Variational Homotopy Perturbation Method for the Fisher s Equation

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1 ISSN (print), (online) International Journal of Nonlinear Science Vol.9() No.3,pp Variational Homotopy Perturbation Method for the Fisher s Equation M. Matinfar, Z. Raeisi, M. Mahdavi Sciences Faculty, Department of Mathematics, Mazandaran University,P.O.Bo , Babolsar, Iran (Received October 9, accepted 8 January ) Abstract: In this paper we use Variational homotopy perturbation method (VHPM) [proposed A. Noor, Mathematical Problems in Engineering, (8), Article ID , pages] for solving the Fisher s equation and also we compare the approimations obtained of variational homotopy perturbation method with the approimations obtained of variational iteration method [proposed M. Matinfar, Int, J. Contemp. Math. Sciences, 9, Vol. 4, No. 7, ] for the Fisher s equation. Keywords: Variational homotopy perturbation method; Variational iteration method; Fisher s equation MSC:47J3, 49S5 Introduction In the paper, we consider the Fisher s equation u t = u + u( u), () where u t = u t, u = u. Fisher s proposed equation of u t = u + u( u) as a model for the propagation of a mutantgene, with u denoting the density of an advantageous. This equation is encountered in chemical kinetics [] and population dynamics which includs problems such as nonlinear evolution of a population in a one-dimentional habitat and neutron population in a nuclear reaction and branching. Moreover, the same equation occurs in logistic population growth models [], flame propagation, neurophysiology, autocatalytic chemical reactions, and branching Brownian motion processes. Ji-Huan He proposed the well-known variational iteration method (VIM) to have a series solution of a nonlinear differential equation using an iterative formula [ 4, 8]. Our aim in this paper is to compare the variational homotopy perturbation method [] which is a technique by combining the variational iteration method[] and the homotopy perturbation method[7, 9], with the Variational iteration method [6] for solving the Fisher s equation. Variational homotopy perturbation method To convey the basic idea of the Variational homotopy perturbation method, we consider the following general differential equation : Lu + Nu = g(), () where L is a linear operator, N is a nonlinear operator, and g() is the forcing term. According to variational iteration method, we can construct a correct functional as follows: u n+ () = u n () + λ(τ)(lu n (τ) + N u n (τ) g(τ))dτ, (3) where λ is a Lagrange multiplier, which can be identified optimally via a variational iteration method. The subscripts n denote the nth approimation, u n is considered as a restricted variation. That is, δ u n = ;. Now, we apply the homotopy perturbation method, Corresponding author. address: m.matinfar@umz.ac.ir Copyright c World Academic Press, World Academic Union IJNS..6.5/36

2 M. Matinfar, Z. Raeisi, M. Mahdavi : Variational Homotopy Perturbation Method for the Fisher s Equation 375 ρ (n) u n = u () + ρ n= λ(τ)( ρ (n) L(u n (t)) + N( ρ (n) u n (τ))dτ n= n= λ(τ)g(τ)dτ, (4) Which is the variational homotopy perturbation method and is formulated by the coupling of variational iteration method and Adomians polynomials. A comparison of like powers of ρ gives solutions of various orders. 3 Implementation of the VHPM In this section, we consider special forms of the Fisher s equation and then take into account a general form. In order to this work, we consider a Fisher s equation with =, in Eq. (). i.e. subject to the initial condition u (, t) = μ u k+ (, t) = u k (, t) + u t = u + u( u), (5) τ u k u k + u k)dτ Making the above functional stationary, the Lagrange multiplier can be determined as λ(τ) =, which yields the following iteration formula: u k+ (, t) = μ + ( u k u u k + u k)dτ +ρu + ρ u + = μ + ρ ( u + ρ u +...)dτ +ρ (u + ρu + )dτ ρ (u + ρu + ) dτ ρ () ρ () ρ () : u (, t) = μ, : u (, t) = μ( μ)t, : u (, t) = μ( μ)[ μ] t!,. (6) μ + μ( μ)t + μ( μ)[ μ] t + = For eample, we also consider a Fisher s equation with = 6, in Eq. (). i. e. μe t μ + μe t (7) u t = u + 6u( u) (8) subject to the initial condition u(, ) = ( + e ). u k+ (, t) = u k (, t) + τ u k 6u k + 6u k)dτ IJNS homepage:

