Constructing Approximate and Exact Solutions for Boussinesq Equations using Homotopy Perturbation Padé Technique
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.
[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.
[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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:51140", author = "Mohamed M. Mousa and Aidarkhan Kaltayev", title = "Constructing Approximate and Exact Solutions for Boussinesq Equations using Homotopy Perturbation Padé Technique", 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.", volume = "3", number = "2", pages = "70-9", }