Expanding the Tanh-Function Method for Solving Nonlinear Equations

Abstract

In this paper, using the tanh-function method, we introduce a new approach to solitary wave solutions for solving nonlinear PDEs. The proposed method is based on adding integration constants to the resulting nonlinear ODEs from the nonlinear PDEs using the wave transformation. Also, we use a transformation related to those integration constants. Some examples are considered to find their exact solutions such as KdV- Burgers class and Fisher, Boussinesq and Klein-Gordon equations. Moreover, we discuss the geometric interpretations of the resulting exact solutions.

Share and Cite:

Abdel-Aal, N. , Abdel-Razek, M. and Seddeek, A. (2011) Expanding the Tanh-Function Method for Solving Nonlinear Equations. Applied Mathematics, 2, 1096-1104. doi: 10.4236/am.2011.29151.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] M. J. Ablowitz and H. Segur, “Solitons, Nonlinear Evolution Equations and Inverse Scattering,” Cambridge University Press, Cambridge, 1991.
[2] R. Hirota, “The Direct Method in Soliton Theory,” Cambridge University Press, Cambridge, 2004.
[3] R. Conte, “Painlevé Property,” Springer, Berlin, 1999.
[4] W. Hereman, P. P. Banerjee, A. Korpel, G. Assanto, A. Van Immerzeele and A. Meerpoel, “Exact Solitary Wave Solutions of NonLinear Evolution and Wave Equations Using a Direct Algebraic Method,” Journal of Physics A: Mathematical and General, Vol. 19, No. 5, 1986, pp. 607-628. doi:10.1088/0305-4470/19/5/016
[5] W. Malfliet, “Solitary Wave Solutions of Nonlinear Wave Equations,” American Journal of Physics, Vol. 60, No. 7, 1992, pp. 650-654. doi:10.1119/1.17120
[6] W. Malfliet and W. Hereman, “The Tanh Method: I Exact Solutions of Nonlinear Evolution and Wave Equations,” Physica Scripta, Vol. 54, No. 6, 1996, pp. 563-568. doi:10.1088/0031-8949/54/6/003
[7] S. A. El-Wakil, S. K. El-labany, M. A. Zahran and R. Sabry, “Modified Extended Tanh Function Method for Solving Nonlinear Partial Differential Equations,” Physics Letters A, Vol. 299, No. 2-3, 2002, pp. 179-188. doi:10.1016/S0375-9601(02)00669-2
[8] E. Fan, “Extended Tanh-Function Method and Its Applications to Nonlinear Equations,” Physics Letters A, Vol. 277, No. 4-5, 2000, pp. 212-218. doi:10.1080/08035250152509726
[9] Y.-T. Gao and B. Tian, “Generalized Tanh Method with Symbolic Computation and Generalized Shallow Water Wave Equation,” Computers & Mathematics with Applications, Vol. 33, No. 4, 1997, pp. 115-118. doi:10.1016/S0898-1221(97)00011-4
[10] C. Yan, “A Simple Transformation for Nonlinear Waves,” Physics Letters A, Vol. 224, No. 1-2, 1996, pp. 77-84. doi:10.1016/S0375-9601(96)00770-0
[11] Z.-Y., Yan and H.-Q. Zhang, “Auto-Darboux Transformation and Exact Solutions of the Brusselator Reaction Diffusion Model,” Applied Mathematics and Mechanics, Vol. 22, No. 5, 2000, pp. 541-546. doi:10.1023/A:1016359331072
[12] W. Hereman, A. Korpel and P. P. Banerjee, “A General Physical Approach to Solitary Wave Construction from Linear Solutions,” Wave Motion, Vol. 7, No. 3, 1985, pp. 283-289. doi:10.1016/0165-2125(85)90014-9
[13] A. A. Soliman, “The Modified Extended Tanh-Function Method for Solving Burgers-Type Equations,” Physica A: Statistical Mechanics and its Applications, Vol. 361, No. 2, 2006, pp. 394-404. doi:10.1016/j.physa.2005.07.008
[14] A.-M. Wazwaz, “The Extended Tanh Method for Abundant Solitary Wave Solutions of Nonlinear Wave Equations,” Applied Mathematics and Computation, Vol. 187, No. 2, 2007, pp. 1131-1142. doi:10.1016/j.amc.2006.09.013
[15] A.-M. Wazwaz, “New Travelling Wave Solutions to the Bous-sinesq and the Klein-Gordon Equations,” Communications in Nonlinear Science and Numerical Simulation, Vol. 13, No. 5, 2008, pp. 889-901. doi:10.1016/j.cnsns.2006.08.005

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.