Trial Equation Method for Solving the Improved Boussinesq Equation

ABSTRACT

Trial equation method is a powerful tool for obtaining exact solutions of nonlinear differential equations. In this paper, the improved Boussinesq is reduced to an ordinary differential equation under the travelling wave transformation. Trial equation method and the theory of complete discrimination system for polynomial are used to establish exact solutions of the improved Boussinesq equation.

Cite this paper

Y. Li, "Trial Equation Method for Solving the Improved Boussinesq Equation,"*Advances in Pure Mathematics*, Vol. 4 No. 2, 2014, pp. 47-52. doi: 10.4236/apm.2014.42007.

Y. Li, "Trial Equation Method for Solving the Improved Boussinesq Equation,"

References

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[2] R. Hirota, “Exact Envelope-Soliton Solutions of a Nonlinear Wave Equation,” Journal of Mathematical Physics, Vol. 14, No. 7, 1973, p. 805. http://dx.doi.org/10.1063/1.1666399

[3] R. Hirota and J. Satsuma, “Soliton Solutions of a Coupled Korteweg-de Vries Equation,” Physical Letters A, Vol. 85, No. 8-9, 1981, pp. 407-408. http://dx.doi.org/10.1016/0375-9601(81)90423-0

[4] E. Fan, “Extended Tank-Function Method and Its Applications to Nonlinear Equations,” Physical Letters A, Vol. 277, No. 4, 2000, pp. 212-218.

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http://dx.doi.org/10.1016/S0375-9601(96)00770-0

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[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 and Fractals, Vol. 42, No. 3, 2009, pp. 1356-1363.

http://dx.doi.org/10.1016/j.chaos.2009.03.043

[9] “Applications,” Communications in Theoretical Physics, Vol. 45, No. 2, 2006, pp. 219-223.

[10] C. S. Liu, “Trial Equation Method and Its Applications to Nonlinear Evolution Equations,” Acta Physical Sinica, Vol. 54, 2005, p. 2505 (in Chinese).

[11] C. S. Liu, “Using Trial Equation Method to Solve the Exact Solutions for Two Kinds of KdV Equations with Variable Coefficients,” Acta Physical Sinica, Vol. 54, No. 10, 2005, p. 4506.

[12] C. S. Liu, “Representations and Classification of Traveling Wave Solutions to Sinh-Gordon Equation,” Communications in Theoretical Physics, Vol. 49, 2008, pp. 153-158. http://dx.doi.org/10.1088/0253-6102/49/1/33

[13] C. S. Liu, “Solution of ODE and Applications to Classifications of All Single Travelling Wave Solutions to Some Nonlinear Mathematical Physics Equations,” Communications in Theoretical Physics, Vol. 49, 2008, pp. 291-296. http://dx.doi.org/10.1088/0253-6102/49/2/07

[14] C. S. Liu, “Applications of Complete Discrimination System for Polynomial for Classifications of Traveling Wave Solutions to Nonlinear Differential Equations,” Computer Physics Communications, Vol. 181, 2010, pp. 317-324.

http://dx.doi.org/10.1016/j.cpc.2009.10.006

[15] C. S. Liu, “Classification of All Single Travelling Wave Solutions to Calogero-Degasperis-Focas Equation,” Communications in Theoretical Physics, Vol. 48, 2007, pp. 601-604. http://dx.doi.org/10.1088/0253-6102/48/4/004

[16] C. S. Liu, “All Single Traveling Wave Solutions to Nizhnok-Novikov-Veselov Equation,” Communications in Theoretical Physics, Vol. 45, 2006, pp. 991-992. http://dx.doi.org/10.1088/0253-6102/45/6/006

[17] P. L. Christiansen and V. Muto, “Physica D 68,” 1993. http://dx.doi.org/10.1016/0167-2789(93)90033-W

[18] Z. Yang and B. Y. C. Hon, “An Improved Modified Extended Tanh-Function Method,” Zeitschrift fur Naturforschung A— Journal of Physical Sciences, Vol. 61, No. 3-4, 2006, pp. 103-115.

