Approximation of the Huxley equation with nonstandard finite-difference scheme

Document Type : Research Article

Authors

Department of Mathematics, Vali{e{Asr University of Rafsanjan, Rafsanjan, Iran.

Abstract

In this paper, an explicit exact finite-difference scheme for the Huxley equation is presented based on the nonstandard finite-difference (NSFD) scheme. Afterwards, an NSFD scheme is proposed for the numerical solution of the Huxley equation. The positivity and boundedness of the scheme is discussed. It is shown through analysis that the proposed scheme is consistent, stable, and convergence. The numerical results obtained by the NSFD scheme is compared with the exact solution and some available methods, to verify the accuracy and efficiency of the NSFD scheme.

Keywords

References

1. Batiha, B., Noorani, M.S.M. and Hashim, I. Application of variational it eration method to the generalized Burgers–Huxley equation, Chaos Solitons Fractals, 36(3) (2008), 660–663.
2. Batiha, B., Noorani, M.S.M. and Hashim, I. Numerical simulation of the generalized Huxley equation by He’s variational iteration method, Appl. Math. Comput. 186(2) (2007), 1322–1325.
3. Biazar, J. and Mohammadi, F. Application of differential transform method to the generalized Burgers–Huxley equation, Appl. Appl. Math. 5(10) (2010), 1726–1740.
4. Erdogan, U. and Ozis, T. A smart nonstandard finite difference scheme for second order nonlinear boundary value problems, J. Comput. Phys. 230(17) (2011), 6464–6474.
5. Gonzalez–Parra, G., Arenas, A.J. and Chen–Charpentier, B.M. Combination of nonstandard schemes and Richardsons extrapolation to improve the numerical solution of population models, Math. Comput. Modelling, 52(7-8) (2010), 1030–1036.
6. Hariharan, G. and Kannan, K. Haar wavelet method for solving FitzHugh– Nagumo equation, World Acad Sci, Eng Technol, 67(43) (2010), 560–574.
7. Hashim, I., Noorani, M.S.M. and Batiha, B. A note on the Adomian decomposition method for the generalized Huxley equation, Appl. Math. Comput. 181(2) (2006), 1439–1445.
8. Hashim, I., Noorani, M.S.M. and Said Al–Hadidi, M.R. Solving the generalized Burgers–Huxley equation using the Adomian decomposition method, Math. Comput. Modelling, 43(11-12) (2006),1404–1411.
9. Inan, B. Finite difference methods for the generalized Huxley and Burgers Huxley equations, Kuwait J. Sci. 44(3) (2017), 20–27.
10. Ismail, H.N.A., Raslan, K. and Abd Rabboh, A.A. Adomian decomposition method for Burgers– Huxley and Burgers–Fisher equations, Appl. Math. Comput. 159(1) (2004), 291–301.
11. Khattak, A.J. A computational meshless method for the generalized Burgers–Huxley equation, Applied Mathematical Modelling 33 (2009), 3718–3729.
12. Mickens, R.E. Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition, Numer. Methods Partial Differential Equations, vol 23(3) (2007),672–691.
13. Mickens, R.E. Advances in the Applications of Nonstandard Finite Difference Schemes, Wiley–Interscience, Singapore, 2005.
14. Murray, J.D. Mathematical Biology I, II, Third edition, Springer, 2003.
15. Namjoo, M. and Zibaei, S. Numerical solutions of FitzHugh–Nagumoequation by exact finite-difference and NSFD schemes, Comp. Appl. Math., (2016) 1–17.
16. Singh, B.K., Arora, G and Singh, M.K. A numerical scheme for the generalized Burgers–Huxley equation, Journal of the Egyptian MathematicalSociety (2016), 1–9.
17. Wang, X.Y., Zhu, Z.S. and Lu, Y.K. Solitary wave solutions of the generalised Burgers–Huxley equation, J. Phys. A: Math. Gen., 23 (1990), 271–274.
18. Wazwaz, A.M. Partial Differential Equations and Solitary WavesTheory–Higher Education Press, Springer, New York, 2009.
19. Zeinadini, M. and Namjoo, M. A Numerical Method for Discrete Fractional–Order Chemostat Model Derived from Nonstandard NumericalScheme, Bull. Iranian Math. Soc., (2016), in press.
20. Zibaei, S. and Namjoo, M. A Nonstandard Finite Difference Scheme for Solving Fractional–Order Model of HIV–1 Infection of CD 4+ T–cells,Iran. J. Math. Chem., vol 6(2) (2015), 145–160.
21. Zibaei, S. and Namjoo, M. A Nonstandard Finite Difference Scheme for Solving Three–Species Food Chain with Fractional–Order Lotka–Volterra Model, Iran. J. Numer. Anal. Optim., vol 6(1) (2016), 53–78.
22. Zibaei, S. and Namjoo, M. A NSFD scheme for Lotka–Volterra food web model, Iran. J. Sci. Technol. Trans. A Sci., vol 38(4) (2014), 399–414.
23. Zibaei, S. and Namjoo, M. Solving fractional–order competitive Lotka–Volterra model by NSFD schemes, TWMS J. App. Eng. Math., vol 6(2)(2016), 264–277.
24. Zibaei, S., Zeinadini, M. and Namjoo, M. Numerical solutions of Burgers–Huxley equation by exact finite difference and NSFD schemes, J. DifferenceEqu. Appl., (2016) 1–16.
Send comment about this article
CAPTCHA Image

Files

Share

How to cite

Statistics