# 臺灣博碩士論文加值系統

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 在本篇論文中，我們利用程式MAPLE幫助我們使用(G’/G, 1/G)-展開法求 Benney-Luke方程其他不一樣的精確行波解以及跟其它篇論文所得出的解做比較。我們還發現到齊次平衡法跟(G’/G, 1/G)-展開法有一定的關聯性。
 In this thesis, the (G’/G, 1/G)-expansion method with the aid of Maple is used to find (some different) the exact travelling wave solutions of the Benney-Luke equation and we compare our solutions with the solutions appearing in the other papers. We notice that there is a connection between the homogeneous balance method and the (G’/G, 1/G)-expansion method.
 ContentsAbstract 1誌謝辭 2Contents 31 Introduction 42 Methods 62.1 The homogeneous balance method 62.2 The (G’/G, 1/G)-expansion method 92.3 Connections 163 Application 184 Conclusion 325 Bibliography 33
 [1] ML Wang, Solitary wave solutions for variant Boussinesq equations, Physics Letters A, 199, 1995, 169-172.[2] ML Wang, XZ Li, Simplified homogeneous balance method and its applications to the Whitham-Broer-Kaup model equations, Journal of Applied Mathematics and Physics, 2014, 2, 823-827.[3] CL Bai, H Zhao, JG Han, Application of an extended homogeneous balance method to new exact solutions of nonlinear evolution equations, Czechoslovak Journal of Physics, 56, 2006, 3, 237-242[4] M Holzer, Anomalous spreading in a system of coupled Fisher-KPP equations, Physica D, 270, 2014, 1–10.[5] ML Wang, Exact solutions for a compound Kdv-Burgers equation, Physics Letters A, 213, 1996, 279-287.[6] EME Zayed, MAM Abdelaziz, The two-variable (G’/G, 1/G)-expansion method for solving the nonlinear Kdv-mKdv equation, Mathematical Problems in Engineering, 2012.[7] B Lu, Bäcklund transformation of fractional Riccati equation and infinite sequence solutions of nonlinear fractional PDEs, Abstract and Applied Analysis, 2014.[8] L Li, E Li, M Wang, The (G′/G, 1/G)-expansion method and its application to travelling wave solutions of the Zakharov equations, Applied Mathematics A Journal of Chinese Universities, 2010, 25, 4, 454–462.[9] S Demiray, Ö Ünsal, A Bekir, Exact solutions of nonlinear wave equations using (G’/G, 1/G)-expansion method, Journal of the Egyptian Mathematical Society, 2015, 23, 78–84.[10] M Wang, X Li, J Zhang, The (G’/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Physics Letters A, 372, 2008, 417-423.[11] EME Zayed, KAE Alurrfi, The homogeneous balance method and its applications for finding the exact solutions for nonlinear evolution equations, Italian journal of pure and applied mathematics, 2014, 33.[12] MS Islam, K Khan, MA Akbar, Application of the improved F-expansion method with Riccati equation to find the exact solution of the nonlinear evolution equations, Journal of the Egyptian Mathematical Society, 2016, 000, 1–6.[13] AM Wazwaz, The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations, Applied Mathematics and Computation, 184, 2007, 1002–1014.[14] JH He, XH Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals, 30, 2006, 700–708.[15] M Eslami, BF Vajargah, M Mirzazadeh, Application of the first integral method to fractional partial differential equations, Indian J Phys, 2014, 88(2), 177–184.[16] M Younis, S Ali, Solitary wave and shock wave solutions of (1+ 1)-dimensional perturbed Klein-Gordon,(1+ 1)-dimensional Kaup-Keperschmidt and (2+1) dimensional ZK-BBM, Open Eng. 2015, 5, 124–130.[17] S Bilige, T Chaolu, An extended simplest equation method and its application to several forms of the fifth-order KdV equation, Applied Mathematics and Computation, 216, 2010, 3146–3153.[18] H Triki, A Yikdirim, T Hayat, OM Aldossary, A Biswas, Shock wave solution of Benney-Luke equation, Rom. Journ. Phys. 57, 2012, 7–8, 1029–1034.[19] J Akter, MA Akbar, Exact solutions to the Benney–Luke equation and the Phi-4 equations by using modified simple equation method, Results in Physics, 5, 2015, 125–130.[20] S Labbe, L Paumond, Numerical comparisons of two long-wave limit models, Mathematical Modeling and Numerical Analysis, 38, 2004, 419-436.[21] TK Yuldashev, Inverse problem for a nonlinear Benney-Luke type integro- differential equations with degenerate kernel, Russian Mathematics, 60, 2016, 9, 53-60.[22] SM Rayhanul Islam, Applications of the exp(-Φ(ξ))-expansion method to find exact traveling wave solutions of the Benney-Luke equation in mathematical physics. American Journal of Applied Mathematics, 3, 2015, 3, 100-105.[23] ÖF Gözükızıl, Ş Akçağıl, Travelling wave solutions to the Benney-Luke and the higher-order improved Boussinesq equations of Sobolev type, Abstract and Applied Analysis, 2012.
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