Scientific Research

An Academic Publisher

**The Basic ( G'/G)-Expansion Method for the Fourth Order Boussinesq Equation** ()

The (

*G'/G*)-expansion method is simple and powerful mathematical tool for constructing traveling wave solutions of nonlinear evolution equations which arise in engineering sciences, mathematical physics and real time application fields. In this article, we have obtained exact traveling wave solutions of the nonlinear partial differential equation, namely, the fourth order Boussinesq equation involving parameters via the (*G'/G*)-expansion method. In this method, the general solution of the second order linear ordinary differential equation with constant coefficients is implemented. Further, the solitons and periodic solutions are described through three different families. In addition, some of obtained solutions are described in the figures with the aid of commercial software Maple.Share and Cite:

H. Naher and F. Abdullah, "The Basic (

*G'/G*)-Expansion Method for the Fourth Order Boussinesq Equation,"*Applied Mathematics*, Vol. 3 No. 10, 2012, pp. 1144-1152. doi: 10.4236/am.2012.310168.Conflicts of Interest

The authors declare no conflicts of interest.

[1] | R. Hirota, “Exact Solution of the KdV Equation for Multiple Collisions of Solutions,” Physics Review Letters, Vol. 27, 1971, pp. 1192-1194. doi:10.1103/PhysRevLett.27.1192 |

[2] | J. Yan, “Soliton Resonances of the Nonisospectral Modified Kadomtsev-Petviashvili Equation,” Applied Mathematics, Vol. 2, No. 6, 2011, pp. 685-693. doi:10.4236/am.2011.26090 |

[3] | M. R. Mimura, “Backlund Transformation,” Springer, Berlin, 1978. |

[4] | C. Rogers and W. F. Shadwick, “Backlund Transformations,” Academic Press, New York, 1982. |

[5] | J. Weiss, M. Tabor and G. Carnevale, “The Painleve Property for Partial Differential Equations,” Journal of Mathematical Physics, Vol. 24, No. 3, 1982, pp. 522-526. doi:10.1063/1.525721 |

[6] | S. L. Zhang, B. Wu and S. Y. Lou, “Painleve Analysis and Special Solutions of Generalized Broer-Kaup Equations,” Physics Letters A, Vol. 300, No. 1, 2002, pp. 4048. doi:10.1016/S0375-9601(02)00688-6 |

[7] | M. J. Ablowitz and P. A. Clarkson, “Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform,” Cambridge University Press, Cambridge, 1991. doi:10.1017/CBO9780511623998 |

[8] | N. A. Kudryashov, “Exact Solutions of the Generalized Kuramoto-Sivashinsky Equation,” Physics Letters A, Vol. 147, No. 5-6, 1990, pp. 287-291. doi:10.1016/0375-9601(90)90449-X |

[9] | 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,” Physics Letters A, Vol. 216, 1996, pp. 67-75. doi:10.1016/0375-9601(96)00283-6 |

[10] | S. Liu, Z. Fu, S. Liu and Q. Zhao, “Jacobi Elliptic Function Expansion Method and Periodic Wave Solutions of Nonlinear Wave Equations,” Physics Letters A, Vol. 289, 2001, pp. 69-74. doi:10.1016/S0375-9601(01)00580-1 |

[11] | Z. Yan, “Abundant Families of Jacobi Elliptic Function Solutions of the (2+1)-Dimensional Integrable DaveyStewartson-Type Equation via a New Method,” Chaos, Solitons and Fractals, Vol. 18, No. 2, 2003, pp. 299-309. doi:10.1016/S0960-0779(02)00653-7 |

[12] | Y. Chen and Q. Wang, “Extended Jacobi Elliptic Function Rational Expansion Method and Abundant Families of Jacobi Elliptic Function Solutions to (1+1)-Dimensional Dispersive Long Wave Equation,” Chaos, Solitons and Fractals, Vol. 24, No. 3, 2005, pp. 745-757. doi:10.1016/j.chaos.2004.09.014 |

