Application of homotopy analysis method for solving nonlinear fractional partial differential equations

Authors

  • Aref Guzali
  • Jalil Manafian Heris Islamic Azad University ahar Branch
  • Jalal Jalali

Keywords:

Analytical solution, Nonlinear fractional heat conduction, Kaup-Kupershmidt, Fisher, Huxley, Burgers–Fisher and Burgers–Huxley equations, Fractional partial differential equations (FPDEs), Homotopy analysis method

Abstract

In this article, the homotopy analysis method is applied to solve nonlinear fractional partial differential equations.Based on the homotopy analysis method, a scheme is developed to obtain the approximate solution of the nonlinearfractional heat conduction, Kaup–Kupershmidt, Fisher and Huxley equations with initial conditions, introduced byreplacing some integer-order time derivatives by fractional derivatives. The solutions of the studied models are calculatedin the form of convergent series with easily computable components. The results of applying this procedure tothe studied cases show the high accuracy and efficiency of the new technique. The fractional derivative is describedin the Caputo sense. Some illustrative examples are presented to observe some computational results.

References

K.S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations,

New York, Wiley, 1993.

M.J. Ablowitz, P.A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, Cambridge

University Press, New York, 1991.

I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional

differential equations, to methods of their solution and some of their applications, New York:

Academic Press, 1999.

S.G. Samko, AA. Kilbas, OI. Marichev, Fractional integrals and derivatives: theory and applications,

Yverdon, Gordon and Breach, 1993.

B.J. West, M. Bolognab, P. Grigolini, Physics of fractal operators. New York: Springer; 2003.

M. Caputo, Linear models of dissipation whose Q is almost frequency independent, J. Royal Astr.

Soc, 13 (1967) 529-539.

L. Debanth, Recents applications of fractional calculus to science and engineering, Int. J. Math.

Math. Sci, 54 (2003) 3413-3442.

H. Jafari, S. Seifi, Solving a system of nonlinear fractional partial differential equations using

homotopy analysis method, Commu. Nonlinear Sci. Num. Simu, 14 (2009) 1962-1969.

S. Kemple, H. Beyer, Global and causal solutions of fractional differential equations, Transform

methods and special functions: Varna 96, Proceedings of 2nd international workshop (SCTP),

Singapore, 96 (1997) 210-216.

A.A. Kilbas, J.J. Trujillo, Differential equations of fractional order: methods, results problems,

Appl. Anal, 78 (2001) 153-192.

S. Momani, NT. Shawagfeh, Decomposition method for solving fractional Riccati differential equations,

Appl. Math. Comput, 182 (2006) 1083-1092.

K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.

M. Dehghan, J. Manafian, A. Saadatmandi, Solving nonlinear fractional partial differential equations

using the homotopy analysis method, Num. Meth. Par. Diff. Equ. J, 26 (2010) 448-479.

S. Abbasbandy, Approximate solution for the nonlinear model of diffusion and reaction in porous

catalysts by means of the homotopy analysis method, Chem. Eng. J, 136 (2008) 144-150.

M. Dehghan, J. Manafian, The solution of the variable coefficients fourth–order parabolic partial

differential equations by homotopy perturbation method, Z. Naturforsch, 64a (2009) 420-430.

M. Dehghan, J. Manafian, A. Saadatmandi, The solution of the linear fractional partial differential

equations using the homotopy analysis method, Z. Naturforsch, 65a (2010) 935-949.

M. Dehghan, J. Manafian, The solution of the variable coefficients fourth–order parabolic partial

differential equations by homotopy perturbation method, Z. Naturforsch, 64a (2009) 420-430.

J. Manafian Heris, M. Bagheri, Exact Solutions for the Modified KdV and the Generalized KdV

Equations via Exp-Function Method, J. Math. Extension, 4 (2010) 77-98.

M. Dehghan, J. Manafian, A. Saadatmandi, Application of semi–analytic methods for the

Fitzhugh–Nagumo equation, which models the transmission of nerve impulses, Math. Meth. Appl.

Sci, 33 (2010) 1384-1398.

A. M. Wazwaz, The tanh method for generalized forms of nonlinear heat conduction and Burgers-

Fisher equations, Appl. Math. Comput, 169 (2005) 321-338.

E. Babolian, J. Saeidian, Analytic approximate solutions to Burgers, Fisher, Huxley equations and

two combined forms of these equations, Commu. Nonlinear Sci. Num. Simu, 14 (2009) 1984-1992.

A. M. Wazwaz, Analytic study on Burgers, Fisher, Huxley equations and combined forms of these

equations, Appl. Math. Comput, 195 (2008) 754-761.

T. ¨ Ozi¸s, C. K¨oroˇglu, A novel approach for solving the Fisher equation using Exp-function method,

Phys. Let. A, 372 (2008) 3836-3840.

A. Borhanifar, M.M. Kabir, New periodic and soliton solutions by application of Exp-function

method for nonlinear evolution equations, J. Compu. Appl. Math, 229 (2009) 158-167.

S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, PhD

thesis, Shanghai Jiao Tong University, 1992.

S.J. Liao, Series solutions of unsteady boundary-layer flows over a stretching flat plate, Stud. Appl.

Math, 117 (2006) 2529-2539.

S.J. Liao, Beyond perturbation: Introduction to the homotopy analysis method, Boca Raton:

Chapman and Hall, CRC Press; 2003.

S.J. Liao, On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a

stretching sheet, J. Fluid Mech, 488 (2003) 189-212.

S.J. Liao, An analytic approximate approach for free oscillations of self-excited systems, Int. J.

NonLinear Mech, 39 (2004) 271-280.

A. Osborne, The inverse scattering transform: Tools for the nonlinear fourier analysis and filtering

of ocean surface waves, Chaos, Solitons and Fractals, 5 (1995) 2623-2637.

C.L. Peter, V. Muto, S. Rionero, Solitary wave solutions to a system of Boussinesq-like equations,

Chaos, Solitons and Fractals, 2 (1992) 529-539.

T.J. Priestly, P.A. Clarkson, Symmetries of a generalized Boussinesq equation, IMS Technical

Report, UKC/IMS/ 59 (1996).

S.S. Ray, RK. Bera, An approximate solution of a nonlinear fractional differential equation by

Adomian decomposition method, Appl. Math. Comput, 167 (2005) 561-571.

P.Rosenau, J.M. Hyman, Compactons: Solitons with finite wavelengths, Phys. Rev. Let, 70 (5)

(1993) 564-567.

N.T. Shawagfeh, Analytical approximate solutions for nonlinear fractional differential equations,

Appl. Math. Comput, 131 (2002) 241-259.

L. Song, H. Zhang, Solving the fractional BBM-Burgers equation using the homotopy analysis

method, Chaos, Solitons and Fractals, 40 (2009) 1616-1622.

L. Song, H. Zhang, Application of homotopy analysis method to fractional KdV-Burgers-Kuramoto

equation, Phys. Let. A, 367 (2007) 88-94.

Downloads

Published

2014-06-15

Issue

Section

Articles

How to Cite

Application of homotopy analysis method for solving nonlinear fractional partial differential equations. (2014). Asian Journal of Fuzzy and Applied Mathematics, 2(3). https://ajouronline.com/index.php/AJFAM/article/view/1064

Similar Articles

41-50 of 53

You may also start an advanced similarity search for this article.