[1] M. J. Ablowitz and P. A. Clarson, “Solitons, Nonlinear Evolution Equations and Inverse Scattering,” Cambridge University Press, New York, 1991. http://dx.doi.org/10.1017/CBO9780511623998

[2] R. Hirota, “Exact Envelope-Soliton Solutions of a Nonlinear Wave Equation,” Journal of Mathematical Physics, Vol. 14, No. 7, 1973, p. 805. http://dx.doi.org/10.1063/1.1666399

[3] R. Hirota and J. Satsuma, “Soliton Solutions of a Coupled Korteweg-de Vries Equation,” Physical Letters A, Vol. 85, No. 8-9, 1981, pp. 407-408. http://dx.doi.org/10.1016/0375-9601(81)90423-0

[4] E. Fan, “Extended Tank-Function Method and Its Applications to Nonlinear Equations,” Physical Letters A, Vol. 277, No. 4, 2000, pp. 212-218.

[5] C. T. Yan, “A Simple Transformation for Nonlinear Waves,” Physical Letters A, Vol. 224, No. 1-2, 1996, pp. 77-84.

http://dx.doi.org/10.1016/S0375-9601(96)00770-0

[6] M. Wang, “Solitary Wave Solutions for Variant Boussinesq Equations,” Physical Letters A, Vol. 199, No. 3, 1995, pp. 169-172.

[7] M. L. Wang, “Solitary Wave Solutions for Variant Boussinesq Equations,” Physical Letters A, Vol. 199, No. 3-4, 1995, pp. 169-172. http://dx.doi.org/10.1016/0375-9601(95)00092-H

[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 and Fractals, Vol. 42, No. 3, 2009, pp. 1356-1363.

http://dx.doi.org/10.1016/j.chaos.2009.03.043

[9] “Applications,” Communications in Theoretical Physics, Vol. 45, No. 2, 2006, pp. 219-223.

[10] C. S. Liu, “Trial Equation Method and Its Applications to Nonlinear Evolution Equations,” Acta Physical Sinica, Vol. 54, 2005, p. 2505 (in Chinese).

[11] C. S. Liu, “Using Trial Equation Method to Solve the Exact Solutions for Two Kinds of KdV Equations with Variable Coefficients,” Acta Physical Sinica, Vol. 54, No. 10, 2005, p. 4506.

[12] C. S. Liu, “Representations and Classification of Traveling Wave Solutions to Sinh-Gordon Equation,” Communications in Theoretical Physics, Vol. 49, 2008, pp. 153-158. http://dx.doi.org/10.1088/0253-6102/49/1/33

[13] C. S. Liu, “Solution of ODE and Applications to Classifications of All Single Travelling Wave Solutions to Some Nonlinear Mathematical Physics Equations,” Communications in Theoretical Physics, Vol. 49, 2008, pp. 291-296. http://dx.doi.org/10.1088/0253-6102/49/2/07

[14] C. S. Liu, “Applications of Complete Discrimination System for Polynomial for Classifications of Traveling Wave Solutions to Nonlinear Differential Equations,” Computer Physics Communications, Vol. 181, 2010, pp. 317-324.

http://dx.doi.org/10.1016/j.cpc.2009.10.006

[15] C. S. Liu, “Classification of All Single Travelling Wave Solutions to Calogero-Degasperis-Focas Equation,” Communications in Theoretical Physics, Vol. 48, 2007, pp. 601-604. http://dx.doi.org/10.1088/0253-6102/48/4/004

[16] C. S. Liu, “All Single Traveling Wave Solutions to Nizhnok-Novikov-Veselov Equation,” Communications in Theoretical Physics, Vol. 45, 2006, pp. 991-992. http://dx.doi.org/10.1088/0253-6102/45/6/006

[17] P. L. Christiansen and V. Muto, “Physica D 68,” 1993. http://dx.doi.org/10.1016/0167-2789(93)90033-W

[18] Z. Yang and B. Y. C. Hon, “An Improved Modified Extended Tanh-Function Method,” Zeitschrift fur Naturforschung A— Journal of Physical Sciences, Vol. 61, No. 3-4, 2006, pp. 103-115.