Research Article

Year 2010,
Volume: 7 Issue: 2, - , 01.04.2010
### Abstract

### Keywords

### References

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.

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

- [1] L. Debtnath, Nonlinear Partial Differential Equations for Scientist and Engineers, Birkhauser, Boston, MA 1997.
- [2] A. M. Wazwaz, Partial Differential Equations: Methods and Applications, Balkema, Rotterdam 2002.
- [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] 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] A. M. Wazwaz, A study of nonlinear dispersive equations with solitary-wave solutions having compact support, Math. Comput. Simulation 56 (2001), 269–276.
- [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] 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] J. F. Zhang, New exact solitary wave solutions of the KS equation, Int. J. Theor. Phys. 38 (1999), 1829–1834.
- [9] M. L. Wang, Exact solutions for a compound KdV-Burgers equation, Phys. Lett. A 213 (1996), 279–287.
- [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] M. L. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys. 60 (1992), 650–657.
- [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] 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] 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] 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] 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] E. G. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277 (2000), 212–218.
- [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] 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] 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] E. G. Fan, Auto-B¨acklund transformation and similarity reductions for general variable coefficient KdV equations, Phys. Lett. A 294 (2002), 26–30.
- [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] 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] 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] 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] 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] 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.

Year 2010,
Volume: 7 Issue: 2, - , 01.04.2010
### Abstract

### References

- [1] L. Debtnath, Nonlinear Partial Differential Equations for Scientist and Engineers, Birkhauser, Boston, MA 1997.
- [2] A. M. Wazwaz, Partial Differential Equations: Methods and Applications, Balkema, Rotterdam 2002.
- [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] 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] A. M. Wazwaz, A study of nonlinear dispersive equations with solitary-wave solutions having compact support, Math. Comput. Simulation 56 (2001), 269–276.
- [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] 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] J. F. Zhang, New exact solitary wave solutions of the KS equation, Int. J. Theor. Phys. 38 (1999), 1829–1834.
- [9] M. L. Wang, Exact solutions for a compound KdV-Burgers equation, Phys. Lett. A 213 (1996), 279–287.
- [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] M. L. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys. 60 (1992), 650–657.
- [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] 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] 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] 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] 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] E. G. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277 (2000), 212–218.
- [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] 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] 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] E. G. Fan, Auto-B¨acklund transformation and similarity reductions for general variable coefficient KdV equations, Phys. Lett. A 294 (2002), 26–30.
- [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] 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] 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] 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] 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] 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.

There are 27 citations in total.

Subjects | Engineering |
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Journal Section | Articles |

Authors | |

Publication Date | April 1, 2010 |

Published in Issue | Year 2010 Volume: 7 Issue: 2 |