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DOI QR Code

SOLUTION OF A NONLINEAR EQUATION WITH RIEMANN-LIOUVILLES FRACTIONAL DERIVATIVES BY HOMOTOPY PERTURBATION METHOD

  • Received : 2010.03.29
  • Accepted : 2010.05.28
  • Published : 2011.01.30

Abstract

The aim of the paper is to apply Homotopy Perturbation Method (HPM) for the solution of a nonlinear fractional differential equation. Finally, the solution obtained by the Homotopy perturbation method has been numerically evaluated and presented in the form of tables and then compared with those obtained by truncated series method. A good agreement of the results is observed.

Keywords

References

  1. I. Podlubny, Fractional Differential Equations, Academic Press, 1999.
  2. R.L. Bagley, P.J. Torvik, On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech. 51 (1984) 294-298. https://doi.org/10.1115/1.3167615
  3. M. Caputo, Elasticita'e Dissipazione, Zanichelli, Bologna, 1969.
  4. L.E. Suarez, A. Shokooh, An eigenvector expansion method for the solution of motion containing fractional derivatives, ASME J. Appl. Mech. 64 (1997) 629-635. https://doi.org/10.1115/1.2788939
  5. M.W. Michalski, Derivatives of non integer order and their applications, Dissertationes Mathematical (Habilitationsschrift), Polska Akademia Nauk, Instytut Matematyczuy Warszawa,1993.
  6. W.G. Glockle, T.F. Nonnenmacher, Fractional integral operators and Foxfunctions in the theory of viscoelasticity, Macromolecules 24 (1991) 6424-6434.
  7. Y. Babenko, Non integer differential equation, Conference Bordeaux 3-8 July (1994a).
  8. Y. Babenko, Non integer differential equation in Engineering: Chemical Engineering, Conference Bordeaux 3-8 July (1994b).
  9. F. Mainardi, Fractional relaxation and fractional diffusion equations: mathematical aspects in: Proceedings of the 14th IMACS World Congress, vol. 1, 1994. pp. 329-332.
  10. L. Gaul, P. Klein, S. Kempfle, Damping description involving fractional operators, Mech.Syst. Signal Process. 5 (1991) 81-88. https://doi.org/10.1016/0888-3270(91)90016-X
  11. M. Ochmann, S. Makarov, Representation of the absorption of nonlinear waves by fractional derivatives, J. Acoust. Soc. Am. 94 (6) (1993).
  12. K. Diethelm, A.D. Ford, On the solution of nonlinear fractional-order differential Equations used in the modeling of viscoplasticity, in: F. Keil, W. Mackens, H. Voss, J. Werther (Eds.), Scientific Computing in Chemical Engineering II. Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties, Springer-Verlag, Heidelberg, 1999, pp. 217-224. 570 S. Saha Ray, R.K. Bera / Appl. Math. Comput. 167 (2005) 561-571. https://doi.org/10.1016/j.amc.2004.07.020
  13. K. Diethelm, N.J. Ford, The numerical solution of linear and nonlinear fractional differential equations involving fractional derivatives of several orders, Numerical analysis Report 379, Manchester Centre for Computational Mathematics, 2001.
  14. S. Abbasbandy, An approximation solution of a nonlinear equation with Riemann-Liouville's fractional derivative by He's variational iteration method
  15. J.H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Engrg. 178 (1999) 257-262. https://doi.org/10.1016/S0045-7825(99)00018-3
  16. J.H. He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Internat. J. Non-Linear Mech. 35 (2000) 37-43. https://doi.org/10.1016/S0020-7462(98)00085-7
  17. K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.