Volume 3, Issue 1, January 2018, Page: 23-45
Some New Traveling Wave Solutions of Modified Camassa Holm Equation by the Improved G'/G Expansion Method
Rida Tassew Redi, Department of Mathematics, Institute of Technology, Dire Dawa University, Dire Dawa, Ethiopia
Akalu Abriham Anulo, Department of Mathematics, Institute of Technology, Dire Dawa University, Dire Dawa, Ethiopia
Received: Jan. 17, 2018;       Accepted: Feb. 16, 2018;       Published: Apr. 9, 2018
DOI: 10.11648/j.mcs.20180301.14      View  1973      Downloads  118
Abstract
In this article, the improved G'/G-expansion method has been implemented to generate travelling wave solutions, where G(ŋ) satisfies the second order nonlinear ordinary differential equation. To show the advantages of the method, the Simplified Modified Camassa Holm (SMCH) equation has been investigated. Nonlinear partial differential equations have many potential applications in mathematical physics and engineering sciences. Some of our solutions are in good agreement with already published results for a special case and others are new. The solutions in this work may express a variety of new features of waves. Furthermore, these solutions can be valuable in the theoretical and numerical studies of the considered equation.
Keywords
Improved G'/G-Expansion Method, The SMCH Equation, Traveling Wave Solutions, Nonlinear Evolution Equations
To cite this article
Rida Tassew Redi, Akalu Abriham Anulo, Some New Traveling Wave Solutions of Modified Camassa Holm Equation by the Improved G'/G Expansion Method, Mathematics and Computer Science. Vol. 3, No. 1, 2018, pp. 23-45. doi: 10.11648/j.mcs.20180301.14
Copyright
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.
Reference
[1]
C. Rogers and W. F. Shadwick, Backlund Transformations, Academic Press, New York (1982).
[2]
He, J. H., Wu, X. H.: Construction of Solitary Solution and Compaction-Like Solution by Variational Iteration Method. Chaos Solitons and Fractals 29, 108-113 (2006).
[3]
Hirota, R.: Exact Solution of the Kdv Equation for Multiple Collisions of Solitons. Physical Review Letters 27, 1192-1194 (1971).
[4]
Borhanifar, A., Jafari, H., Karimi, S. A.: New solitary wave solutions for the bad Boussinesq and good Boussinesq equa-tions, Numer. Methods for Partial Differential Equations 25, 1231–1237 (2009).
[5]
Borhanifar, A., Jafari, H., Karimi, S. A.: New solitons and periodic solutions for the Kadomtsev–Petviashvili equation. Nonlinear Sci. Appl. 4, 224–229. (2008).
[6]
Liu, G. T., Fan, T. Y.: New applications of developed Jacobi elliptic function expansion methods. Phys. Lett. A 345, 161–166 (2005).
[7]
Wazwaz, A. M.: Exact solutions to the double sinh-Gordan equation by tanh method and a variable separated ode method. Compt. Math. Appl. 501685-1696 (2005).
[8]
Malfliet, W., Hereman, W.: The tanh method: exact solutions of nonlinear evolution and wave equations. Phys. Scr. 54, 563–568 (1996).
[9]
Wang, M. L.: Exact solution for a compound KdV-Burgers equations. Phys. Lett. A 213, 279– 287 (1996).
[10]
Malfliet, W.: Solitary wave solutions of nonlinear wave equations. Am. J. Phy. 60, 650–658 (1992).
[11]
Malfliet, W., Hereman, W.: The tanh method: II. Perturbation technique for conservative systems. Phys Scr. 54, 569–575 (1996).
[12]
Zhou, X. W., Wen, Y. X., He J. H.: Exp-function method to solve the nonlinear dispersive k (m, n) equations, Int. J. Non-linear Sci. Numer. Simul. 9, 301–306 (2008).
[13]
Liu, G. T, Fan, T. Y.: New applications of developed Jacobi elliptic function expansion methods. Phys. Lett. A 345, 161–166 (2005).
[14]
Ablowitz, M. J., Segur, H.: Solitons and Inverse Scattering Transform. SIAM, Philadelphia (1981).
[15]
Wang, M. L., Li, X. Zhang, J.: The (G’/G)-expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 372 417–423 (2008).
[16]
Ozis, T. and Aslan, I., “Application of the G'/G -expansion method to Kawahara type equations using symbolic computation,” Appl. Math. Computation, 216, 2360-2365, 2010.
[17]
Naher, H., Abdullah, F. A. and Akbar, M. A., “The G'/G -expansion method for abundant traveling wave solutions of Caudrey-Dodd-Gibbon equation,” Math. Prob. Eng., Article ID: 218216, 11 pages, 2011.
[18]
Jabbari, A., Kheiri, H. and Bekir, A., “Exact solutions of the coupled Higgs equation and the Miccari system using He’s semi-inverse method and G'/G -expansion method,” Computers Math. Appli. 62, 2177-2186, 2011.
[19]
Naher, H. and Abdullah, F. A., “The basic G'/G -expansion method for the fourth order Boussinesq equation,” Appl. Math., 3, 1144-1152, 2012.
[20]
Zhang, J. Jiang, F. and Zhao, X., “An improved (G’/G)-expansion method for solving nonlinear evolution equations,” Int. J. Computer Math., 87, 1716-1725, 2010.
[21]
Zhao, Y. M., Yang, Y. J. and Li, W., “Application of the improved (G’/G)-expansion method for the Variant Boussinesq equations,” Appl. Math. Sci., 5, 2855-2861, 2011.
[22]
Nofel, T. A, Sayed, M., Hamad, Y. S. and Elagan, S. K., “The improved (G’/G)-expansion method for solving the fifth-order KdV equation,” Annals of Fuzzy Math. Informatics, 3, 9- 17, 2011.
[23]
Naher, H., Abdullah, F. A. and Akbar, M. A., “New traveling wave solutions of the higher dimensional nonlinear evolution equation by the improved (G’/G)-expansion method,” World Appl. Sci. J., 16, 11-21, 2012.
[24]
Md. Nur Alam and M. Ali Akbar Traveling Wave Solutions of Nonlinear Evolution Equations Via the New Generalized (G'/G)-Expansion Method 1 (4) (2013):129-136, DOI: 10.13189/ujcmj.2013.010403.
[25]
Abdollah Borhanifar and Reza Abazari, General Solution of Two Generalized Form of Burgers Equation by Using the (G’/G)-Expansion Method Applied Mathematics, 3 (2012), 158- 168 http://dx.doi.org/10.4236/am.2012.32025.
[26]
Abaker A. Hassaballa et al, Applications of the Improved G_/G Expansion Method for Solve Burgers- Fisher Equation. Journal of Computational and Theoretical Nano science Vol. 14, 4664–4668, 2017.
[27]
A. M. Wazwaz, Appl. Math. Comput. 163, 1165-1179 (2005). X. Liu, L. Tian and, Y. Wu, Appl. Math. Comput. 217, 1376-1384 (2010).
[28]
Hasibun Naher et al. Some New Solutions of The (1+1)-Dimensional PDE via The Improved (G '/G)-Expansion Method.
Browse journals by subject