[13] | C. Xiang, “Jacobi Elliptic Function Solutions for (2+1) Dimensional Boussinesq and Kadomtsev-Petviashvili Equation,” Applied Mathematics, Vol. 2, No. 11, 2011, pp. 1313-1316. doi:10.4236/am.2011.211183 |

[14] | Z. Yan and H. Q. Zhang, “New Explicit Solitary Wave Solutions and Periodic Wave Solutions for WhithamBroer-Kaup Equation in Shallow Water,” Physcis Letters A, Vol. 285, No. 5-6, 2001, pp. 355-362. doi:10.1016/S0375-9601(01)00376-0 |

[15] | 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 |

[16] | A. M. Wazwaz, “The tanh-coth Method for Solitons and Kink Solutions for Nonlinear Parabolic Equations,” Applied Mathematics and Computation, Vol. 188, 2007, pp. 1467-1475. doi:10.1016/j.amc.2006.11.013 |

[17] | A. Bekir and A. C. Cevikel, “Solitary Wave Solutions of Two Nonlinear Physical Models by tanh-coth Method,” Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 5, 2009, pp. 1804-1809. doi:10.1016/j.cnsns.2008.07.004 |

[18] | F. Fan, “Extended tanh-Function Method And Its Applications To nonlinear equations,” Physics Letters A, Vol. 277, No. 4-5, 2000, pp. 212-218. doi:10.1016/S0375-9601(00)00725-8 |

[19] | N. H. Abdel-All, M. A. Abdel Razek and A. K. Seddeek, “Expanding the tanh-Function Method for Solving Nonlinear Equations,” Applied Mathematics, Vol. 2, No. 9, 2011, pp. 1096-1104. doi:10.4236/am.2011.29151 |

[20] | M. L. Wang and X. Z. Li, “Applications of F-Expansion to Periodic Wave Solutions for a New Hamiltonian Amplitude Equation,” Chaos, Solitons and Fractals, Vol. 24, No. 5, 2005, pp. 1257-1268. doi:10.1016/j.chaos.2004.09.044 |

[21] | M. A. Abou, “The Extended F-Expansion Method and Its Applications for a Class of Nonlinear Evolution Equation,” Chaos, Solitons and Fractals, Vol. 31, 2007, pp. 95-104. doi:10.1016/j.chaos.2005.09.030 |

[22] | A. A. Soliman and H. A. Abdo, “New Exact Solutions of Nonlinear Variants of the RLW, the PHI-Four and Boussinesq Equations Based on Modified Extended Direct Algebraic Method,” International Journal of Nonlinear Science, Vol. 7, No. 3, 2009, pp. 274-282. |

[23] | A. H. Salas, and C. A. Gomez, “Application of the ColeHopf Transformation for Finding Exact Solutions to Several Forms of the Seventh-Order KdV Equation,” Mathematical Problems in Engineering, 2010, Article ID 194329, 14 p. |

[24] | J. H. He and X. H. Wu, “Exp-Function Method for Nonlinear Wave Equations,” Chaos Solitons and Fractals, Vol. 30, No. 3, 2006, pp. 700-708. doi:10.1016/j.chaos.2006.03.020 |

[25] | H. Naher, F. A. Abdullah and M. A. Akbar, “New Traveling Wave Solutions of the Higher Dimensional Nonlinear Partial Differential Equation by the Exp-Function Method,” Journal of Applied Mathematics, 2012, Article ID: 575387, 14 p. |

[26] | S. T. Mohyud-din, M. A. Noor and K. I. Noor, “ExpFunction Method for Traveling Wave Solutions of Modified Zakharov-Kuznetsov Equation,” Journal of King Saud University, Vol. 22, No. 4, 2010, pp. 213-216. doi:10.1016/j.jksus.2010.04.015 |

