跳到主要內容

臺灣博碩士論文加值系統

(44.222.64.76) 您好!臺灣時間:2024/06/17 09:08
字體大小: 字級放大   字級縮小   預設字形  
回查詢結果 :::

詳目顯示

: 
twitterline
研究生:郭峻谷
研究生(外文):Chun-KuKuo
論文名稱:以SEM及extended SEM求解非線性偏微分方程式的精確孤波解
論文名稱(外文):Exact soliton solutions of nonlinear partial differential equations by the simplest and the extended simplest equation method
指導教授:李森墉
指導教授(外文):Sen-Yung Lee
學位類別:博士
校院名稱:國立成功大學
系所名稱:機械工程學系
學門:工程學門
學類:機械工程學類
論文種類:學術論文
論文出版年:2015
畢業學年度:104
語文別:英文
論文頁數:84
中文關鍵詞:KdVBurgers可線性化孤波滑移完全可積分方程式the potential KdVHirota’s method.
外文關鍵詞:KdVBurgerslinearitysoliton-slidingcompletely integrable equationthe potential KdVHirota’s method.
相關次數:
  • 被引用被引用:0
  • 點閱點閱:470
  • 評分評分:
  • 下載下載:116
  • 收藏至我的研究室書目清單書目收藏:0
本文中介紹6種方法求解KdV方程式及Burgers方程式的精確孤波解,其所有解皆與非線性項係數成反比,而使得解本身無法與方程式同步線性化。經深入研究並分析各種方法後,發現可利用simplest equation method (SEM) with Bernoulli equation as the simplest equation求得KdV 方程式及Burgers方程式的可線性化精確孤波解。新解克服了存在已久的非線性項係數趨近於零而解不連續的問題。藉由與文獻中的精確解比較,可發現新解呈現出新的物理現象,吾人暫稱為”孤波滑移”。
除此之外,我們延伸了SEM的運用層面。我們利用Burgers equation屬於完全可積分方程式的性質,將它設定為simplest equation來求解其他完全可積分方程式。經由KdV及potential KdV 兩方程式的驗證,該方法確實可行。有別於 Cole-Hopf transformation 及 Hirota’s method兩種方法,我們所提出的方法更簡單、直接且效率更高。
因此,我們相信對於具有孤波及多重孤波解的非線性物理模態皆可利用SEM及延伸型SEM求解。
In this paper, six different exact solution methods have been presented and applied to solve Korteweg and de Vries (KdV) and Burgers equations, whose solutions are proportional to the nonlinear term coefficients. The new exact solutions of KdV and Burgers equations, with the nonlinear term coefficients being arbitrary constants, are derived by the simplest equation method (SEM) with Bernoulli equation as the simplest equation. It is shown that the proposed exact solutions overcome the long existing problem of discontinuity and can be successfully reduced to linearity when the nonlinear term coefficients approach to zero. Comparison of the existing and new soliton solutions is presented. A new phenomenon, named soliton sliding, is observed.
Moreover, we extend the SEM by choosing Burgers equation as the simplest equation. The reason of setting Burgers equation as the simplest equation is due it being completely integrable equation. The extended SEM is applied to handle two completely integrable equations, KdV and the potential KdV equations. The general forms of the multi-soliton solutions are formally established. Unlike Hirota’s method, the results confirm the extended SEM is concise and effective for constructing multi-soliton solutions.
Accordingly, we believe that solitary solutions and multi-soliton solutions existing for other classes of nonlinear mathematic physics models can be easily solved by the SEM and the extended SEM. Further work on these aspects is recommended.
摘 要. …………………I
Abstract II
Acknowledgments IV
Contents V
List of Tables VIII
List of Figures IX
Nomenclature XII
Chapter
1 Introduction 1
1.1 Introduction 1
1.2 Literature review 3
1.2.1 KdV equation 3
1.2.2 Burgers equation 7
1.3 Purpose of present study 8
1.4 Scope 9
2 Six exact solution methods 12
2.1 Analysis of the methods 12
2.2 The tanh-coth method 12
2.3 The sine-cosine method 14
2.4 The simplest equation method 15
2.5 The Exp-function method 16
2.6 The Cole-Hopf transformation 17
2.7 Hirota’s method 18
3 Exact solutions 20
3.1 KdV equation 20
3.2 Existing exact solutions of KdV equation 20
3.2.1 By the tanh-coth method 20
3.2.2 By the sine-cosine method 22
3.2.3 By the Exp-function method 23
3.2.4 By Hirota’s method 26
3.3 Existing exact solutions of Burgers equation 29
3.3.1 By the tanh-coth method 29
3.3.2 By the SEM with Riccati as the simplest equation 30
3.3.3 By the Cole-Hopf transformation 31
3.4 The linearized solutions by the SEM 33
3.4.1 Linearized solutions of KdV equation 33
3.4.2 The linearized solution of Burgers equation 36
3.5 Numerical results and discussions 37
3.6 Conclusions 42
4 The general forms of the multi-soliton solutions for the completely integrable equations by the extended SEM 61
4.1 The extended SEM 61
4.2 Multi-soliton solutions of KdV equation 63
4.3 Multi-soliton solutions of the potential KdV equation 65
4.4 Conclusions 67
5 Summary and Future Prospects 70
5.1 General Conclusions 70
