A Generalized Tanh-Function Type Method and the(G'/G) -Expansion Method for Solving

Weimin Zhang

School of Mathematics, Jiaying University, Meizhou, China.

DOI: 10.4236/am.2013.410A1003                

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.

Keywords

The Generalized Tanh-Function Method; (G'/G) -Expansion Method; mKdV Equation; The Transformed Rational Function

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	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
[2]	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
[3]	M. R. Miurs, “Backlund Transformation,” Springer, Berlin, 1978.
[4]	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
[5]	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
[6]	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
[7]	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
[8]	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
[9]	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
[10]	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
[11]	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
[12]	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
[13]	N. A. Kudryashov, “A Note on the (G'/G)-Expansion Method,” Applied Mathematics and Computationh, Vol. 217, No. 4, 2010, pp. 1755-1758. http://dx.doi.org/10.1016/j.amc.2010.03.071
[14]	J. M. Zuo, “Application of the Extended (G'/G)-Expansion Method to Solve the Pochhammer-Chree Equations,” Applied Mathematics and Computationh, Vol. 217, No. 1, 2010, pp. 376-383. http://dx.doi. org/10.1016/j.amc.2010.05.072
[15]	S. Kutluay, A. Esen and O. Tasbozan, “The (G'/G)-Expansion Method for Some Nonlinear Evolution Equations,” Applied Mathematics and Computation, Vol. 217, No. 1, 2010, pp. 384-391. http://dx.doi .org/10.1016/j.amc.2010.05.073
[16]	Y. M. Chen, S. H. Ma and H. Y. Ma, “New Exact Solutions of a (3 + 1)-Dimensional Jimbo-Miwa System”, Chinese Physics B, Vol. 22, No. 5, 2013, Article ID: 0505101-5. http://dx.doi.org/10.1088/1674-1056/22/5/050510
[17]	W. X. Ma and B. Fuchssteiner, “Explicit and Exact Solutions to Kolmogorov-Petrovsk II-Piskunov Equation,” Journal of Non-Linear Mechanics, Vol. 31, No. 3, 1996, pp. 329-338. http://dx.doi.org/. 10.1016/0020-7462(95)00064-X
[18]	S. A. El-Wakil and M. A. Abdou, “Modified Extended Tanh-Function Method for Solving Nonlinear Partial Differential Equations,” Chaos, Solitons & Fractals, Vol. 31, No. 5, 2007, pp. 1256-1264. http://dx.doi .org/10.1016/j.chaos.2005.10.072