[27] | H. Naher, F. Abdullah and M. A. Akbar, “The Exp-Function Method for New Exact Solutions of the Nonlinear Partial Differential Equations,” International Journal of the Physical Sciences, Vol. 6, No. 29, 2011, pp. 6706-6716. |

[28] | A. Yildirim and Z. Pinar, “Application of the Exp-Function Method for Solving Nonlinear Reaction-Diffusion Equations Arising in Mathematical Biology,” Computers & Mathematics with Applications, Vol. 60, No. 7, 2010, pp. 1873-1880. doi:10.1016/j.camwa.2010.07.020 |

[29] | I. Aslan, “Application of the Exp-Function Method to Nonlinear Lattice Differential Equations for Multi-Wave and Rational Solutions,” Mathematical Methods in the Applied Sciences, Vol. 60, No. 7, 2011, pp. 1707-1710. doi:10.1002/mma.1476 |

[30] | A. Bekir and A. Boz, “Exact Solutions for Nonlinear Evolution Equations Using Exp-Function Method,” Physics Letters A, Vol. 372, No. 4, 2008, pp. 1619-1625. doi:10.1016/j.physleta.2007.10.018 |

[31] | Q. Liu and R. Xu, “Periodic Solutions of a CohenGrossberg-Type BAM Neural Networks with Distributed Delays and Impulses,” Journal of Applied Mathematics, 2012, Article ID: 643418, 17 p. doi:10.1155/2012/643418 |

[32] | D. Feng and K. Li, “On Exact Traveling Wave Solutions for (1+1) Dimensional Kaup-Kupershmidt Equation,” Applied Mathematics, Vol. 2, No. 6, 2011, pp. 752-756. doi:10.4236/am.2011.26100 |

[33] | K. A. Gepreel, S. Omran and S. K. Elagan, “The Travelling Wave Solutions for Some Nonlinear PDEs in Mathematical Physics,” Applied Mathematics, Vol. 2, No. 3, 2011, pp. 343-347. doi:10.4236/am.2011.23040 |

[34] | M. Noor, K. Noor, A. Waheed and E. A. Al-Said, “An Efficient Method for Solving System of Third-Order Nonlinear Boundary Value Problems,” Mathematical Problems in Engineering, 2011, Article ID: 250184, 14 p. doi:10.1155/2011/250184 |

[35] | J. Wang, L. Wang and K. Yang, “Some New Exact Travelling Wave Solutions for the Generalized Benny-Luke (GBL) Equation with Any Order,” Applied Mathematics, Vol. 3, 2012, pp. 309-314. doi:10.4236/am.2012.34046 |

[36] | J. F. Alzaidy, “Exact Travelling Wave Solutions of Nonlinear PDEs in Mathematical Physics,” Applied Mathematics, Vol. 3, No. 7, 2012, pp. 738-745. doi:10.4236/am.2012.37109 |

[37] | M. Wang, X. Li and J. Zhang, “The G'/G-Expansion Method and Travelling Wave Solutions of Nonlinear Evolution Equations in Mathematical Physics,” Physics Letters A, Vol. 372, No. 4, 2008, pp. 417-423. doi:10.1016/j.physleta.2007.07.051 |

[38] | J. Feng, W. Li and Q. Wan, “Using G'/G-Expansion Method to Seek Traveling Wave Solution of Kolmo-gorov-Petrovskii-Piskunov Equation,” Applied Mathematics and Computation, Vol. 217, 2011, pp. 5860-5865. doi:10.1016/j.amc.2010.12.071 |

[39] | H. Naher, F. A. Abdullah and M. A. Akbar, “The G'/G-Expansion Method for Abundant Traveling Wave Solutions of Caudrey-Dodd-Gibbon Equation,” Mathematical Problems in Engineering, 2011, Article ID: 218216, 11 p. |

[40] | R. Abazari and R. Abazari, “Hyperbolic, Trigonometric and Rational Function Solutions of Hirota-Ramani Equation via G'/G-Expansion Method,” Mathematical Problems in Engineering, 2011, Article ID: 424801, 11 p. |

