An Analytic Study on Time-Fractional Fisher Equation using Homotopy Perturbation Method

Authors

  • Xindong ZHANG College of Mathematics Sciences, Xinjiang Normal University, Urumqi, Xinjiang 830054
  • Juan LIU College of Mathematics Sciences, Xinjiang Normal University, Urumqi, Xinjiang 830054

Keywords:

Time-fractional Fisher’s equation, homotopy perturbation method, Caputo fractional derivative

Abstract

In this paper, the homotopy perturbation method (HPM) is effectively applied to obtain the approximate analytic solutions of the time-fractional Fisher equation (TFFE) with initial conditions. The fractional derivatives are described in the Caputo sense. The initial approximation can be determined by imposing the initial conditions. Some examples are given. Numerical results show that the HPM is easy to implement and is accurate when applied to TFFE.

doi:10.14456/WJST.2014.72

Downloads

Download data is not yet available.

Metrics

Metrics Loading ...

Author Biography

Xindong ZHANG, College of Mathematics Sciences, Xinjiang Normal University, Urumqi, Xinjiang 830054

Mathematics and System Sciences

References

KS Miller and B Ross. An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York, 1993.

AA Kilbas, HM Srivastava and JJ Trujillo. Theory and Applications of Fractional Differential Equations. Elsevier, San Diego, 2006.

RA Fisher. The wave of advance of advantageous genes. Ann. Eugen. 1936; 7, 355-69.

D Olmos and BD Shizgal. A pseudo-spectral method of solution of Fisher’s equation. J. Comput. Appl. Math. 2006 ; 193, 219-42.

N Parekh and S Puri. A new numerical scheme for the Fisher equation. J. Phys. A 1990; 23, L1085-L1091.

EH Twizell, Y Wang and WG Price. Chaos free numerical solutions of reaction-diffusion equations. Proc. Roy. Soc. London Sci. A 1990; 430, 541-76.

RE Mickens. A best finite-difference scheme for Fisher’s equation. Numer. Meth. Part. Differ. Equat. 1994; 10, 581-5.

RE Mickens. Relation between the time and space step-sizes in nonstandard finite difference schemes for the Fisher equation. Numer. Meth. Part. Differ. Equat. 1997; 13, 51-5.

R Uddin. Comparison of the nodal integral method and non standard finite-difference schemes for the Fisher equation. SIAM J. Sci. Comput. 2001; 22, 1926-42.

S Tang and RO Weber. Numerical study of Fisher’s equation by a Petrov-Galerkin finite element method. J. Austr. Math. Sot. 1991; 33, 27-38.

GF Carey and Y Shen. Least-squares finite element approximation of Fisher’s reaction-diffusion equation. Numer. Meth. Part. Differ. Equat. 1995; 11, 175-86.

J Roessler and H Hüssner. Numerical solution of the 1+2 dimensional Fisher’s equation by finite elements and the Galerkin method. Math. Comput. Model. 1997; 25, 57-67.

UG Abdullaev. Stability of symmetric traveling waves in the Cauchy problem for the Kolmogorov-Petrovskii-Piskunor equation. Diff. Equat. 1994; 30, 377-86.

DJ Logan. An Introduction to Nonlinear Partial Differential Equations. Wiley, New York, 1994.

A Öğün and C Kart. Exact solutions of Fisher and generalized Fisher equations with variable coefficients. Acta Math. Appl. Sin. 2007; 23, 563-68.

MJ Ablowitz and A Zeppetella. Explicit solutions of Fisher’s equations for a special wave speed. Bull. Math. Biol. 1979; 41, 835-40.

JF Hammond and DM Bortz. Analytical solutions to Fisher’s equation with time-variable coefficients. Appl. Math. Comput. 2011; 218, 2497-508.

K Al-Khaled. Numerical study of Fisher’s reaction-diffusion equation by the Sinc collocation method. J. Comput. Appl. Math. 2001; 137, 245-55.