5.2 Future Prospects 71
References 72
Vita 84
[1] G. B. Whitham, “Linear and nonlinear waves, vol. 42, John Wiley & Sons, 2011.
[2] A. M. Wazwaz, “Partial differential equations and solitary waves theory, Springer Science & Business Media, 2010.
[3] S. Liu, et al., “Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Physics Letters A, vol. 289, no.1, pp. 69-74, 2001.
[4] Z. Yan, H. Zhang, “New explicit solitary wave solutions and periodic wave solutions for Whitham–Broer–Kaup equation in shallow water, Physics Letters A, vol. 285, no. 5, pp. 355-362, 2001.
[5] W. Yuan, et al., “The general solutions of an auxiliary ordinary differential equation using complex method and its applications, Advances in Difference Equations, vol. 1, pp. 1-9, 2014.
[6] A. M. Wazwaz, “New solitary-wave special solutions with compact support for the nonlinear dispersive K (m, n) equations, Chaos, Solitons & Fractals, vol. 13, no. 2, pp. 321-330, 2002.
[7] A. Esen, S. Kutluay, “Solitary wave solutions of the modified equal width wave equation, Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 8, pp. 1538-1546, 2008.
[8] D. J. Evans, K. R. Raslan, “Solitary waves for the generalized equal width (GEW) equation, International Journal of Computer Mathematics, vol. 82, no. 4, pp. 445-455, 2005.
[9] N. A. Kudryashov, M. V. Demina, “Travelling wave solutions of the generalized nonlinear evolution equations, Applied Mathematics and Computation, vol. 210, no. 2, pp. 551-557, 2009.
[10] Y. Chen, B. Li, “General projective Riccati equation method and exact solutions for generalized KdV-type and KdV–Burgers-type equations with nonlinear terms of any order, Chaos, Solitons & Fractals, vol. 19, no. 4, pp. 977-984, 2004.
[11] S. Shen, Z. Pan, “A note on the Jacobi elliptic function expansion method, Physics Letters A, vol. 308, no .2, pp. 143-148, 2003.
[12] D. Lu, B. Hong, L. Tian, “New solitary wave and periodic wave solutions for general types of KdV and KdV–Burgers equations, Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 1, pp. 77-84, 2009.
[13] M. Wang, Y. Wang, Y. Zhou, “An auto-Backlund transformation and exact solutions to a generalized KdV equation with variable coefficients and their applications, Physics Letters A, vol. 303, no. 1, pp. 45-51, 2002.
[14] K. A. Gepreel, T. A. Nofal, F. M. Alotaibi, “Exact solutions for nonlinear differential difference equations in mathematical physics, Abstract and Applied Analysis, Hindawi Publishing Corporation, 2013.
[15] A. M. Wazwaz, A. Gorguis, “Exact solutions for heat-like and wave-like equations with variable coefficients, Applied Mathematics and Computation, vol. 149, no. 1, pp. 15-29, 2004.
[16] W. Zhang, Q. Chang, B. Jiang, “Explicit exact solitary-wave solutions for compound KdV-type and compound KdV–Burgers-type equations with nonlinear terms of any order, Chaos, Solitons & Fractals, vol. 13, no. 2, pp. 311-319, 2002.
[17] A. M. Wazwaz, “New solitons and kink solutions for the Gardner equation, Communications in Nonlinear Science and Numerical Simulation, vol. 12, no. 8, pp. 1395-1404, 2007.
[18] N. A. Kudryashov, “Exact solitary waves of the Fisher equation, Physics Letters A, vol. 342, no. 1, pp. 99-106, 2005.
[19] P. M. Kruglyakov et al., “Foam drainage, Current Opinion in Colloid & Interface Science, vol. 13, no. 3, pp. 163-170, 2008.
[20] S. Zhang, H. Q. Zhang, “Fractional sub-equation method and its applications to nonlinear fractional PDEs, Physics Letters A, vol. 375, no. 7, pp. 1069-1073, 2011.
[21] M. J. Ablowitz, H. Segur, “Solitons and the inverse scattering transform, vol. 4, Philadelphia: Siam, 1981.
[22] R. Hirota, “The direct method in soliton theory., Cambridge, Cambridge University Press 2004.
[23] M. Khalfallah, “New exact travelling wave solutions of the (3+ 1) dimensional Kadomtsev–Petviashvili (KP) equation, Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 4, pp. 1169-1175, 2009.
[24] M. Wang, “Solitary wave solutions for variant Boussinesq equations, Physics Letters A, vol. 199, no. 3, pp. 169-172, 1995.
[25] M. Wang, X. Li, J. Zhang, “The ( )-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Physics Letters A, vol. 372, no. 4, pp. 417-423, 2008.