[41] | E. M. E. Zayed, “Traveling Wave Solutions for Higher Dimensional Nonlinear Evolution Equations Using the G'/G-Expansion Method,” Journal of Applied Mathematics & Informatics, Vol. 28, No. 1-2, 2010, pp. 383-395. |

[42] | A. Jabbari, H. Kheiri and A. Bekir, “Exact Solutions of the Coupled Higgs Equation and the Miccari System Using He’s Semi-Inverse Method and G'/G-Expansion Method,” Computers and Mathematics with Applications, Vol. 62, 2011, pp. 2177-2186. doi:10.1016/j.camwa.2011.07.003 |

[43] | T. Ozis and I. Aslan, “Application of the G'/G-Expansion Method to Kawahara Type Equations Using Symbolic Computation,” Applied Mathematics and Computation, Vol. 216, 2010, pp. 2360-2365. doi:10.1016/j.amc.2010.03.081 |

[44] | K. A. Gepreel, “Exact Solutions for Nonlinear PDEs with the Variable Coefficients in Mathematical Physics,” Journal of Information and Computing Science, Vol. 6, No. 1, 2011, pp. 003-014. |

[45] | S. K. Elagan, M. Sayed and Y. S. Hamed, “An Innovative Solutions for the Generalized FitzHugh-Nagumo Equation by Using the Generalized G'/G-Expansion Method,” Applied Mathematics, Vol. 2, 2011, pp. 470-474. doi:10.4236/am.2011.24060 |

[46] | A. Borhanifar and A. Z. Moghanlu, “Application of the G'/G-Expansion Method for the Zhiber-Sabat Equation and Other Related Equations,” Mathematical and Computer Modelling, Vol. 54, 2011, pp. 2109-2116. doi:10.1016/j.mcm.2011.05.020 |

[47] | A. Borhanifar and R. Abazari, “General Solution of Two Generalized Form of Burgers Equation by Using the -Expansion Method,” Applied Mathematics, Vol. 3, 2012, pp. 158-168. doi:10.4236/am.2012.32025 |

[48] | G. Wang, X. Liu and Y. Zhang, “New Explicit Solutions of the Generalized (2+1)-Dimensional Zakharov-Kuznetsov Equation,” Applied Mathematics, Vol. 3, No. 6, 2012, pp. 523-527. doi:10.4236/am.2012.36079 |

[49] | H. Zhang, “New Exact Travelling Wave Solutions for Some Nonlinear Evolution Equations,” Chaos, Solitons and Fractals, Vol. 26, No. 3, 2005, pp. 921-925. doi:10.1016/j.chaos.2005.01.035 |

[50] | A. M. Wazwaz, “New Travelling Wave Solutions to the Boussinesq and 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 |

[51] | Y. Guo and S. Lai, “New Exact Solutions for an (n+1)Dimensional Generalized Boussinesq Equation,” Nonlinear Analysis, Vol. 72, 2010, pp. 2863-2873. doi:10.1016/j.na.2009.11.030 |

[52] | Sirendaoreji, “A New Auxiliary Equation and Exact Travelling Wave Solutions of Nonlinear Equations,” Physics Letters A, Vol. 356, No. 2, 2006, pp. 124-130. doi:10.1016/j.physleta.2006.03.034 |

[53] | A. Parker and J. M. Dye, “Boussinesq-Type Equations and Switching Solitons,” Proceedings of Institute of Mathematics of NAS of Ukraine, Vol. 43, No. 1, 2002, pp. 344-351. |

[54] | Z. Yan, “Constructing Exact Solutions for Two-Dimensional Nonlinear Dispersion Boussinesq Equation. II: Solitary Patterns Solutions,” Chaos, Solitons & Fractals, Vol. 18, No. 4, 2003, pp. 869-880. doi:10.1016/S0960-0779(03)00059-6 |

Copyright © 2020 by authors and Scientific Research Publishing Inc.

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