3 376 International Journal of NonlinearScience,Vol.9(),No.3,pp Making the above functional stationary, the Lagrange multiplier can be determined as λ(τ) =, which yields the following iteration formula: u k+ (, t) = ( + e ) + ( u k 6u k + 6u k)dτ. u +ρu + ρ u + = (+e ) + ρ ( u + ρ u +...)dτ +ρ 6(u + ρu + )dτ ρ 6(u + ρu + ) dτ. ρ () : u (, t) = ( + e ), e ρ () : u (, t) = ( + e ) 3 t, ρ () : u (, t) = 5 e (e ) t ( + e ) 4!,. (9) u(, t) = ( + e ) + e ( + e ) 3 t + (e ) t 5e ( + e ) 4! + = ( + e 5t ) () Now, we consider the Fisher s equation u t = u + u( u) subject to the initial condition u(, ) =. ( + e 6 ) u k+ (, t) = u k (, t) + τ u k u k + u k)dτ. Making the above functional stationary, the Lagrange multiplier can be determined as λ(τ) =, which yields the following iteration formula: u k+ (, t) = + ( + e 6 ) ( u k u k + u k)dτ. u +ρu + ρ u + = 6 + ρ (+e ( u + ρ u + )dτ ) +ρ (u + ρu + )dτ ρ (u + ρu + ) dτ IJNS for contribution: editor@nonlinearscience.org.uk

4 M. Matinfar, Z. Raeisi, M. Mahdavi : Variational Homotopy Perturbation Method for the Fisher s Equation 377 ρ () :, ( + e 6 ) ρ () 5 e 6 : 3 t, ( + e 6 ) 3. () u(, t) = + ( + e 5 e 6 6 ) 3 t + = ( + e 6 ) 3 () ( + e t ) Some values of the si terms approimations to the solutions for the Eq. (9) are shown in Table. Table : The absolute error for comparison of the results of the VHPM and VIM with the eact solution at = t V HP M u u V IM u u. 5.e e e-..853e e e e-.4.4e-.4e e-.46e+.6.4 VHPM Eact.8 VIM Eact u().8 u() and the absolute error be- Figure : The absolute error between the eact solution and u V HP M tween the eact solution and u V IM at = and = 6. Fig. shows the absolute error between the eact solution and u V HP M and the absolute error between the eact solution and u V IM at = and = 6. Comparison of error shows that the VHPM results are better than the VIM results. Therefore by observing the results obtained by VHPM and VIM, we found that the series solution obtained by VHPM converges faster than series solution obtained by VIM in the studied case. IJNS homepage:

5 378 International Journal of NonlinearScience,Vol.9(),No.3,pp Conclusions In this paper, variational homotopy perturbation method was employed successfully for solving the Fisher s equation. The application of Variational iteration method to the Fisher s equation leads to calculation of unneeded terms for series solution.therefore the approimations obtained by VHPM converg to its eact solution faster than approimations obtained by Variational iteration method in the studied case. References [] P. Brazhnic and J. Tyson. On traveling wave solutions of Fishers equation in two spatial dimentions. SIAM j. Appl. Math.,6 ():(999), [] J. H. He.A new approach to nonlinear partial differential equations. Commun. Nonlinear Sci. Numer. Simulation, (4):(997),3 35. [3] J. H. He. Variational iteration method-a kind of non-linear analytical technique: some eamples. Int. J. Non-Linear Mech., 34 (4):(999), [4] J. H. He. Some asymptotic methods for strongly nonlinear equations. Int. J. Mod. Phys. B., ():(6),4 99. [5] D. S. Jone and B. D. Sleeman. Differential equations and Mathemetical Biology.Chapman and Hall/ CRC, New York(3). [6] M. Matinfar and M. Ghanbari. Solving the Fisher equation by means of variational iteration method. Int, J. Contemp. Math. Sciences,4(7):(9), [7] M. Matinfar and M. Ghanbari. Homotopy Perturbation Method for the Fishers Equation and Its Generalized. International Journal of Nonlinear Science,8(4):(9), [8] M. Matinfar, A. Fereidoon, A. Aliasghartoyeh and M.Ghanbari. Variational Iteration Method for Solving Nonlinear WBK Equations. International Journal of Nonlinear Science,8(4):(9), [9] M. Aslam Noor and Syed Tauseef Mohyud-Din. Homotopy Perturbation Method for Solving Thomas-Fermi Equation Using Pade Approimants. International Journal of Nonlinear Science, 8():(9),7 3. [] M. Laila and B. Assas. Numerical Simulation of the Generalized Hirota-Satsuma Coupled KdV Equations by Variational Iteration Method. International Journal of Nonlinear Science, 7():(9), [] W. Malfliet. Solitary wave solutions of nonlinear wave equations. Am. J. Phys., 6(7):(99), [] A. Noor and S. T. Mahyud-Din. Variational Homotopy perturbation Method for solving Higher Dimentional initial boundary value problems, Hindawi publishing corporation. Mathematical Problems in Engineering. Article ID, (8), , pages. IJNS for contribution: editor@nonlinearscience.org.uk