International Journal of Bioinformatics and Biomedical Engineering

Articles Information

Numerical Solution of the Generalized Burgers-Huxley Equation by Exponential Time Differencing Scheme

Pages: 43-52 Views: 1512 Downloads: 1400

[01]
Kolade M. Owolabi, Department of Mathematical Sciences, Federal University of Technology, Ondo-State, Nigeria; Department of Mathematics and Applied Mathematics, University of the Western Cape, Bellville, Republic of South Africa.

Numerical solutions of nonlinear partial differential equations, such as the generalized and extended Burgers-Huxley equations which combine effects of advection, diffusion, dispersion and nonlinear transfer are considered in this paper. Such system can be divided into linear and nonlinear parts, which allow the use of two numerical approaches. Higher order finite difference schemes are employed for the spatial discretization, the resulting nonlinear system of ordinary differential equation is advanced with the modified fourth-order exponential time differencing Runge-Kutta (ETDRK4) method designed to generate the scheme with a smaller truncation error and better stability properties. The stability region of this scheme is shown and computed via its amplification factor. Numerical simulations with comparisons are presented to address any queries that may arise.

Burgers-Huxley Equation, Exponential Time Differencing, Nonlinear, PDEs, Reaction-Diffusion, Stability

[01]
B. Batiha, M.S.M. Noorani and I. Hashim, Application of variational iteration method to the generalized Burgers-Huxley equation, Chaos Solitons and Fractals, 36 (2008) 660- 663.
[02]
G. Beylkin, J.M. Keiser and L. Vozovoi, A new class of time discretization schemes for the solution of nonlinear PDEs, Journal of Computational Physics 147 (1998) 362- 387.
[03]
A.G. Bratsos, A fourth-order numerical scheme for solving the modified Burgers equation, Computers and Mathematics with Applications, 60 (2010) 1393-1400.
[04]
A.G. Bratsos, A fourth order improved numerical scheme for the generalized Burgers-Huxley equation, American Journal of Computational Mathematics, 1 152-158.
[05]
O. Cornejo-Perez and H.C. Rosu, Nonlinear second order ODE: Factorizations and particular solutions, Progress of Theoretical Physics, 114 (2005) 533-538.
[06]
S.M. Cox and P.C. Matthews, Exponential time differencing for stiff systems, Journal of Computational Physics, 176 (2002) 430-455.
[07]
Q. Du and W. Zhu, Analysis and applications of the exponential time differencing schemes and their contour integration modifications, BIT Numerical Mathematics 45 (2005) 307-328.
[08]
X. Deng, Travelling wave solutions for the generalized Burgers-Huxley equation, Applied Mathematics and Computation, 204 (2008) 733-737.
[09]
R. Fitzhugh, Mathematical Models of Excitation and Propagation in Nerve, In: H.P. Schwan Ed., Biological Engineering, McGraw Hill, New York, (1969) p. 1-85.
[10]
G.W. Griffiths and W.E. Schiesser, Traveling Wave Analysis of Partial Differential Equations, Academic Press, Burlington, USA, 2012.
[11]
I. Grooms and K. Julien, Linearly implicit methods for nonlinear PDEs with linear dispersion and dissipation, Journal of Computational Physics, 230 (2011) 3630-3560.
[12]
M. Hochbruck and A. Ostermann, Explicit exponential Runge-Kutta methods for semilinear parabolic problems, SIAM Journal on Numerical Analysis, 43 (2005) 1069-1090.
[13]
Hochbruck and A. Ostermann, Exponential integrators, Acta Numerica 19 (2010) 209-286.
[14]
A.L. Hodgkin and A.F. Huxley, A quantitative desciption of ion currents and its applications to conduction and excitation in nerve membranes, Journal of Physiology, 117 (1952) 500-544.
[15]
F. de la Hoz and F. Vadilo, An exponential time differencing method for the nonlinear schrodinger equation, Computer Physics Communications, 179 (2008) 449-456.
