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AN IMPROVED SECOND-ORDER NUMERICAL METHOD FOR THE GENERALIZED BURGERS–FISHER EQUATION

Published online by Cambridge University Press:  12 June 2013

A. G. BRATSOS*
Affiliation:
Department of Mathematics, Technological Educational Institution (TEI) of Athens, 122 10 Egaleo, Athens, Greece
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Abstract

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A second-order in time finite-difference scheme using a modified predictor–corrector method is proposed for the numerical solution of the generalized Burgers–Fisher equation. The method introduced, which, in contrast to the classical predictor–corrector method is direct and uses updated values for the evaluation of the components of the unknown vector, is also analysed for stability. Its efficiency is tested for a single-kink wave by comparing experimental results with others selected from the available literature. Moreover, comparisons with the classical method and relevant analogous modified methods are given. Finally, the behaviour and physical meaning of the two-kink wave arising from the collision of two single-kink waves are examined.

Type
Research Article
Copyright
Copyright ©2013 Australian Mathematical Society 

References

Ablowitz, M. J. and Zeppetella, A., “Explicit solutions of Fisher’s equation for a special wave speed”, Bull. Math. Biol. 41 (1979) 835840; doi:10.1007/BF02462380.CrossRefGoogle Scholar
Babolian, E. and Saeidian, J., “Analytic approximate solutions to Burgers, Fisher, Huxley equations and two combined forms of these equations”, Comm. Nonlinear Sci. Numer. Simulation 14 (2009) 19841992; doi:10.1016/j.cnsns.2008.07.019.CrossRefGoogle Scholar
Bratsos, A. G., “A modified predictor–corrector scheme for the two-dimensional sine-Gordon equation”, Numer. Algorithms 43 (2006) 295308; doi:10.1007/s11075-006-9061-3.CrossRefGoogle Scholar
Bratsos, A. G., “A second order numerical scheme for the solution of the one-dimensional Boussinesq equation”, Numer. Algorithms 46 (2007) 4558; doi:10.1007/s11075-007-9126-y.CrossRefGoogle Scholar
Bratsos, A. G., “A third order numerical scheme for the two-dimensional sine-Gordon equation”, Math. Comput. Simulation 76 (2007) 271282; doi:10.1016/j.matcom.2006.11.004.CrossRefGoogle Scholar
Bratsos, A. G., “A modified explicit numerical scheme for the two-dimensional sine-Gordon equation”, Int. J. Comput. Math. 85 (2008) 241252; doi:10.1080/00207160701417415.CrossRefGoogle Scholar
Bratsos, A. G., “A numerical method for the one-dimensional sine-Gordon equation”, Numer. Methods Partial Differential Equations 24 (2008) 833844; doi:10.1002/num.20292.CrossRefGoogle Scholar
Bratsos, A. G., “A fourth-order numerical scheme for solving the modified Burgers equation”, Comput. Math. Appl. 60 (2010) 13931400; doi:10.1016/j.camwa.2010.06.021.CrossRefGoogle Scholar
Bratsos, A. G., “A modified numerical scheme for the cubic Schrödinger equation”, Numer. Methods Partial Differential Equations 27 (2011) 608620; doi:10.1002/num.20541.CrossRefGoogle Scholar
Chen, H. and Zhang, H., “New multiple soliton solutions to the general Burgers–Fisher equation and the Kuramoto–Sivashinsky equation”, Chaos Solitons Fractals 19 (2004) 7176; doi:10.1016/S0960-0779(03)00081-X.CrossRefGoogle Scholar
El-Wakil, S. A. and Abdou, M. A., “Modified extended tanh-function method for solving nonlinear partial differential equations”, Chaos Solitons Fractals 31 (2007) 12561264; doi:10.1016/j.chaos.2005.10.072.CrossRefGoogle Scholar
Fahmy, H., “Travelling wave solutions for some time-delayed equations through factorizations”, Chaos Solitons Fractals 38 (2008) 12091216; doi:10.1016/j.chaos.2007.02.007.CrossRefGoogle Scholar
Fisher, R. A., “The wave of advance of advantageous genes”, Ann. Eugenics 7 (1937) 353369; doi:10.1111/j.1469-1809.1937.tb02153.x.CrossRefGoogle Scholar
Golbabai, A. and Javidi, M., “A spectral domain decomposition approach for the generalized Burgers–Fisher equation”, Chaos Solitons Fractals 39 (2009) 385392; doi:10.1016/j.chaos.2007.04.013.CrossRefGoogle Scholar
