B-spline collocation algorithm for numerical solution of the generalized Burger’s-Huxley equation. (English) Zbl 1276.65062

The cubic B-spline collocation scheme is implemented to find numerical solutions of the generalized Burger’s-Huxley equation. The scheme is based on the finite-difference formulation for time integration and cubic B-spline functions for space integration. Convergence of the scheme is discussed through standard convergence analysis. The proposed scheme is of second order convergent. The accuracy of the proposed method is demonstrated by four test problems. The numerical results are found to be in good agreement with the exact solutions. Results are compared with other results given in literature.


65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
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[1] A. L.Hodgkin and A. F.Huxley, A Quantitative description of ion currents and its applications to conduction and excitation in nerve membranes, J Physiol117 ( 1952), 500-544.
[2] J.Satuma, Topics in soliton theory and exactly solvable nonlinear equations, World Scientific, Singapore, 1987. · Zbl 0721.00016
[3] X.Wang, Nerve propagation and wall in liquid crystals, Phys Lett112A ( 1985), 402-406.
[4] X. Y.Wang, Z. S.Zhu, and Y. K.Lu, Solitary wave solutions of the generalised Burger’s-Huxley equation, J Phys A: Math Gen23 ( 1990), 271-274. · Zbl 0708.35079
[5] A. G.Bratsos, A fourth‐order numerical scheme for solving the modified Burger’s equation, Comput Math Appl60 ( 2010), 1393-1400. · Zbl 1201.65151
[6] R.Fitzhugh, Mathematical models of excitation and propagation in nerve, 1969.
[7] P. G.Estévez, Non‐classical symmetries and the singular modified the Burger’s and Burger’s-Huxley equation, J Phy A‐Math Gen27 ( 1994), 2113-2127. · Zbl 0838.35114
[8] O. Y. U.Efimova and N. A.Kudryashov, Exact solutions of the Burger’s-Huxley equation, J Appl Math Mech68 ( 2004), 413-420. · Zbl 1092.35084
[9] H. N. A.Ismail, K.Raslan, and A. A. A.Rabboh, Adomian decomposition method for Burger’s-Huxley and Burger’s‐Fisher equations, Appl Math Comput159 ( 2004), 291-301. · Zbl 1062.65110
[10] I.Hashim, M. S. M.Noorani, and M. R.Said Al‐Hadidi, Solving the generalized Burger’s-Huxley equation using the Adomian decomposition method, Math Comput Model43 ( 2006), 1404-1411. · Zbl 1133.65083
[11] I.Hashim, M. S. M.Noorani, and B.Batiha, A note on the Adomian decomposition method for the generalized Huxley equation, Appl Math Comput181 ( 2006), 1439-1445. · Zbl 1173.65340
[12] B.Batiha, M. S. M.Noorani, and I.Hashim, Numerical simulation of the generalized Huxley equation by He’s variational iteration method, Chaos Soliton Fractals186 ( 2007), 1322-1325. · Zbl 1118.65367
[13] B.Batiha, M. S. M.Noorani, and I.Hashim, Application of variational iteration method to the generalized Burger’s-Huxley equation, Chaos Soliton Fractals36 ( 2008), 660-663. · Zbl 1141.49006
[14] X.Deng, Travelling wave solutions for the generalized Burger’s-Huxley equation, Appl Math Comput204 ( 2008), 733-737. · Zbl 1160.35515
[15] E. S.Fahmy, Travelling wave solutions for some time‐delayed equations through factorizations, Chaos Soliton Fractals38 ( 2008), 1209-1216. · Zbl 1152.35438
[16] A. M.Wazwaz, Analytic study on Burger’s, Fisher, Huxley equations and combined forms of these equations, Appl Math Comput195 ( 2008), 754-761. · Zbl 1132.65098
[17] A.Molabahrami and F.Khani, The homotopy analysis method to solve the Burger’s-Huxley equation, Nonlinear Analysis: Real World Appl10 ( 2009), 589-600. · Zbl 1167.35483
[18] A. S.Bataineh, M. S. M.Noorani, and I.Hashim, Analytical treatment of generalized Burger’s-Huxley equation by homotopy analysis method, Bull Malays Math Sci Soc32 ( 2009), 233-243. · Zbl 1173.35650
