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Homotopy Perturbation Method for Time-Fractional Shock Wave Equation

Published online by Cambridge University Press:  03 June 2015

Mithilesh Singh  and
Praveen Kumar Gupta
Mithilesh Singh*
Affiliation:
Department of Mathematics, Dehradun Institute of Technology, Dehradun, India
Praveen Kumar Gupta
Affiliation:
Department of Applied Mathematics, Institute of Technology, Banaras Hindu University, Varanasi, India
*
*Corresponding author. URL: http://msinghitbhu04.wetpaint.com/ Email: [email protected]
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Abstract

A scheme is developed to study numerical solution of the time-fractional shock wave equation and wave equation under initial conditions by the homotopy perturbation method (HPM). The fractional derivatives are taken in the Caputo sense. The solutions are given in the form of series with easily computable terms. Numerical results are illustrated through the graph.

Keywords

76L9976K9935L10Partial differential equationfractional derivativeshock wave equationhomotopy perturbation method
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 3 , Issue 6 , December 2011 , pp. 774 - 783
Copyright
Copyright © Global-Science Press 2011

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References

[1] Oldham, K. B. and Spanier, J., The Fractional Calculus, New York, Academic Press, 1974.Google Scholar
[2] Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.Google Scholar
[3] Podlubny, I., Fractional Differential Equations, New York, Academic Press, 1999.Google Scholar
[4] He, J. H., Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng., 178 (1999), pp. 257–262.Google Scholar
[5] He, J. H., A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int. J. Nonlinear Mech., 35 (2000), pp. 37–43.Google Scholar
[6] Das, S. and Gupta, P. K., An approximate analytical solution of the fractional diffusion equation with absorbent term and external force by homotopy perturbation method, Zeitschrift-für-Naturforschung, 65a(3) (2010), pp. 182–190.CrossRefGoogle Scholar
[7] Tian, L. and Gao, Y., The global attractor of the viscous Fornberg-Whitham equation, Nonlinear Anal. Theory Method Appl., 71 (2009), pp. 5176–5186.Google Scholar
[8] Mallan, F. and Al-Khaled, K., An approximation of the analytic solution of the shock wave equation, Comput. Appl. Math., 192 (2006), pp. 301–330.Google Scholar
[9] Berberler, M. and Yildrim, E. A., He’s homotopy perturbation method for solving shock wave equation, Appl. Anal., 88 (2009), pp. 997–1004.Google Scholar
[10] Golbabai, A. and Sayevand, K., The homotopy perturbation method for multi-order time fractional differential equations, Nonlinear Sci. Lett. A, 1 (2010), pp. 147–154.Google Scholar
[11] Singh, J., Gupta, P. K. and Rai, K. N., Homotopy perturbation method to space-time fractional solidification in a finite slab, Appl. Math. Model., 35 (2010), pp. 1937–1945.Google Scholar
[12] Gupta, P. K. and Singh, M., Homotopy perturbation method for fractional Fornberg-Whitham equation, Comput. Math. Appl., 61 (2011), pp. 250–254.Google Scholar
[13] Zhang, S., Zong, Q.-A., D. L., and Gao, Q., A generalized exp-function method for fractional riccati differential equations, Commun. Fractional Calculus, 1 (2010), pp. 48–51.Google Scholar
[14] Das, S., Gupta, P. K. and Kumar, R., The homotopy analysis method for fractional Cauchy reaction-diffusion problems, Int. J. Chem. React. Eng., 9 (2011), pp. A15.Google Scholar
[15] Gupta, P. K., Approximate analytical solutions of fractional Benney-Lin equation by reduced differential transformation and homotopy perturbation method, Comput. Math. Appl., 61 (2011), pp. 2829–2842.Google Scholar
[16] Chow, C. Y., An Introduction to Computational Fluid Mechanics, Wiley, New York, 1979.Google Scholar
[17] Kevorkian, J., Partial Differential Equations, Analytical Solution Techniques, Wadsworth and Brooks, New York, 1990.Google Scholar
[18] Al-Khaled, K., Theory and Computations in Hyperbolic Model Problems, Ph.D. Thesis. University of Nebraska-Lincoln, USA, 1996.Google Scholar
[19] He, J. H., Periodic solutions and bifurcations of delay-differential equations, Phys. Lett. A, 347 (2005), pp. 228–230.Google Scholar
[20] He, J. H., Application of homotopy perturbation method to nonlinear wave equations, Chaos Soliton. Fract., 26 (2005), pp. 695–700.Google Scholar
[21] He, J. H., Limit cycle and bifurcation of nonlinear problems, Chaos Soliton. Fract., 26 (2005), pp. 827–833.Google Scholar
[22] Abbaoui, K. and Cherruault, Y., New ideas for proving convergence of decomposition methods, Comput. Math. Appl., 29 (1995), pp. 103–108.Google Scholar