International Journal of Nonlinear Analysis and Applications

Numerical solutions of nonlinear Burgers‒Huxley equation through the Richtmyer type nonstandard finite difference method

Document Type : Research Paper

Authors

1 Department of Applied Mathematics, Faculty of Mathematical Sciences, Lahijan Branch, Islamic Azad University, Lahijan, Iran

2 Department of applied mathematics, Faculty of mathematical sciences, Lahijan branch, Islamic Azad University, Lahijan, Iran

3 Department of Applied Mathematics, Faculty of Mathematical Sciences, Lahijan Branch, Islamic Azad University, Lahijan, Iran.

Abstract

The Burger‒Huxley equation as a well-known nonlinear physical model is studied numerically in the present paper. In this respect, the nonstandard finite difference (NSFD) scheme in company with the Richtmyer’s (3, 1, 1) implicit formula is formally adopted to accomplish this goal. Moreover, the stability, convergence, and consistency analyses of nonstandard finite difference schemes are investigated systematically. Several case studies with comparisons are provided, confirming that the current numerical scheme is capable of resulting in highly accurate approximations.

Keywords

[1] I. Hashim, M.S.M. Noorani and M.R.S. AL-Hadidi, Solving the generalized Burgers-Huxley equation using the Adomian decomposition method, Math. Computer Model. 43 (2006) 1404–1411.
[2] M. Javidi, A numerical solution of the generalized Burger-Huxley equation by pseudospectral method and Darvishi’s preconditioning, Appl. Math. Comput. 175 (2006) 1619–1628.
[3] B. Batiha, M.S.M. Noorani and I. Hashim, Application of variaَtional iteration method to the generalized BurgerHuxley equation, Chaos, Solutions and Fractals 36 (2008) 660–663.
[4] M. Javidi and A. Gollbabai, A new domain decomposition algorithm for generalized Burger-Huxley equation based on Chebyshev polynomials and preconditioning, Chaos, Solutions and Fractals 39(2) (2009) 849–857.
[5] J. Biazar and F. Mahmoodi, Application of differential transform method to the generalized Burger-Huxley equation, Applications and Applied mathematics, Appl. Appl. Math. Int. J. 5(10) (2010) 1726–1740.
[6] I. Celik, Haar wavelet method for solving generalized Burger-Huxley equation, Arab J. Math. Sci. 18 (2012) 25–37.
[7] A.M. AL-Rozbayani, Discrete Adomian decomposition method for solving Burger-Huxley equation, Int. J. Contemp. Math. Sci. 8(13) (2013) 623–631.
[8] A.G. Bratsos, A fourth order improved numerical scheme for the generalized Burger-Huxley equation, Amer. J. Comput. Math. 1 (2011) 152–158.
[9] B. Inan, finite difference methods for generalized Huxley and Burger-Huxley equations, Kuwait J. Sci. 44(3) (2017) 20–27.
[10] J. Noye, Numerical Solution of Partial Differential Equations Finite Difference Methods, Honours Applied Mathematics, 1993.
[11] D. Hoff, stability and convergence of finite difference methods for systems of nonlinear reaction-diffusion, SIAM J. Numerical Anal. 15(6) (1978) 1161–1177.
[12] J. Noye and H.H. Tan, Finite difference methods for solving the two-dimensional advection-diffusion equation, Int. J. Numerical Meth. Fluids 9 (1989) 75–89.
[13] A.R. Mitchell and D.F. Griffiths, Hand Book of Finite Difference Method in Partial Differential Equation, WileyInter Science publication, 1980.
[14] H. Naja, S. Edalatpanah and A.H. Refahi Sheikhani, Convergence Analysis of Modified Iterative Methods to solve Linear Systems, Mediter. J. Math. 11(3) (2014) 1019–1032.
[15] H. Saberi, H. Naja and A. Refahi, FOM-inverse vector iteration method for computing a few smallest (largest) eigenvalues of pair (A, B), Appl. Math. Comput. 184(3) (2007) 421–428.
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 1
March 2022
Pages 1507-1518
  • Receive Date: 24 August 2021
  • Revise Date: 19 October 2021
  • Accept Date: 28 October 2021