Skip to main content
×
×
Home

Homotopy Perturbation Method for Time-Fractional Shock Wave Equation

  • Mithilesh Singh (a1) and Praveen Kumar Gupta (a2)
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.

Copyright
Corresponding author
* Corresponding author. URL: http://msinghitbhu04.wetpaint.com/ Email: msingh.rs.apm@itbhu.ac.in
References
Hide All
[1] Oldham, K. B. and Spanier, J., The Fractional Calculus, New York, Academic Press, 1974.
[2] Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
[3] Podlubny, I., Fractional Differential Equations, New York, Academic Press, 1999.
[4] He, J. H., Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng., 178 (1999), pp. 257262.
[5] He, J. H., A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int. J. Nonlinear Mech., 35 (2000), pp. 3743.
[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. 182190.
[7] Tian, L. and Gao, Y., The global attractor of the viscous Fornberg-Whitham equation, Nonlinear Anal. Theory Method Appl., 71 (2009), pp. 51765186.
[8] Mallan, F. and Al-Khaled, K., An approximation of the analytic solution of the shock wave equation, Comput. Appl. Math., 192 (2006), pp. 301330.
[9] Berberler, M. and Yildrim, E. A., He’s homotopy perturbation method for solving shock wave equation, Appl. Anal., 88 (2009), pp. 9971004.
[10] Golbabai, A. and Sayevand, K., The homotopy perturbation method for multi-order time fractional differential equations, Nonlinear Sci. Lett. A, 1 (2010), pp. 147154.
[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. 19371945.
[12] Gupta, P. K. and Singh, M., Homotopy perturbation method for fractional Fornberg-Whitham equation, Comput. Math. Appl., 61 (2011), pp. 250254.
[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. 4851.
[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.
[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. 28292842.
[16] Chow, C. Y., An Introduction to Computational Fluid Mechanics, Wiley, New York, 1979.
[17] Kevorkian, J., Partial Differential Equations, Analytical Solution Techniques, Wadsworth and Brooks, New York, 1990.
[18] Al-Khaled, K., Theory and Computations in Hyperbolic Model Problems, Ph.D. Thesis. University of Nebraska-Lincoln, USA, 1996.
[19] He, J. H., Periodic solutions and bifurcations of delay-differential equations, Phys. Lett. A, 347 (2005), pp. 228230.
[20] He, J. H., Application of homotopy perturbation method to nonlinear wave equations, Chaos Soliton. Fract., 26 (2005), pp. 695700.
[21] He, J. H., Limit cycle and bifurcation of nonlinear problems, Chaos Soliton. Fract., 26 (2005), pp. 827833.
[22] Abbaoui, K. and Cherruault, Y., New ideas for proving convergence of decomposition methods, Comput. Math. Appl., 29 (1995), pp. 103108.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Advances in Applied Mathematics and Mechanics
  • ISSN: 2070-0733
  • EISSN: 2075-1354
  • URL: /core/journals/advances-in-applied-mathematics-and-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed