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Toward a New Algorithm for Nonlinear Fractional Differential Equations

Published online by Cambridge University Press:  03 June 2015

Fadi Awawdeh  and
S. Abbasbandy
Fadi Awawdeh*
Affiliation:
Department of Mathematics, Hashemite University, Jordan
S. Abbasbandy*
Affiliation:
Department of Mathematics, Imam Khomeini International University, Ghazvin 34149-16818, Iran
*
Email: [email protected]
Corresponding author. Email: [email protected]
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Abstract

This paper is concerned with the development of an efficient algorithm for the analytic solutions of nonlinear fractional differential equations. The proposed algorithm Laplace homotopy analysis method (LHAM) is a combined form of the Laplace transform method with the homotopy analysis method. The biggest advantage the LHAM has over the existing standard analytical techniques is that it overcomes the difficulty arising in calculating complicated terms. Moreover, the solution procedure is easier, more effective and straightforward. Numerical examples are examined to demonstrate the accuracy and efficiency of the proposed algorithm.

Keywords

34A0874G1035C1044A10Homotopy analysis methodLaplace transformfractional differential equations
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Issue 2 , April 2013 , pp. 222 - 234
Copyright
Copyright © Global-Science Press 2013

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References

[1]Bagley, R. L. and Torvik, P. J., On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech., 51 (1984), pp. 294–298.Google Scholar
[2]Miller, K. S. and Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.Google Scholar
[3]Ghorbani, A., Toward a new analytical method for solving nonlinear fractional differential equations, Comput. Methods Appl. Mech. Eng., 197 (2008), pp. 4173–4179.CrossRefGoogle Scholar
[4]Odibat, Z. M. and Momani, S., Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear Sci. Numer. Simul., 7 (2006), pp. 27–34.Google Scholar
[5]Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston and London, 1994.Google Scholar
[6]Abbasbandy, S., Application of He’s homotopy perturbation method to functional integral equations, Chaos Soliton. Fract., 31 (2007), pp. 1243–1247.CrossRefGoogle Scholar
[7]Abbasbandy, S., Solitary wave solutions to the Kuramoto-Sivashinsky equation by means of the homotopy analysis method, Nonlinear Dyn., 52 (2008), pp. 35–40.Google Scholar
[8]Abbasbandy, S., Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method, Chem. Eng. J., 136(2-3) (2008), pp. 144–150.Google Scholar
[9]Awawdeh, F., Jaradat, H. M. and Alsayyed, O., Solving system of DAEs by homotopy analysis method, Chaos Soliton. Fract., 42(3) (2009), pp. 1422–1427.Google Scholar
[10]Awawdeh, F., Adawi, A. and Mustafa, Z., Solutions of the SIR models of epidemics using HAM, Chaos Soliton. Fract., 42(5) (2009), pp. 3047–3052.Google Scholar
[11]Awawdeh, F., Khandaqji, M. and Mustafa, Z., A new approach for the solution of the electrostatic potential differential equations, Math Probl. Eng., 2009 (2009), doi: 10.1155/2009/980987.Google Scholar
[12]Hashim, I., Abdulaziz, O. and Momani, S., Homotopy analysis method for fractional IVPs, Commun. Nonlinear. Sci. Numer. Simul., 14(3) (2009), pp. 674–684.CrossRefGoogle Scholar
[13]Liao, S. J., Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC Press, Boca Raton, 2003.Google Scholar
[14]Liao, S. J., On the homotopy analysis method for nonlinear problems, Appl. Math. Comput., 147 (2004), pp. 499–513.Google Scholar
[15]Liao, S. J., Notes on the homotopy analysis method: Some definitions and theorems, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), pp. 983–997.Google Scholar
[16]Liao, S. J., An optimal homotopy-analysis approach for strongly nonlinear differential equations, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), pp. 2003–2016.Google Scholar
[17]Momani, S. and Odibat, Z., Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method, Appl. Math. Comput., 177(2) (2006), pp.488–494.Google Scholar
[18]Momani, S. and Odibat, Z., Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys. Lett. A, 355 (2006), pp.271–279.CrossRefGoogle Scholar
[19]Odibat, Z., On Legendre polynomial approximation with the VIM or HAM for numerical treatment of nonlinear fractional differential equations, J. Comput. Appl. Math., (2011), doi:10.1016/j.cam.2010.12.013.Google Scholar
[20]Odibat, Z., Momani, S. and Xu, H., A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations, Appl. Math. Model., 34 (2010), pp. 593–600.Google Scholar
[21]Saha Ray, S. and Bera, R. K., An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method, Appl. Math. Comput., 167 (2005), pp. 561–571.Google Scholar
[22]Liao, S. J., Comparison between the homotopy analysis method and homotopy perturbation method, Appl. Math. Comput., 169 (2005), pp. 1186–1194.Google Scholar
[23]Lin, R. and Liu, F., Fractional high order methods for the nonlinear fractional ordinary differential equation, Nonlinear Anal., 66 (2007), pp. 856–869.Google Scholar