[16]
H.N.A. Ismail, K. Raslan and A.A.A. Rabboh, Adomian decomposition method for Burger’s Huxley and Burger’s-Fisher equations, Applied Mathematics and Computation, 159 (2004) 291-301.
[17]
M. Javidi, A numerical solution of the generalized Burgers-Huxley equation by pseudospectral method and Darvishi’s preconditioning, Applied Mathematics and Computation, 175 (2006) 1619-1628.
[18]
M. Javidi, A numerical solution of the generalized Burgers-Huxley equation by spectral collocation method, Applied Mathematics and Computation, 178 (2006) 338-344.
[19]
A. Kassam and L.N. Trefethen, Fourth-order time-stepping for stiff PDEs, SIAM Journal on Scientific Computing 26 (2005) 1214-1233.
[20]
S. Krogstad, Generalized integrating factor methods for stiff PDEs, Journal of Computational Physics 203 (2005) 72-88.
[21]
B.V.R Kumar, V. Sangwan, S.V.S.S.N.V.G.K. Murthy and M. Nigam, A numerical study of singularly perturbed generalized Burgers-Huxley equation using three-step Taylor-Galerkin method, Computers and Mathematics with Applications, 62 (2011) 776-786.
[22]
Y.N. Kyrychko, M.V. Bartuccelli and K.B. Blyuss, Persistence of travelling wave solutions of a fourth order diffusion system, Journal of Computation and Applied Mathematics, 176 (2005) 433-443.
[23]
D. Li, C. Zhang, W. Wang and Y. Zhang, Implicit-explicit predictor-corrector schemes for nonlinear parabolic differential equations, Applied Mathematical Modelling, 35 (2011) 2711-2722.
[24]
B.Q. Lu, B.Z. Xiu, Z.L. Pang and X.F. Jiang, Exact traveling wave solution of one class of nonlinear diffusion equation, Physics Letters A, 175 (1993) 113-115.
[25]
V.T. Luan and A. Ostermann, Explicit exponential Runge-Kutta methods of high order for parabolic problems, Journal of Computational and Applied Mathematics, 256 (2014) 168-179.
[26]
A. Molabahrami and F. Khani, The homotopy analysis method to solve the Burgers- Huxley equation, Nonlinear Analysis: Real World Applications, 10 (2009) 589-600.
[27]
J.D. Murray, Mathematical Biology I: An Introduction, Springer-Verlag, New York,2002.
[28]
H. Munthe-Kaas, High order Runge-Kutta methods on manifolds, Applied Numerical Mathematics, 29 (1999) 115-127.
[29]
K.M. Owolabi and K.C. Patidar, Higher-order time-stepping methods for time dependent reaction-diffusion equations arising in biology, Applied Mathematics and Computation 240 (2014), 30-50.
[30]
A.R. Soheili, A. Kerayechian and N. Davoodi, Adaptive numerical method for Burgerstype nonlinear equations, Applied Mathematics and Computation, 219 (2012) 3486-3495.
[31]
M. Tokman, A new class of exponential propagation iterative methods of Runge-Kutta type (EPIRK) Journal of Computational Physics, 230 (2011) 8762-8778.
[32]
M. Tokman, J. Loffeld and P. Tranquilli, New adaptive exponential propagation iterative methods of Runge-Kutta type, SIAM Journal of Scientific Computing, 34 (2012)A2650-A2669.
[33]
X.Y. Wang, Z.S. Zhu and Y.K. Lu, Solitary wave solutions of the generalized Burgers-Huxley equation, Journal of Physics A, 23 (1990) 271-274.
[34]
A.M. Wazwaz, Traveling wave solutions of the generalized forms of Burgers, Burgers-KdV and Burgers-Huxley equations, Applied Mathematics and Computation, 169 (2005) 639-656.
[35]
A.M. Wazwaz, Analytical study on Burgers, Fisher, Huxley equations and combinedforms of these equations, Applied Mathematics and Computation, 195 (2008) 754-761.

Vol. 4, Issue 4, December Submit a Manuscript Join Editorial Board Join Reviewer Team

About This Journal |

All Issues |

Open Access |

Indexing |

Payment Information |

Author Guidelines |

Review Process |

Publication Ethics |

Editorial Board |

Peer Reviewers |

Copyright © 2014 - 2017 American Institute of Science except certain content provided by third parties.