A coupling method of homotopy perturbation and Laplace transformation for fractional models

UPB Scientific Bulletin, Series A: Applied Mathematics and Physics

Volumn 74, Issue 1, 2012, Pages 57-68

  Khan, Yasir ^a   Faraz, Naeem ^b   Kumar, Sunil ^c   Yildirim, Ahmet ^d,e  

^a ZHEJIANG UNIVERSITY   (China)

^b DONGHUA UNIVERSITY   (China)

^c Dehradun Institute of Tecgnology   (India)

^d EGE UNIVERSITY   (Turkey)

^e UNIVERSITY OF SOUTH FLORIDA   (United States)

Author keywords

Homotopy perturbation method; Laplace transform; Modified Riemann Liouville derivative

Indexed keywords

COUPLING METHODS; FRACTIONAL MODEL; FRACTIONAL ORDER; HOMOTOPIES; HOMOTOPY PERTURBATION METHOD (HPM); HOMOTOPY PERTURBATION TRANSFORM METHODS; INITIAL CONDITIONS; LAPLACE TRANSFORMATIONS; MODIFIED RIEMANN-LIOUVILLE DERIVATIVE;

DIFFERENTIAL EQUATIONS; LAPLACE TRANSFORMS;

PERTURBATION TECHNIQUES;

EID: 84863282629     PISSN: 12237027     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (29)

1. K. B. Oldham, J. Spanier, The fractional calculus, Academic Press, New York, 1999.
2. I. Podlubny, Fractional Differential Equations, Academic Press, California, USA, 1999.
3. K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Willey and Sons, Inc., New York, 2003.
4. M. Caputo, Linear models of dissipation whose Q is almost frequency independent: Part II, J Roy Astronom Soc., vol. 13, 1967, pp. 529-539.
5. H. Beyer, S. Kempfle, Definition of physically consistent damping laws with fractional derivatives, Z. Angew Math. Mech., vol. 75, 1995, pp. 623-635.
6. W. Schneider, W. Wyss, Fractional diffusion and wave equations, J. Math. Phys., vol. 30, 1989, pp. 134-144.
7. F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Solitons and Fractals, vol. 7, 1996, pp. 1461-1477.
8. F. Huang, F. Liu, The time fractional diffusion equation and fractional advection dispersion equation, ANZIAM J. 46 (2005), 1-14.
9. J.H. He, Nonlinear oscillation with fractional derivative and its applications, International Conference on Vibrating Engineering '98, Dalian, pp. 288-291.
10. J.H. He, Approximate analytical solution for seepage flow with fractional derivative in porous media, Comput. Methd. Appl. Mech. Eng. 167 (1998), 57-68.
11. N. Faraz, Y. Khan, D. S. Sankar, Decomposition-transform method for fractional differential equations, Internat. J. Nonlinear Sci. Numer. Simulation, vol. 11, 2010, pp. 305-310.
12. Z. Odibat, S. Momani, The variational iteration method: An efficient scheme for handling fractional partial differential equations in fluid mechanice, Comput. Math. Appl., vol. 58, 2009, pp. 2199-2208.
13. Y. Khan, An Effective Modification of the Laplace Decomposition Method for Nonlinear Equations, Internat. J. Nonlinear Sci. Numer. Simulation, vol. 10, 2009, pp. 1373-1376.
14. Y. Khan, N. Faraz, A. Yildirim, Q. Wu, Fractional variational iteration method for fractional initial-boundary value problems arising in the application of nonlinear Science Comput. Math. Appl., vol. 62, 2011, pp. 2273-2278.
15. Z. Odibat, S. Momani, Application of variational iteration method to Nonlinear differential equations of fractional order, Int. J. Nonlinear Sci. Numer. Simulation, vol. 7, 2006, pp. 27-34. (Pubitemid 43246579)
16. S. Das, Solution of fractional vibration equation by the variational iteration method and modified decomposition method, Int. J. Nonlinear Sci. Numer. Simulation, vol. 9, 2008, pp. 361-366.
17. N. Faraz, Y. Khan, H. Jafari, A. Yildirim, M. Madani, Fractional variational iteration method via modified Riemann-Liouville derivative, Journal of King Saud University (Science), vol. 23, 2011, 413-417.
18. H. L. Arora, F. I. Abdelwahid, Solutions of non-integer order differential equations via the Adomian Decomposition Method, Appl. Math. Lett., vol. 6, 1993, pp. 21-23.
19. N. T. Shawagfeh, The decomposition method for fractional differential equations, J Frac Calc., vol. 16, 1999, pp. 27-33.
20. S. Saha Ray, R. K. Bera, Analytical solution of a fractional diffusion equation by Adomian decomposition method, Appl. Math. Comput., vol. 174, 2006, pp. 329-336. (Pubitemid 43288955)
21. S. Momani, Z. Odibat, Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method, Appl. Math. Comput., vol. 177, 2006, pp. 488-494. (Pubitemid 43866876)
22. H. Jafari, V. D. Gejji, Solving a system of nonlinear fractional differential equations using Adomian decomposition, J. Comput. Appl. Math., vol. 196, 2006, pp. 644-651. (Pubitemid 44093887)
23. Y. Khan, N. Faraz, Modified fractional decomposition method having integral w.r.t (dξ )α, Journal of King Saud University (Science), vol. 23, 2011, pp. 157-161.
24. S. Momani, Z. Odibat, Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. Lett. A, vol. 365, 2007, pp. 345-350.
25. Z. Z. Ganji, D. D. Ganji, H. Jafari, Application of the homotopy perturbation Method to coupled system of partial differential equations with time fractional derivatives, Topol. Methods Nonlinear Anal., vol. 31, 2008, pp. 341-348.
26. A. Yildirim, Solution of BVPs for fourth-order integro-differential equation by using homotopy perturbation method, Comp. Math. Appl., vol. 56, 2008, 3175-3180.
27. J. H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech., vol. 178, 1999, pp. 257-262.
28. J. H. He, A coupling method of homotopy technique and perturbation technique for nonlinear problems, Internat. J. Non-Linear Mech., vol. 35, 2000, pp. 37-43.
29. G. Jumarie, Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions, Appl. Math. Lett., vol. 22, 2009, 378-385 22, 2009, 378-385