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系統識別號 U0026-1207201818033100
論文名稱(中文) 一些非線性系統的精確旅行波解
論文名稱(英文) Exact Traveling Wave Solutions for Some Nonlinear Systems
校院名稱 成功大學
系所名稱(中) 數學系應用數學碩博士班
系所名稱(英) Department of Mathematics
學年度 106
學期 2
出版年 107
研究生(中文) 黃冠傑
研究生(英文) Kuan-Chieh Huang
電子信箱 pjtm1368@hotmail.com
學號 L16031019
學位類別 碩士
語文別 英文
論文頁數 42頁
口試委員 指導教授-方永富
口試委員-林育竹
口試委員-劉聚仁
中文關鍵字 Boussinesq 方程組  Boussinesq-Kadomtsev-Petviashili(BKP) 方程組  非線性薛丁格-KdV 方程組  Klein-Gordon-Zakharov 方程組  指數擴充方法 
英文關鍵字 Boussinesq equation  Boussinesq-Kadomtsev-Petviashili(BKP) equation  Nonlinear Schrodinger-KdV equation  Klein-Gordon-Zakharov equation  exponential expansion method 
學科別分類
中文摘要 在這篇論文中,我們主要探討及整理兩篇論文。首先是由M. G. Hafez及R. Sakthivel所撰寫的論文,題目是「一些重要的非線性物理模型的精確旅行波的解」;其次是M. G. Hafez的著作,題目是「運用指數擴充方法得到(3+1)維度的Klein-Gordon-Zakharov方程式的精確解」。此二篇均是利用「指數擴充方法」去得到方程式的精確旅行波解,兩篇論文探討的常微分方程不一樣,然後代入一些參數,再將其解的圖形用Maple畫出,這些告訴我們方程式有不同類型的解,他們有其代表的物理意義。我們研讀這幾篇論文,探討了四種偏微分方程系統,我們補充了一些作者沒有寫出來的解,讓結果更完整。
英文摘要 In this thesis, we mainly discuss and summarize two papers. The first is a paper written by M. G. Hafez and R. Sakthivel which is entitled “Exact Traveling Wave Solutions for Some Important Coupled Nonlinear Physical Models.” The second is a paper written by M. G. Hafez which is entitled “Exact Solutions to the (3+1)-Dimensional Coupled Klein-Gordon-Zakharov Equation Using Exponential Expansion Method.”
Both of these two articles use the “exponential expansion method” to get the exact traveling wave solutions of the equations, and substitute some proper parameters, and then use the Maple to draw the solution graphs. These tell us that the equations have different types of solutions and they have their own physical meanings.
We studied these several papers and discussed four systems of partial differential equations. We added some solutions that were not written by the authors to make the results more complete.
論文目次 摘要/Abstract I
誌謝 II
Contents III
List of Figures V
1 Introduction 1
2 The Methodology 3
3 Applications to Some Important Nonlinear Coupled Physical Models 8
3.1 The (1+1)-dimensional classical Boussinesq equation..8
3.2 The (2+1)-dimensional Boussinesq and Kadomtsev-Petviashvili (BKP) equation.......................15
3.3 The nonlinear coupled Schrodinger-KdV equation....20
4 Method for Another ODE 27
4.1 Explanation of the method.................27
4.2 The coupled Klein-Gordon-Zakharov equation......33
5 Results and Discussions 41
References 42
參考文獻 [1] Md. Nur Alam, Md. Ali Akbar and Syed Tauseef Mohyud-Din, A novel (G'/G)-expansion method and its application to the Boussinesq equation. Chin. Phys. B Vol. 23, No. 2 (2014) 020203.
[2] Hasibun Naher and Farah Aini Abdullah, New approach of (G'/G)-expansion method and new approach of generalized (G'/G)-expansion method for nonlinear evolution equation. AIP Advances 3, 032116 (2013); doi: 10.1063/1.4794947.
[3] Mohammed K. Elboree, The Jacobi elliptic function method and its application for two component BKP hierarchy equations. Computers and Mathematics with Applications 62 (2011) 4402–4414.
[4] S. A. El-Wakil and M. A. Abdou, Modified extended tanh-function method for solving nonlinear partial differential equations. Chaos, Solitons and Fractals 31 (2007) 1256–1264.
[5] M. G. Hafez, Exact solutions to the (3+1)-dimensional coupled Klein-Gordon-Zakharov equation using exp-expansion method.
[6] Harun-Or-Roshid and Md. Azizur Rahman, The exp-expansion method with application in the (1+1)-dimensional classical Boussinesq equations. Results in Physics 4 (2014) 150–155.
[7] M. G. Hafez and M. A. Akbar, An exponential expansion method and its application to the strain wave equation in microstrutured solids. Ain Shams Engineering Journal 6 (2015), 683–690.
[8] M. G. Hafez and M. A. Akbar, New exact traveling wave solutions to the (1+1)-dimensional Klein-Gordon-Zakharov equation for wave propagation in plasma using the exp-expansion method. Propulsion and Power Research 4(1) 2015, 31–39.
[9] M. G. Hafez, M. N. Alam and M. A. Akbar, Traveling wave solutions for some important coupled nonlinear physical models via the coupled Higgs equation and the Maccari system. Journal of King Saud University–Science 27 (2015), 105–112.
[10] Ji-Huan He and Xu-Hong Wu, Exp-function method for nonlinear wave equations. Chaos, Solitons and Fractals 30 (2006) 700–708.
[11] H. Jafari, N. Kadkhoda and Anjan Biswas, The (G'/G)-expansion method for solutions of evolution equations from isothermal magnetostatic atmospheres. Journal of King Saud University–Science (2013) 25, 57–62.
[12] W. Malfliet, The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations. Journal of Computational and Applied Mathematics 164–165 (2004) 529–541.
[13] Abdul-Majid Wazwaz, The extended tanh method for abundant solitary wave solutions of nonlinear wave equations. Applied Mathematics and Computation 187 (2007) 1131–1142.
[14] Jin-Liang Zhang, Ming-Liang Wang, Yue-Ming Wang and Zong-De Fang, The improved F-expansion method and its applications. Physics Letters A 350 (2006) 103–109.
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