Volume 6, Issue 5, October 2018, Page: 149-158
Advance Exp (-Φ(ξ)) Expansion Method and Its Application to Find the Exact Solutions for Some Important Coupled Nonlinear Physical Models
M. Mashiur Rahhman, Department of Mathematics, Begum Rokeya University, Rangpur, Bangladesh
Ayrin Aktar, Department of Mathematics, Begum Rokeya University, Rangpur, Bangladesh
Kamalesh Chandra Roy, Department of Mathematics, Begum Rokeya University, Rangpur, Bangladesh
Received: Oct. 4, 2018;       Accepted: Nov. 7, 2018;       Published: Dec. 18, 2018
DOI: 10.11648/j.ajam.20180605.11      View  40      Downloads  32
The theoretical investigations of resonance physical phenomena by nonlinear coupled evolution equations are become important in currently. Hence, the purpose of this paper is to represent an advance exp (-Φ(ξ))-expansion method with nonlinear ordinary differential equation for finding exact solutions of some nonlinear coupled physical models. The present method is capable of evaluating all branches of solutions simultaneously and this difficult to distinguish with numerical technique. To verify its computational efficiency, the coupled classical Boussineq equation and (2+1)-dimensional Boussinesq and Kadomtsev-Petviashili equation are considered. The obtained solutions in this paper reveal that the method is a very effective and easily applicable of formulating the exact traveling wave solutions of the nonlinear coupled evolution equations arising in mathematical physics and engineering.
Coupled Classical Boussinesq Equation, Boussinesq-Kadomtsev-Petviashili Equation, Solitary Wave Solution, Periodic Wave Solution
To cite this article
M. Mashiur Rahhman, Ayrin Aktar, Kamalesh Chandra Roy, Advance Exp (-Φ(ξ)) Expansion Method and Its Application to Find the Exact Solutions for Some Important Coupled Nonlinear Physical Models, American Journal of Applied Mathematics. Vol. 6, No. 5, 2018, pp. 149-158. doi: 10.11648/j.ajam.20180605.11
Copyright © 2018 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Wang ML, Li XZ, Zhang J. The (G'/G)-expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 2008; 372; 417-423.
Kim H, Sakthivel R. New exact traveling wave solutions of some nonlinear higher dimensional physical models. Reports Math. Phys. 2012; 70 (1); 39-50.
Alam MN, Akbar MA, Mohyud-Din ST. A novel (G'/G)-expansion method and its application to the Boussinesq equation. Chin. Phys. B 2014; 23 (2); 020202-020210.
Hafez MG, Alam MN, Akbar MA, Exact traveling wave solutions to the Kelein_Gordon equation using the novel (G'/G)-expansion method, Results in Phys. 2014; 4; 177-184.
Naher H, Abdullah FA. New approach of (G'/G)-expansion method for nonlinear evolution equation. AIP Advan. 2013; 3; 032116. DOI: 10.1063/1.479447.
Zayed EME. A note on the modified simple equation method applied to Sharma-Tasso-Olver equation. Appl. Math. Comput. 2011; 218 (7); 3962–3964.
Zayed EME and Hoda Ibrahim SA. Modified simple equation method and its applications for some nonlinear evolution equations in mathematical physics. Int. J. Comput. Appli. 2013; 67 (6); 39-44.
Wazwaz AM. The extended tanh-method for new compact and non-compact solutions for the KP-BBM and the ZK-BBM equations. Chaos, Solitons Fract. 2008; 38: 1505-1516.
Abdou MA. The extended tanh-method and its applications for solving nonlinear physical models. Appl. Math. Comput. 2007; 190: 988-996.
Malfliet W. The tanh method: A tool for solving certain classes of nonlinear evolution and wave equations. J. Comput. Appl. Math. 2004; 164: 529-541.
Chun C, Sakthivel R. Homotopy perturbation technique for solving two point boundary value problems-comparison with other methods. Computer Phys. Commun. 2010; 181: 1021-1024.
Zhao X, Tang D. A new note on a homogeneous balance method. Phys. Lett. A 2002; 297 (1–2); 59–67.
