Constructing Approximate and Exact Solutions for Boussinesq Equations using Homotopy Perturbation Padé Technique

Authors: Mohamed M. Mousa, Aidarkhan Kaltayev

Abstract:

Based on the homotopy perturbation method (HPM) and Padé approximants (PA), approximate and exact solutions are obtained for cubic Boussinesq and modified Boussinesq equations. The obtained solutions contain solitary waves, rational solutions. HPM is used for analytic treatment to those equations and PA for increasing the convergence region of the HPM analytical solution. The results reveal that the HPM with the enhancement of PA is a very effective, convenient and quite accurate to such types of partial differential equations.

Keywords: Homotopy perturbation method, Padé approximants, cubic Boussinesq equation, modified Boussinesq equation.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1056272

References:

[1] J.H. He, "Homotopy perturbation technique," Comput. Methods Appl. Mech. Eng., vol. 178, pp. 257-262, 1999.
[2] J.H. He, "Homotopy perturbation method: a new nonlinear analytical technique," Appl. Math. Comput., vol. 135, pp. 73-79, 2003.
[3] J.H. He, "Comparison of homotopy perturbation method and homotopy analysis method," Comput. Methods Appl. Mech. Eng., vol. 156, pp. 527-539, 2004.
[4] J.H. He, "Application of homotopy perturbation method to nonlinear wave equations," Chaos, Solitons Fractals, vol. 26, pp. 695-700, 2005.
[5] T. Ozis and A.Yildirim, "A comparative study of He's homotopy perturbation method for determining frequency-amplitude relation of a nonlinear oscillator with discontinuities," Int. J. Nonlinear Sci. Numer. Simul., vol. 8(2), pp. 243-248, 2007.
[6] Q.K. Ghori, M. Ahmed and A.M. Siddiqui, "Application of homotopy perturbation method to squeezing flow of a Newtonian fluid," Int. J. Nonlinear Sci. Numer. Simul., vol. 8(2), pp. 179-184, 2007.
[7] M.M. Mousa and S.F. Ragab, "Application of the homotopy perturbation method to linear and nonlinear schrödinger equations," Z.Naturforsch. (a journal Physical Sciences), 63a, pp. 140-144, 2008.
[8] G.A. Baker, Essentials of Padé Approximants, Academic press, New York, 1975.
[9] B. Li, Y. Chen and H.Q. Zhang, "Explicit exact solutions for some nonlinear partial differential equations with nonlinear terms of any order," Czech. J. Phys., vol. 53, pp. 283-295, 2003.
[10] B. Li and Y. Chen, "Nonlinear Partial Differential Equations Solved by Projective Riccati Equations Ansatz," Z.Naturforsch. (a journal Physical Sciences), 58a, pp. 511-519, 2003.
[11] A.M. Wazwaz, "The variational iteration method for rational solutions for KdV, K(2,2), Burgers, and cubic Boussinesq equations," J. Comput. Appl. Math., vol. 207(1), pp. 18-23, 2007.