Haq, S., Hussain, A. and Uddin, M., “On the numerical solution of nonlinear Burgers’-type equations using meshless method of lines”, Appl. Math. Comput. 218 (2012) 62806290; doi:10.1016/j.amc.2011.11.106.Google Scholar
Ismail, H. N. A. and Abd Rabboh, A. A., “A restrictive Padé approximation for the solution of the generalized Fisher and Burger–Fisher equations”, Appl. Math. Comput. 154 (2004) 203210; doi:10.1016/S0096-3003(03)00703-3.Google Scholar
Ismail, H. N. A., Raslan, K. and Abd Rabboh, A. A., “Adomian decomposition method for Burger’s–Huxley and Burger’s–Fisher equations”, Appl. Math. Comput. 159 (2004) 291301; doi:10.1016/j.amc.2003.10.050.Google Scholar
Javidi, M., “Spectral collocation method for the solution of the generalized Burger–Fisher equation”, Appl. Math. Comput. 174 (2006) 345352; doi:10.1016/j.amc.2005.04.084.Google Scholar
Kaya, D. and El–Sayed, S. M., “A numerical simulation and explicit solutions of the generalized Burgers–Fisher equation”, Appl. Math. Comput. 152 (2004) 403413; doi:10.1016/S0096-3003(03)00565-4.Google Scholar
Kolmogorov, A. N., Petrovsky, I. G. and Piscounov, N. S., “Étude de l’équation de la diffusion avec croisance de la quantité de matiére et son application á un probléme biologique”, Bull. Univ. Moskau Ser. Intl. Sec. A 1 (1937) 125.Google Scholar
Moghimi, M. and Hejazi, F. S. A., “Variational iteration method for solving generalized Burger–Fisher and Burger equations”, Chaos Solitons Fractals 33 (2007) 17561761; doi:10.1016/j.chaos.2006.03.031.CrossRefGoogle Scholar
Sari, M., “Differential quadrature solutions of the generalized Burgers–Fisher equation with a strong stability preserving high-order time integration”, Math. Comput. Appl. 16 (2011) 477486; http://mcajournal.cbu.edu.tr/volume16/vol16no2/v16no2p477.pdf.Google Scholar
Sari, M., Gürarslan, G. and Dağ, I., “A compact finite difference method for the solution of the generalized Burgers–Fisher equation”, Numer. Methods Partial Differential Equations 26 (2010) 125134; doi:10.1002/num.20421.CrossRefGoogle Scholar
Sari, M., Gürarslan, G. and Zeytinoğlu, A., “High-order finite difference schemes for the solution of the generalized Burgers–Fisher equation”, Int. J. Numer. Methods Biomed. Eng. 27 (2011) 12961308; doi:10.1002/cnm.1360.CrossRefGoogle Scholar
Twizell, E. H., Computational methods for partial differential equations (Ellis Horwood, Chichester, 1984).Google Scholar
Tyson, J. J. and Brazhnik, P. K., “On traveling wave solutions of Fisher’s equation in two spatial dimensions”, SIAM J. Appl. Math. 60 (1999) 371391; doi:10.1137/S0036139997325497.CrossRefGoogle Scholar
Wang, X. Y., “Exact and explicit solitary wave solutions for the generalised Fisher equation”, Phys. Lett. A 131 (1988) 277279; doi:10.1016/0375-9601(88)90027-8.CrossRefGoogle Scholar
Wazwaz, A.-M., “The tanh method for generalized forms of nonlinear heat conduction and Burgers–Fisher equations”, Appl. Math. Comput. 169 (2005) 321338; doi:10.1016/j.amc.2004.09.054.Google Scholar
Wazwaz, A.-M., “The tanh–coth method for solitons and kink solutions for nonlinear parabolic equations”, Appl. Math. Comput. 188 (2007) 14671475; doi:10.1016/j.amc.2006.11.013.Google Scholar
Wazzan, L., “A modified tanh–coth method for solving the general Burgers–Fisher and the Kuramoto–Sivashinsky equations”, Comm. Nonlinear Sci. Numer. Simulation 14 (2009) 26422652; doi:10.1016/j.cnsns.2008.08.004.CrossRefGoogle Scholar
Xu, Z.-H. and Xian, D.-Q., “Application of exp-function method to generalized Burgers–Fisher equation”, Acta Math. Appl. Sin. English Ser. 26 (2010) 669676; doi:10.1007/s10255-010-0031-0.CrossRefGoogle Scholar
Zhao, T., Li, C., Zang, Z. and Wu, Y., “Chebyshev–Legendre pseudo-spectral method for the generalised Burgers–Fisher equation”, Appl. Math. Model. 36 (2012) 10461056; doi:10.1016/j.apm.2011.07.059.CrossRefGoogle Scholar
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AN IMPROVED SECOND-ORDER NUMERICAL METHOD FOR THE GENERALIZED BURGERS–FISHER EQUATION
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