Zhao X, Wang L, Sun W. The repeated homogeneous balance method and its applications to nonlinear partial differential equations. Chaos, Solitons Fract. 2006; 28 (2); 448–453.
Zhaosheng F. Comment on the extended applications of homogeneous balance method. Appl. Math. Comput. 2004; 158 (2); 593–596.
Hirota R. Exact solution of the KdV equation for multiple collisions of solutions. Phys. Rev. Lett. 1971; 27; 1192-1194.
Lee J, Sakthivel R. Exact travelling wave solutions for some important nonlinear physical models. Pramana - J. Phys. 2013; 80: 757-769.
Kudryashov NA. Exact solutions of the generalized Kuramoto-Sivashinsky equation. Phys. Lett. A 1990; 147: 287-291.
Noor MA, Mohyud-Din ST, Waheed A. Exp-function method for travelling wave solutions of nonlinear evolution equations. Appl. Math. Comput. 2010; 216: 477-483.
Noor MA, Mohyud-Din ST, Waheed A. Exp-function method for solving Kuramoto-Sivashinsky and Boussinesq equations. J. Appl. Math. Computing 2008; 29: 1-13.
Wang D, Zhang HQ. Further improved F-expansion method and new exact solutions of Konopelchenko-Dubrovsky equation. Chaos, Solitons Fract. 2005; 25: 601-610.
Akbar MA, Ali NHM. Solitary wave solutions of the fourth order Boussinesq equation through the exp(-Φ(η))-expansion method. SpringerPlus, 2014; 3-344. doi:10.1186/2193-1801-3-344.
Hafez MG, Alam MN, Akbar MA. Traveling wave solutions for some important coupled nonlinear physical models via the coupled Higgs equation and the Maccari system. J. King Saud Uni.-Sci. in press, 2014. http://dx.doi.org/10.1016/j.jksus.2014.09.001.
Hafez MG, Kauser MA, Akter MT. Some new exact traveliing wave solutions of the cubic nonlinear Schrodinger equation using the exp(-Φ(ξ))-expansion method. Int. J. Sci. Eng. Tech. 2014; 3 (7); 848-851.
Roshid OR, Rahman MA. The exp(-Φ(η))-expansion method with application in the (1+1)-dimensional classical Boussinesq equations. Results Phy. 2014; 4; 150-155.
Hafez MG, Kauser MA, Akter MT. Some new exact travelling wave solutions for the Zhiber-Shabat equation. British J. Math. Com. Sci. 2014; 4 (18); 2582-2593.
Wu TY, Zhang JE. In: Cook P, Roytburd V, Tulin M, editors. Mathematics is for solving problems. Philadelphia, PA, USA: SIAM; 1996. p. 233.
Gepreel KA. J. Partial Differ. Equ. 2011; 24: 55.
Zayed EME, Joudi SA. Math Probl Eng 2010. http://dx.doi.org/10.1155/2010/ 768573. Article ID 768573, 19pp.
Date E, Jimbo M, Kashiwara M, Miwa T. Transformation groups for soliton equations. IV. A new hierarchy of soliton equations of KP type. Physica D 1982; 4: 343–365.
Jimbo M, Miwa T. Solitons and infinite-dimensional Lie algebras. Publ. RIMS. Kyoto. Univer. 1983; 19: 943–1001.
Ma HC, Wang Y, Qin ZY. New exact complex traveling wave solutions for (2+1)-dimensional BKP equation. Appl. Math. Comput. 2009; 208: 564–568.
Zheng B. Travelling wave solutions of two nonlinear evolutions equations by using the G’/G–expansion method. Appl. Math. Comput. 2011; 217: 5743–5753.
Jabbari A, Kheiri H, Bekir A. Comput. Math. Appl. 2011; 62: 2177.
Bahrami BS., Abdollahzadeh H, Berijani IM, Ganji DD, Abdollahzadeh M. Pramana – J. Phys. 2011; 77: 263.
Kudryashov NA. A note on the (G'/G)-expansion method. Appl. Math. Comput. 2010; 217 (4):1755-1758.
Xiang C. Jacobi Elliptic function solutions for (2+1) dimensional Boussinesq and Kadomtsev-Petviashili equation. Appl. Math. 2011; 2; 1313-1316.
Browse journals by subject