[26] E. Fan, “Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics, Chaos, Solitons & Fractals, vol. 16, no. 5, pp. 819-839, 2003.
[27] N. Taghizadeh, M. Mirzazadeh, “Exact Travelling Wave Solutions for Konopelchenko-Dubrovsky Equation by the First Integral Method, Appl. Appl. Math, vol. 6, pp. 1893-1901, 2011.
[28] A. Bekir, O. Unsal, “Exact solutions for a class of nonlinear wave equations by using first integral method, International Journal of Nonlinear Science, vol. 15, no. 2, pp. 99-110, 2013.
[29] F. Tascan, A. Bekir, M. Koparan, “Travelling wave solutions of nonlinear evolution equations by using the first integral method, Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 1810-1815, 2009.
[30] F. Khani, et al., “New solitary wave and periodic solutions of the foam drainage equation using the Exp-function method, Nonlinear Analysis: Real World Applications, vol. 10, no. 3, pp. 1904-1911, 2009.
[31] Z. Zheng, W. R. Shan, “Application of Exp-function method to the Whitham–Broer–Kaup shallow water model using symbolic computation, Applied Mathematics and Computation, vol. 215, no. 6, pp. 2390-2396, 2009.
[32] A. Ebaid, “Exact solitary wave solutions for some nonlinear evolution equations via Exp-function method, Physics Letters A, vol. 365, no. 3, pp. 213-219, 2007.
[33] F. Xu, et al., “Evaluation of two-dimensional ZK-MEW equation using the Exp-function method, Computers & Mathematics with Applications, vol. 58, no. 11, pp. 2307-2312, 2009.
[34] W. Malfliet, W. Hereman, “The tanh method: I. Exact solutions of nonlinear evolution and wave equations, Physica Scripta, vol. 54, no. 6, pp. 563, 1996.
[35] W. Malfliet, W. Hereman, “The tanh method: II. Perturbation technique for conservative systems, Physica Scripta, vol. 54, no. 6, pp. 569, 1996.
[36] A. A. Soliman, “The modified extended tanh-function method for solving Burgers-type equations, Physica A: Statistical Mechanics and its Applications, vol. 361, no. 2, pp. 394-404, 2006.
[37] A. M. Wazwaz, “The tanh method for travelling wave solutions of nonlinear equations, Applied Mathematics and Computation, vol. 154, no. 3, pp. 713-723, 2004.
[38] Z. Lü, H. Zhang, “On a further extended tanh method, Physics Letters A, vol. 307, no. 5, pp. 269-273, 2003.
[39] H. Chen, H. Zhang, “New multiple soliton solutions to the general Burgers–Fisher equation and the Kuramoto–Sivashinsky equation, Chaos, Solitons & Fractals, vol. 19, no. 1, pp. 71-76, 2004.
[40] A. M. Wazwaz, “The tanh and the sine–cosine methods for a reliable treatment of the modified equal width equation and its variants, Communications in Nonlinear Science and Numerical Simulation, vol. 11, no. 2, pp. 148-160, 2006.
[41] A. Gorguis, “A comparison between Cole–Hopf transformation and the decomposition method for solving Burgers’ equations, Applied Mathematics and Computation, vol. 173, no. 1, pp. 126-136, 2006.
[42] B. M. Vaganan, “Cole‐Hopf Transformations for Higher Dimensional Burgers Equations With Variable Coefficients, Studies in Applied Mathematics, vol. 129, no. 3, pp. 300-308, 2012.
[43] P. M. Jordan, “On the application of the Cole–Hopf transformation to hyperbolic equations based on second-sound models, Mathematics and Computers in Simulation, vol. 81, no. 1, pp. 18-25, 2010.
[44] A. H. Salas, C. A. Gómez S., “Application of the Cole-Hopf transformation for finding exact solutions to several forms of the seventh-order KdV equation, Mathematical Problems in Engineering 2010.
[45] P. L. Sachdev, “A generalised Cole-Hopf transformation for nonlinear parabolic and hyperbolic equations, Zeitschrift für angewandte Mathematik und Physik ZAMP, vol. 29, no. 6, pp. 963-970, 1978.
[46] N. K. Vitanov, “On modified method of simplest equation for obtaining exact and approximate solutions of nonlinear PDEs: the role of the simplest equation, Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 11, pp. 4215-4231, 2011.
[47] İ. Aslan, “A discrete generalization of the extended simplest equation method, Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 8, pp. 1967-1973, 2010.
[48] N. A. Kudryashov, N. B. Loguinova, “Extended simplest equation method for nonlinear differential equations, Applied Mathematics and Computation, vol. 205, no. 1, pp. 396-402, 2008.