[19] H.Gao and R.Zhao, New exact solutions to the generalized Burger’s-Huxley equation, Appl Math Comput217 ( 2010), 1598-1603. · Zbl 1202.35220
[20] M.Javidi, A numerical solution of the generalized Burger’s-Huxley equation by spectral method, Appl Math Comput178 ( 2006), 338-344. · Zbl 1100.65081
[21] M.Javidi and A.Golbabai, A new domain decomposition algorithm for generalized Burger’s-Huxley equation based on Chebyshev polynomials and preconditioning, Chaos Soliton Fractals39 ( 2009), 849-857. · Zbl 1197.65153
[22] M. T.Darvishi, S.Kheybari, and F.Khani, Spectral collocation method and Darvishi’s preconditionings to solve the generalized Burger’s-Huxley equation, Commun Nonlinear Sci Numer Simul13 ( 2008), 2091-2103. · Zbl 1221.65261
[23] A.Kaushik and M. D.Sharma, A uniformly convergent numerical method on non‐uniform mesh for singularly perturbed unsteady Burger’s-Huxley equation, Appl Math Comput195 ( 2008), 688-706. · Zbl 1136.65085
[24] M.Sari and G.Gürarslan, Numerical solutions of the generalized Burger’s-Huxley equation by a differential quadrature method (DQM), Math Prob Eng2009. DOI: 10.1155/2009/370765. · Zbl 1184.65094
[25] A. J.Khattak, A computational meshless method for the generalized Burger’s-Huxley equation, Appl Math Model33 ( 2009), 3718-3729. · Zbl 1185.65191
[26] S.Tomasiello, Numerical solutions of the Burger’s-Huxley equation by the IDQ method, Int J Comput Math87 ( 2010), 129-140. · Zbl 1182.65162
[27] V.Gupta and M. K.Kadalbajoo, A singular perturbation approach to solve Burger’s-Huxley equation via monotone finite difference scheme on layer‐adaptive mesh, Commun Nonlinear Sci Numer Simul16 ( 2011), 1825-1844. · Zbl 1221.65221
[28] B. V.Rathish Kumar, V.Sangwan, S. V. S. S. N. V. G. K.Murthy, and M.Nigam, A numerical study of singularly perturbed generalized Burger’s-Huxley equation using three‐step Taylor‐Galerkin method, Comput Math Appl62 ( 2011), 776-786. · Zbl 1228.65189
[29] J. E.Macías‐Díaz, J.Ruiz‐Ramírez, and J.Villa, The numerical solution of a generalized Burger’s-Huxley equation through a conditionally bounded and symmetry‐preserving method, Comput Math Appl62 ( 2011), 3330-3342. · Zbl 1222.65095
[30] M.Sari, G.Gürarslan, and A.Zeytino&011F;lu, High‐order finite difference schemes for numerical solutions of the generalized Burger’s-Huxley equation, Numer Methods Partial Diff Eq27 ( 2011), 1313-1326. · Zbl 1226.65078
[31] A. G.Bratsos, A fourth order improved numerical scheme for the Generalized Burger’s-Huxley equation, Am J Comput Math1 ( 2011), 152-158.
[32] S.Zhou and X.Cheng, A linearly semi‐implicit compact scheme for the Burger’s-Huxley equation, Int J Comput Math88 ( 2011), 795-804. · Zbl 1213.65122
[33] M. K.Jain, Numerical solution of differential equations, 2nd Ed., Wiley Eastern, New Delhi, 1984. · Zbl 0536.65004
[34] P. M.Prenter, Splines and variational methods, John Wiley, New York, 1975. · Zbl 0344.65044
[35] P.Henrici, Discrete variable methods in ordinary differential equations, Wiley, New York, 1962. · Zbl 0112.34901
[36] M. K.Kadalbajoo, V.Gupta, and A.Awasthi, A uniformly convergent B‐spline collocation method on a nonuniform mesh for singularly perturbed one‐dimensional time‐dependent linear convection‐diffusion problem, J Comput App Math220 ( 2008), 271-289. · Zbl 1149.65085
[37] J.Stoer and R.Bulrisch, An introduction to numerical analysis, Springer‐Verlag, New York, 1991.
[38] W.Rudin, Principles of mathematical analysis, 3rd ed., McGraw‐Hill Inc, New York, 1976. · Zbl 0148.02903
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