A Fourth Order Improved Numerical Scheme for the Generalized Burgers—Huxley Equation

A. G. Bratsos
DOI: 10.4236/ajcm.2011.13017                

Abstract

A fourth order finite-difference scheme in a two-time level recurrence relation is proposed for the numerical solution of the generalized Burgers--Huxley equation. The resulting nonlinear system, which is analysed for stability, is solved using an improved predictor-corrector method. The efficiency of the proposed method is tested to the kink wave using both appropriate boundary values and conditions. The results arising from the experiments are compared with the relevant ones known in the available bibliography.

Keywords

Burgers-Huxley, Finite-Difference Method, Modified Predictor-Corrector

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	A. L. Hodgkin and A. F. Huxley, “A Quantitative Description of Ion Currents and Its Applications to Conduction and Excita-tion in Nerve Membranes,” Journal of Physiology, Vol. 117, No. 4, 1952, pp. 500-544.
[2]	J. Satsuma, “Topics in Soliton Theory and Exactly Solvable Nonlinear Equations,” In: M. Ablowitz, B. Fuchssteiner and M. Kruskal, Eds., World Scien-tific, Singapore City, 1987.
[3]	X. Wang, “Nerve Propagation and Wall in Liquid Crystals,” Physics Letters, Vol. 112A, No. 8, 1985, pp. 402-406.
[4]	X. Y. Wang, Z. S. Zhu and Y. K. Lu, “Solitary Wave Solutions of the Generalised Burgers-Huxley Equation,” Journal of Physics A: Mathematical and General, Vol. 23, No. 3, 1990, pp. 271-274. doi:10.1088/0305-4470/23/3/011
[5]	A. G. Bratsos, “A Fourth-Order Numerical Scheme for Solving the Modified Burgers Equation,” Computers and Mathematics with Applica-tions, Vol. 60, No. 5, 2010, pp. 1393-1400. doi:10.1016/j.camwa.2010.06.021
[6]	R. Fitzhugh, “Mathe-matical Models of Excitation and Propagation in Nerve,” In: H. P. Schwan. Ed., Biological Engineering, McGraw Hill, New York, 1969, pp. 1-85.
[7]	P. G. Estévez and P. R. Gordoa, “Painlevé Analysis of the generalized Burgers-Huxley Equa-tion,” Journal of Physics A: Mathematical and General, Vol. 23, No. 21, 1990, pp. 4831-4837.
[8]	P. G. Estévez, “Non-Classical Symmetries and the Singular Manifold Method: The Burgers and the Burgers-Huxley Equations,” Journal of Physics A: Mathematical and General, Vol. 27, No. 6, 1994, pp. 2113- 2127.
[9]	O. Yu. Yefimova and N. A. Kudryashov, “Exact Solutions of the Burgers-Huxley Equation,” Journal of Applied Mathematics and Mechanics, Vol. 68, No. 3, 2004, pp. 413-420. doi:10.1016/S0021-8928(04)00055-3
[10]	X. Deng, “Travelling Wave Solutions for the Generalized Bur-gers-Huxley Equation,” Applied Mathematics and Computation, Vol. 204, No. 1, 2008, pp. 733-737. doi:10.1016/j.amc.2008.07.020
[11]	A. M. Wazwaz, “Analytic Study on Burgers, Fisher, Huxley Equations and Combined Forms of These Equations,” Applied Mathematics and Compu-tation, Vol. 195, No. 2, 2008, pp. 754-761. doi:10.1016/j.amc.2007.05.020
[12]	H. N. A. Ismail, K. Ra-slan and A. A. A. Rabboh, “Adomian Decomposition Method for Burger’s Huxley and Burger’s-Fisher Equations,” Applied Mathematics and Computation, Vol. 159, No. 1, 2004, pp. 291-301. doi:10.1016/j.amc.2003.10.050
[13]	I. Hashim, M. S. M. Noorani and M. R. S. Al-Hadidi, “Solving the Generalized Burgers-Huxley Equation Using the Adomian Decomposition Method,” Mathematical and Computer Modelling, Vol. 43, No. 11-12, 2006, pp. 1404-1411. doi:10.1016/j.mcm.2005.08.017
[14]	M. Javidi, “A Numerical Solution of the Generalized Burger’s-Huxley Equation by Pseudospectral Method and Darvishi’s Preconditioning,” Ap-plied Mathematics and Computation, Vol. 175, No. 1, 2006, pp. 1619-1628. doi:10.1016/j.amc.2005.09.009
[15]	M. Javidi, “A Numerical Solution of the Generalized Burgers-Huxley Equation by Spec-tral Collocation Method,” Applied Mathematics and Computa-tion, Vol. 178, No. 2, 2006, pp. 338-344.
[16]	M. Javidi and A. Golbabai, “A New Domain Decomposition Algorithm for Generalized Burgers-Huxley Equation Based on Chebyshev Polynomials and Preconditioning,” Chaos Soliton & Fractals, Vol. 39, 2009, pp. 849-857.
[17]	B. Batiha, M. S. M. Noorani and I. Hashim, “Application of Variational Iteration Method to the Generalized Burgers-Huxley Equation,” Chaos Solitons & Fractals, Vol. 36, No. 3, 2008, pp. 660-663. doi:10.1016/j.chaos.2006.06.080
[18]	A. Khattak, “A Compu-tational Meshless Method for the Generalized Burgers-Huxley Equation,” Applied Mathematical Modelling, Vol. 33, No. 9, 2009, pp. 3718-3729. doi:10.1016/j.apm.2008.12.010
[19]	E. Babolian and J. Saeidian, “Analytic Approximate Solutions to Burgers, Fisher, Huxley Equations and Two Combined Forms of These Equa-tions,” Communications in Nonlinear Science and Numerical Simulation, Vol. 14, 2009, pp. 1984-1992. doi:10.1016/j.cnsns.2008.07.019
[20]	B. Fornberg, “A Practi-cal Guide to Pseudospectral Methods,” Cambridge University Press, Cambridge, 1998.
[21]	E. H. Twizell, “Computational Methods for Partial Differential Equations,” Ellis Horwood Limited, Chichester, 1984.