying to develop different other methods for solving these problems.
In this study, the applicability of the homogeneous balance method in solving partial differential equations was examined by comparing the analytical results with the numerical simulation which based on finite - difference time - domain method.
This work can provide a guideline for the application of the homogeneous balance method in solving partial differential equations with various parameter space and initial conditions. |
| 關鍵字(中) |
★ 非線性系統
★ 孤波 |
關鍵字(英) |
★ Nonlinear system
★ Soliton |
| 論文目次 |
目錄
中文摘要 ii
Abstract iii
目錄 iv
圖目錄 v
致謝 vii
第一章 緒論 1
1.1 非線性系統 6
第二章 物理模型與解析方法
2.1 多粒子電漿系統理論推導 3
2.2 單模態光纖系統理論推導 12
2.3 齊次平衡解析法 20
第三章 數值計算與結果
3.1 數值計算驗證 29
3.2 多例子電漿系統 32
3.3 單模態光纖系統 43
第四章 結論與未來展望 48
參考文獻 49
Appendix A 50
Appendix B 55 |
| 參考文獻 |
[1] V. O. Vakhnenko, E. J. Parkes, and A. J. Morrison, Chaos, Solitons & Fractals 17, 683 (2003)—P.2
[2] V. E. ZAKHAROVand A. B. SHABAT, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of wave in nonlinear media (1970)—P.2
[3] R. Sabry, M. A. Zahran, and E. Fan, Phys. Lett. A 326, 326 (2004)—P.2
[4] M. L. Wang, Phys. Lett. A 199, 169 (1995)—P.2
[5] Anwar Ja′afar Mohamad Jawad, The sin-cos function method for the exact solution of nonlinear partial differential equation (2012)—P.2
[6] Elsayed M. E. Zayed, Exact Solutions for Fifth-Order KdV Equations with Variable Coefficients by Using the Modified Sine- Cosine Method (2014)—P.2
[7] U M ABDELSALAM, F M ALLEHIANY, Nonlinear structures for extended Korteweg–de Vries equation in multicomponent plasma (2014)—P.9
[8] Mathematics of soliton transmission in optical fiber—P.19
[9] Tesi di Dottorato, A Study of a Nonlinear Schro ?dinger Equation for Optical Fibers (2016)—P.19
[10] George B. Arfken, Mathematical method for physicists (2005)—P.33
[11] Ling-Hsiao Lyu, Nonlinear space plasma physics (2005)—P.44
[12] Abdul-Majid Wazwaz, Reliable analysis for nonlinear Schro ?dinger equations with a cubic nonlinearity and a power law nonlinearity (2005)—P.49
[13] W. Malfliet, The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations (2002)—P.55 |
| 指導教授 |
陳仕宏(CHEN, Shih-Hung)
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審核日期 |
2018-8-22 |
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