Solving Fisher’s Equation by Using Modified Variational Iteration Method

[1]

**Yaseen Ul Rehman**, Department of Mathematics, National College of Business Administration and Economics, Gujrat, Pakistan.

[2]

**Memmona Yaqub**, Department of Mathematics, Allama Iqbal Open University, Islamabad, Pakistan.

[3]

**Qazi Mahmood Ul-Hassan**, Department of Mathematics, University of Wah, Wah Cantt., Pakistan.

[4]

**Kamran Ayub**, Department of Mathematics, Riphah International University, Islamabad, Pakistan.

[5]

**Ayesha Siddiqa**, Department of Mathematics, University of Wah, Wah Cantt., Pakistan.

Fisher equation is nonlinear partial differential equation which is used in various biological, chemical processes in engineering and sciences. In this paper, we use the numerical technique, Variational iteration method and its some sub sequenced modification to solve the Fisher’s equation. Lagrange multipliers for identification of optimal value of parameters in a functional are the basis of Variational iteration method. This method can be used to find possible approximate or series solution of problem. This paper also contains the numerical solution of Fisher equation. The Important purpose of this paper is to check the reliability of these techniques for the solution of Fisher equation as compared to other techniques.

Fisher Equation, Variational Iteration Method, Maple 18, Series Solution

[1]

Fisher RA. The wave of advance of advantageous genes. Ann Eugen 1937; 7: 355-69.

[2]

Canosa J. On a nonlinear diffusion equation describing population growth. IBM J Res Dev 1937; 17: 307-13.

[3]

Aggarwal SK. Some numerical experiments on Fisher’s Equation. Int Commun heat mass Transf 1985; 12: 417-30.

[4]

Aronson DG, Weinberger HF. Nonlinear diffusion in population genetics combustion and never pules propagation. In Partial differential equations and related topics. Lecture notes in mathematics, vol. 446. P 5-49.

[5]

Maini PK. McElwain DLS, Leavesley D. Traveling waves in a wound healing assay. App Math Lett 2004; 17: 575-80.

[6]

Sengers BG, Please CP, Oreffo ROC. Experimental characterization and computational modelling of two-dimensional cell spreading for skeletal regeneration. J R Soc Interf 2007; 4: 1107-17.

[7]

N. F. Briton, Reaction–Diffusion Equations and their Applications to Biology, Academic, New York, 1986.

[8]

J. D. Murray, Mathematical Biology, second ed., Springer, Berlin, 1993.

[9]

K. P. Hadeler, Reaction transport systems in mathematics inspired by biology, in: V. Capasso, O. Diskmann (Eds.), Cime Lectures, Florence, Springer Verlag, Berlin, 1998.

[10]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenic. 7 (1937) 353.

[11]

J. H. He, A new approach to nonlinear partial di_erential equations. Comm. Nonlinear Sci. Numer. Simul, 2, (1997), PP. 203-205.

[12]

J. H. He and X. H. Wu, Variational iteration method: new development and applications. Comput. Math. Appl, 54, (2007), PP. 881-894.

[13]

J. H. He, Homotopy perturbation technique. Comput. Meth. Appl. Eng, vol. 178, no. 3-4, (1999), p. 257-262.

[14]

J. H. He, Application of homotopy perturbation method to nonlinear wave equations. Chaos Solitons Fractals, 26, (2005), pp. 695-700.

[15]

J. H. He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems. Int. J. of Non-Linear Mechanics, 35, (2000), pp. 37-43.

[16]

J. H. He, Some asymptotic methods for strongly nonlinear equations. Int. J. Modern Phys, B 20, (2006), pp. 1141-1199.

[17]

M. Matinfar, M. Mahdavi and Z. Rafeisi, Variational Homotopy Perturbation Method for Analytic Treatment for Linear and Nonlinear Ordinary Di_eretial Equations. J. Appl. Math. Inf, 28, (2010), pp. 845-862.

[18]

M. Matinfar, M. Mahdavi and Z. Raeisy, The implementation of Variational Homotopy Perturbation Method for Fisher's equation. Int. J. Non- linear Sci, 9, (2010), pp. 188-194.

[19]

Y. Liu, Variational Homotopy Perturbation Method for Solving Fractional Initial Boundary Value Problems. Abstract Appl. Anal, A. ID 727031, (2012), pp. 1-10.

[20]

F. H. Easif, S. A. Manaa, B. A. Mahmood and M. A. Yousif, Variational Homotopy Perturbation Method for Solving Benjamin-Bona- Mahony Equation. Appl. Math, 6, (2015), pp. 675-683.

[21]

P. Brazhnik, J. Tyson, On traveling wave solutions of Fisher’s equation in two spatial dimensions, SIAM J. Appl. Math. 60 (2) (1999) 371-391.

[22]

W. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys. 7 (1992) 650-654.

[23]

J. H. He, A new approach to nonlinear partial differential equations, Commun. Nonlinear Sci. Numer. Simulation, 2 (4) (1997) 230-235.

[24]

J. H. He, Variational iteration method-a kind of non-linear analytical technique: some examples, Int. J. Non-Linear Mech. 34 (4) (1999) 699-708.

[25]

J. H. He, Some asymptotic methods for strongly nonlinear equations, Int. J. Mod. Phys. B, 20 (10) (2006) 1141-1199.

[26]

M. Inokuti, H. Sekine, T. Mura, General use of the Lagrange multiplier in non-linear mathematical physics, in: S. Nemat-Nasser (Ed.), Variational Method in the Mechanics of Solids, Pergamon Press, Oxford, 1978, pp. 156-162.

[27]

D. S. Jone, B. D. Sleeman, Differential Equations and Mathematical Biology, Chapman and Hall/ CRC, New York, 2003.