T Mavounugou and Y Cerrault. Numerical study of Fisher’s equation by Adomian’s method. Math. Comput. Model. 1994; 19, 89-95.

G Hariharan, K Kannan and KR Sharma. Haar wavelet method for solving Fisher’s equation. Appl. Math. Comput. 2009; 211, 284-92.

XY Wang. Exact and explicit solitary wave solutions for the generalized Fisher equation. Phys. Lett. A 1988; 131, 277-9.

AM Wazwaz and A Gorguis. An analytic study of Fisher’s equation by using Adomian decomposition method. Appl. Math. Comput. 2004; 154, 609-20.

AM Wazwaz. Analytic study on Burgers, Fisher, Huxley equations and combined forms of these equations. Appl. Math. Comput. 2008; 195, 754-61.

JH He. Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Meth. Appl. Mech. Eng. 1998; 167, 57-68.

JH He. Application of homotopy perturbation method to nonlinear wave equations. Chaos Soliton. Fract. 2005; 26, 695-700.

S Abbasbandy. Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian’s decomposition method. Appl. Math. Comput. 2006; 172, 485-90.

M Rafie, DD Ganji and H Mohammadi. Solution of the epidemic model by homotopy perturbation method. Appl. Math. Comput. 2007; 187, 1056-62.

A Rajabi, DD Ganji and H Taherian. Application of homotopy perturbation method in nonlinear heat conduction and convection equations. Phys. Lett. A. 2007; 360, 570-3.

XL Feng, LQ Mei and GL He. An efficient algorithm for solving Troesch’s problem. Appl. Math. Comput. 2007; 189, 500-7.

S Momani and Z Odibat. Homotopy perturbation method for nonlinear partial differential equations of fractional order. Phys. Lett. A 2007; 365, 345-50.

ZZ Ganji, DD Ganji and H Jafari and M Rostamian. Application of the homotopy perturbation method to coupled system of partial differential equations with time fractional derivatives. Topol. Meth. Nonlinear Anal. 2008; 31, 341-8.

A Yıldırım. An algorithm for solving the fractional nonlinear Schrödinger equation by means of the homotopy perturbation method. Int. J. Nonlinear Sci. Numer. Simulat. 2009; 10, 445-50.

A Yıldırım and K Hüseyin. Homotopy perturbation method for solving the space-time fractional advection-dispersion equation. Adv. Water Resour. 2009; 32, 1711-6.

S Momani and Z Odibat. Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations. Comput. Math. Appl. 2007; 54, 910-9.

H Jafari and S Momani. Solving fractional diffusion and wave equations by modified homotopy perturbation method. Phys. Lett. A 2007; 370, 388-96.

KA Gepreel. The homotopy perturbation method applied to the nonlinear fractional Kolmogorov-Petrovskii-Piskunov equations. Appl. Math. Lett. 2011; 24, 1428-34.

Q Wang. Homotopy perturbation method for fractional KdV equation. Appl. Math. Comput. 2007; 190, 1795-802.

XC Li, MY Xu and XY Jiang. Homotopy perturbation method to time-fractional diffusion equation with a moving boundary condition. Appl. Math. Comput. 2009; 208, 434-9.

Y Nawaz. Variational iteration method and homotopy perturbation method for fourth-order fractional integro-differential equations. Comput. Math. Appl. 2011; 61, 2330-41.

I Podlubny. The Laplace Transform Method for Linear Differential Equations of Fractional Order. Slovac Academy of Science, Slovak Republic, 1994.

I Podlubny. Fractional Differential Equations. Academic Press, New York, 1999.

Downloads

Published

2014-01-20

How to Cite

ZHANG, X., & LIU, J. (2014). An Analytic Study on Time-Fractional Fisher Equation using Homotopy Perturbation Method. Walailak Journal of Science and Technology (WJST), 11(11), 975–985. Retrieved from https://wjst.wu.ac.th/index.php/wjst/article/view/844