[49] N. K. Vitanov, “Application of simplest equations of Bernoulli and Riccati kind for obtaining exact travelling-wave solutions for a class of PDEs with polynomial nonlinearity, Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 8, pp. 2050-2060, 2010.
[50] N. K. Vitanov, “Modified method of simplest equation: powerful tool for obtaining exact and approximate travelling-wave solutions of nonlinear PDEs, Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1176-1185, 2011.
[51] N. A. Kudryashov, “One method for finding exact solutions of nonlinear differential equations, Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 6, pp. 2248-2253, 2012.
[52] N. A. Kudryashov, “Seven common errors in finding exact solutions of nonlinear differential equations, Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 9, pp. 3507-3529, 2009.
[53] K. Brauer, “The Korteweg-de Vries equation: history, exact solutions, and graphical representation, University of Osnabrück/Germany1, 2000.
[54] J. M. Curry, “Soliton Solutions of Integrable Systems and Hirota’s Method, Mathematics Review, vol. 43, 2007.
[55] A. Jeffrey, T. Kakutani, “Weak nonlinear dispersive waves: a discussion centered around the Korteweg-de Vries equation, Siam Review, vol. 14, no. 4, pp. 582-643, 1972.
[56] J. W. Miles, “The Korteweg-de Vries equation: a historical essay, Journal of Fluid Mechanics, vol. 106, pp. 131-147, 1981.
[57] M. J. Ablowitz, P. A. Clarkson, “Solitons, nonlinear evolution equations and inverse scattering, vol. 149, Cambridge University Press, 1991.
[58] J. Hietarinta, “Introduction to the Hirota bilinear method: Integrability of Nonlinear Systems, Springer Berlin Heidelberg, pp. 95-103, 1997.
[59] H. Bateman, “Some recent researches on the motion of fluids, Monthly Weather Review, vol. 43, pp. 163-170, 1915.
[60]J. M. Burgers, “A mathematical model illustrating the theory of turbulence, Advances in Applied Mechanics, vol. 1, pp. 171-199, 1948.
[61] E. Hopf, “The partial differential equation , Commun. Pure Appl. Math., vol. 3, pp. 201-230, 1950.
[62] J. D. Cole, “On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math., vol. 9, pp. 225-236, 1951.
[63] N. Sugimoto, “Burgers’ equation with a fractional derivative; hereditary effects on nonlinear acoustic waves, J. Fluid Mech., vol. 225, pp. 631-653, 1991.
[64] E. Weinan, et al, “Invariant measure for Burgers equation with stochastic forcing, Annals of Mathematics-Second Series, vol. 151, pp. 877-960, 2000.
[65] A. M. Wazwaz, “Travelling wave solutions of generalized forms of Burgers, Burgers-KdV and Burgers-Huxly equations, Applied Mathematics and Computation, vol. 169, pp. 639-656, 2005.
[66]A. George, W. Blum, J. D. Cole, “The general similarity solution of the heat equation, Journal of Mathematics and Mechanics, vol. 18, no. 11, 1969.
[67]J. Bec, K. Khanin, “Burgers turbulence, Physics Reports, vol. 447, pp. 1-66, 2007.
[68]B. Bell et al., “A second-order projection method for the incompressible Navier-Stokes equations, Journal of Computational Physics, vol. 85, pp. 257-283, 1989.
[69] J. H. He, X. H. Wu, “Exp-function method for nonlinear wave equations, Chaos, Solitons & Fractals, vol.30, no. 3, pp. 700-708, 2006.
[70] W. Hereman, A. Nuseir, “Symbolic methods to construct exact solutions of nonlinear partial differential equations, Mathematics and Computers in Simulation, vol.43, no. 1, pp. 13-27, 1997.
[71] R. S. Johnson, “A modern introduction to the mathematical theory of water waves, vol. 19, Cambridge University Press, 1997.
[72] M. R. Spiegel, “Schaum’s Outline Series Mathematical handbook of formulas and tables, Schaum Pub. Co., 1968.
QRCODE
 
 
 
 
 
                                                                                                                                                                                                                                                                                                                                                                                                               
第一頁 上一頁 下一頁 最後一頁 top