Complex Solutions for the Fisher Equation and the Benjamin-Bona-Mahony Equation

Abstract

In this article, we give direct algebraic method for the complex solutions of the Fisher equation and Benjamin-Bona-Mahony equation. We get some complex solutions of the Fisher equation and Benjamin-Bona-Mahony equation by this method. 

Keywords

• Fisher equation,Benjamin-Bona-Mahony equation,direct algebraic method,complex solutions,traveling wave solutions

References

1. [1] L. Debtnath, Nonlinear Partial Differential Equations for Scientist and Engineers, Birkhauser, Boston, MA 1997.
2. [2] A. M. Wazwaz, Partial Differential Equations: Methods and Applications, Balkema, Rotterdam 2002.
3. [3] 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, J. Phys. A 19 (1986), 607–628.
4. [4] A. H. Khater, M. A. Helal and O. H. El-Kalaawy, B¨acklund transformations: exact solutions for the KdV and the Calogero-Degasperis-Fokas mKdV equations, Math. Meth. Appl. Sci. 21 (1998), 719–731.
5. [5] A. M. Wazwaz, A study of nonlinear dispersive equations with solitary-wave solutions having compact support, Math. Comput. Simulation 56 (2001), 269–276.
6. [6] S. A. Elwakil, S. K. El-Labany, M. A. Zahran and R. Sabry, Modified extended tanh-function method for solving nonlinear partial differential equations, Phys. Lett. A 299 (2002), 179–188.
7. [7] Y. Lei, Z. Fajiang and W. Yinghai, The homogeneous balance method, Lax pair, Hirota transformation and a general fifth-order KdV equation, Chaos, Solitons Fractals 13 (2002), 337–340.
8. [8] J. F. Zhang, New exact solitary wave solutions of the KS equation, Int. J. Theor. Phys. 38 (1999), 1829–1834.

9. [9] M. L. Wang, Exact solutions for a compound KdV-Burgers equation, Phys. Lett. A 213 (1996), 279–287.
10. [10] M. L. Wang, Y. B. Zhou and Z. B. Li, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. Lett. A 216 (1996), 67–75.
11. [11] M. L. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys. 60 (1992), 650–657.
12. [12] E. J. Parkes and B. R. Duffy, An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations, Comput. Phys. Commun. 98 (1996), 288–300.
13. [13] B. R. Duffy and E. J. Parkes, Travelling solitary wave solutions to a seventh-order generalized KdV equation, Phys. Lett. A 214 (1996), 271–272.
14. [14] E. J. Parkes and B. R. Duffy, Travelling solitary wave solutions to a compound KdV-Burgers equation, Phys. Lett. A 229 (1997), 217–220.
15. [15] N. Bildik and H. Bayramo˘glu, The solution of two dimensional nonlinear differential equation by the Adomian decomposition method, Appl. Math. and Comput. 163 (2005), 519–524.
16. [16] T. Ozi¸s and A. Yıldırım, Traveling wave solution of Korteweg-de Vries equation using He’s homotopy perturbation method, Int. J. Nonlinear Sci. and Numer. Simul. 8 (2007), 239–242.
17. [17] E. G. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277 (2000), 212–218.
18. [18] H. Chen and H. Zhang, New multiple soliton solutions to the general Burgers-Fisher equation and the Kuramoto-Sivashinsky equation, Chaos, Solitons Fractals 19 (2004), 71–76.
19. [19] H. Chen and H. Zhang, New multiple soliton-like solutions to the generalized (2 + 1)- dimensional KP equation, Appl. Math. Comput. 157 (2004), 765–773.
20. [20] Z. Y. Yan and H. Q. Zhang, New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water, Phys. Lett. A 285 (2001), 355–362.
21. [21] E. G. Fan, Auto-B¨acklund transformation and similarity reductions for general variable coefficient KdV equations, Phys. Lett. A 294 (2002), 26–30.
22. [22] M. L. Wang and Y. M. Wang, A new B¨acklund transformation and multi-soliton solutions to the KdV equation with general variable coefficients, Phys. Lett. A 287 (2001), 211–216.
23. [23] E. G. Fan and H. Q. Zhang, New exact solutions to a system of coupled KdV equations, Phys. Lett. A 245 (1998), 389–392.
24. [24] H. Zhang, A direct algebraic method applied to obtain complex solutions of some nonlinear partial differential equations, Chaos, Solitons Fractals 39 (2009), 1020–1026.
25. [25] S. T. Mohyud-Din, A. Yıldırım and G. Demirli, Traveling wave solutions of Whitham-BroerKraup equations by homotopy perturbation method, Journal of King Saud University-Science. 22 (2010), 173–176.
26. [26] B. Raliari and A. Yıldırım, The Application of homotopy perturbation method for MHD Flows of UCM Fluids above Porous Stretching Sheets, Comput. Math. Appl. 59 (2010), 3328–3337.
27. [27] A. Yıldırım, S. T. Mohyud-Din and D. H. Zhang, Analytical solutions to the pulsed KleinGordon equation using Modified Variational Iteration (MVIM) and Boubaker Polynomials Expansion Scheme (BPES), Comput. Math. Appl. 59 (2010), 2473–2477.