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A Generalized Tanh-Function Type Method and the(G'/G) -Expansion Method for Solving
School of Mathematics, Jiaying University, Meizhou, China
- 1 School of Mathematics, Jiaying University, Meizhou, China
Applied Mathematics·Volume 04 (2013)·Pages 11–16·Published 26 September 2013·DOI10.4236/am.2013.410A1003
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Abstract
In this paper a generalized tanh-function type method is proposed by using the idea of the transformed rational function method. We show that the ( G' / G ) -expansion method is a special case of the generalized tanh-function type method, so the ( G'/G ) -expansion method is considered as a special deformation application of the transformed rational function method. We demonstrate that all solutions obtained by the ( G'/G ) -expansion method were found by the generalized tanh-function type method. As applications, we consider mKdV equation. Compared with the ( G'/G ) -expansion method, the generalized tanh-function type method gives new and more abundant solutions.
KeywordsThe Generalized Tanh-Function Method(<i>G'/G</i>) -Expansion MethodmKdV EquationThe Transformed Rational Function
- M. J. Ablowitz and P. A. Clarkson, “Soliton, Nonlinear Evolution Equations and Inverse Scattering,” Cambridge University Press, New York, 1991. http://dx.doi.org/10.1017/CBO9780511623998
- P. G. Esteevez, “Darboux Transformation and Solutions for an Equation in 2 + 1 Dimensions,” Journal Mathematical Physics, Vol. 40, No. 3, 1999, pp. 1406-1419. http://dx.doi.org/10.1063/1.532811
- M. R. Miurs, “Backlund Transformation,” Springer, Berlin, 1978.
- H. W. Tam, W. X. Ma, X. B. Hu and D. L Wang, “The Hirota-Satsuma Coupled KdV Equation and a Coupled Ito System Revisite,” Journal of the Physical Society of Japan, Vol. 69, No. 1, 2000, pp. 45-51. http://dx.doi.org/10.1143/JPSJ.69.45
- P. J. Olver, “Applications of Lie Groups to Differential Equations,” Springer, New York, 1993. http://dx.doi.org/10.1007/978-1-4612-4350-2
- A. M. Wazwaz, “The Variational Iteration Method for Solving Linear and Nonlinear Systems of PDEs,” Computers and Mathematics with Applications, Vol. 54, No. 7-8, 2007, pp. 895-902. http://dx.doi.org/ 10.1016/j.camwa.2006.12.059
- S. Pamuk, “An Application for Linear and Nonlinear Heat Equations by Adomian’s Decomposition Method,” Applied Mathematics and Computation, Vol. 163, No. 1, 2005, pp. 89-96. http://dx.doi.org/ 10.1016/j.amc.2003.10.051
- W. X. Ma and J. H. Lee, “A Transformed Rational Function Method and Exact Solutions to the 3 + 1 Dimensional Jimbo-Miwa Equation,” Chaos, Solitons & Fractals, Vol. 42, No. 3, 2009, pp. 1356-1363. http://dx.doi.org/10.1016/j.chaos.2009.03.043
- M. L. Wang, X. Li and W. Wang, “The (G’/G)-Expansion Method and Traveling Wave Solutions of Nonlinear Evolutions in Mathematical Physics,” Physics Letter A, Vol. 372, No. 4, 2008, pp. 417-423. http://dx.doi.org/10.1016/j.physleta.2007.07.051
- S. Zhang, J. L. Tong and W. Wang, “A Generalized (G'/G)-Expansion Method for the mKdV Equation with Variable Coefficients,” Physics Letter A, Vol. 372, No. 13, 2008, pp. 2254-2257. http://dx.doi.org/ 10.1016/j.physleta.2007.11.026
- A. Bekir, “Application of the (G’/G)-Expansion Method for Nonlinear Evolution Equations,” Physics Letter A, Vol. 372, No. 19, 2008, pp. 3400-3406 http://dx.doi.org/10.1016/j.physleta.2008.01.057
- D. D. Ganji and M. Abdollahzadeh, “Exact Traveling Solutions of Nonlinear Evolution Equation by (G'/G)-Expansion Method,” Journal of Mathematical Physics, Vol. 50, No. 1, 2009, Article ID: 013519-1-12. http://dx.doi.org/10.1063/1.3052847