A numerical Approximation for the One-dimensional Burger–Fisher Equation
Asian Research Journal of Mathematics,
Page 22-30
DOI:
10.9734/arjom/2022/v18i530375
Abstract
In this paper, an implicit finite difference method based on the Crank–Nicolson method is proposed for the numerical solution of the one-dimensional Burger–Fisher equation. The Crank–Nicolson scheme provides a system of nonlinear difference equations, which is solved by an integration of the Jacobian-Free-Newton-Krylov (JFNK) and GMRES methods. Various numerical examples are given to demonstrate the efficiency of the proposed scheme. Comparison of the computed solutions with the analytical ones demonstrates the accuracy of this proposed method.
Keywords:
- Crank–Nicolson scheme
- burger–fisher equation
- jacobian-free-newton-krylov method
- GMRES method
How to Cite
Hajinezhad, H. (2022). A numerical Approximation for the One-dimensional Burger–Fisher Equation. Asian Research Journal of Mathematics, 18(5), 22-30. https://doi.org/10.9734/arjom/2022/v18i530375
References
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Pirdawood MA, Sabawi YA. High-order solution of Generalized Burgers–Fisher Equation using compact finite difference and DIRK methods. in Journal of Physics: Conference Series; 2021.
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Gürbüz B, Sezer M. A Modified Laguerre Matrix Approach for Burgers–Fisher Type Nonlinear Equations, in Numerical Solutions of Realistic Nonlinear Phenomena. 2020;Springer:107-123.
Singh A, Dahiya S, Singh S. A fourth-order B-spline collocation method for nonlinear Burgers–Fisher equation. Mathematical Sciences. 2020;14(1):75-85.
Mohanty RK, Sharma S. A high-resolution method based on off-step non-polynomial spline approximations for the solution of Burgers-Fisher and coupled nonlinear Burgers’ equations. Engineering Computations; 2020.
Wasim I, Abbas M, Amin M. Hybrid B-spline collocation method for solving the generalized Burgers-Fisher and Burgers-Huxley equations. Mathematical Problems in Engineering; 2018.
Yadav OP, Jiwari R. Finite element analysis and approximation of Burgers’‐Fisher equation. Numerical Methods for Partial Differential Equations. 2017;33(5):1652-1677.
Triki H, Wazwaz AM. Trial equation method for solving the generalized Fisher equation with variable coefficients. Physics Letters A. 2016;380(13):1260-1262.
Chandraker V, Awasthi A, Jayaraj S. Numerical treatment of Burger-Fisher equation. Procedia Technology. 2016;25:1217-1225.
Zhang L, Wang L, Ding, X. Exact finite difference scheme and nonstandard finite difference scheme for Burgers and Burgers-Fisher equations. Journal of Applied Mathematics; 2014.
Zhang R, Yu X, Zhao G. The local discontinuous Galerkin method for Burger’s–Huxley and Burger’s–Fisher equations. Applied Mathematics and Computation. 2012;218(17):8773-8778.
Kocacoban D, et al. A better approximation to the solution of Burger-Fisher equation. in Proceedings of the World Congress on Engineering; 2011.
Zhang J, Yan G. A lattice Boltzmann model for the Burgers–Fisher equation. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2010;20(2):023129.
Sari M, Gürarslan G, Dağ İ. A compact finite difference method for the solution of the generalized Burgers–Fisher equation. Numerical Methods for Partial Differential Equations: An International Journal. 2010;26(1):125-134.
Rashidi M, Ganji D, Dinarvand S. Explicit analytical solutions of the generalized Burger and Burger–Fisher equations by homotopy perturbation method. Numerical Methods for Partial Differential Equations: An International Journal. 2009;25(2):409-417.
Khattak AJ. A computational meshless method for the generalized Burger’s–Huxley equation. Applied Mathematical Modelling. 2009;33(9):3718-3729.
Golbabai A, Javidi M. A spectral domain decomposition approach for the generalized Burger’s–Fisher equation. Chaos, Solitons & Fractals. 2009;39(1):385-392.
Ismail HN, Raslan K, Abd Rabboh AA. Adomian decomposition method for Burger's–Huxley and Burger's–Fisher equations. Applied mathematics and computation. 2004;159(1):291-301.
Javidi M. Spectral collocation method for the solution of the generalized Burger–Fisher equation. Applied Mathematics and Computation. 2006;174(1):345-352.
Wazwaz AM. The tanh method for generalized forms of nonlinear heat conduction and Burgers–Fisher equations. Applied Mathematics and Computation. 2005;169(1):321-338.
Ismail HN, Abd Rabboh AA. A restrictive Padé approximation for the solution of the generalized Fisher and Burger–Fisher equations. Applied Mathematics and Computation. 2004;154(1):203-210.
Kaya D, El-Sayed SM. A numerical simulation and explicit solutions of the generalized Burgers–Fisher equation. Applied Mathematics and computation. 2004;152(2):403-413.
Mickens R, Gumel A. Construction and analysis of a non-standard finite difference scheme for the Burgers-Fisher equation. Journal of sound and vibration. 2002;257(4):791-797.
Knoll DA, Keyes DE. Jacobian-free Newton–Krylov methods: A survey of approaches and applications. Journal of Computational Physics. 2004;193(2):357-397.
Heyouni M. Newton Generalized Hessenberg method for solving nonlinear systems of equations. Numerical Algorithms. 1999;21(1-4):225-246.
Mousseau VA, Knoll DA, Rider WJ. A Multigrid Newton-Krylov Solver for Non-linear Systems. Lecture Notes in Computational Science and Engineering. 2000;14:200-206.
Brown PN, Saad Y. Hybrid Krylov methods for nonlinear systems of equations. SIAM Journal on Scientific and Statistical Computing. 1990;11(3):450-481.
Chan TF, Jackson KR. Nonlinearly preconditioned Krylov subspace methods for discrete Newton algorithms. SIAM Journal on Scientific and Statistical Computing.1984;5(3):533-542.
Hajinezhad H, et al. Numerical solution and convergence analysis of steam injection in heavy oil reservoirs. Computational Geosciences. 2018;22(6): 1433-1444.
Chen BK, et al. Exp-function method for solving the Burgers-Fisher equation with variable coefficients. arXiv preprint arXiv:1004.1815; 2010.
Pirdawood MA, Sabawi YA. High-order solution of Generalized Burgers–Fisher Equation using compact finite difference and DIRK methods. in Journal of Physics: Conference Series; 2021.
IOP Publishing.
Gürbüz B, Sezer M. A Modified Laguerre Matrix Approach for Burgers–Fisher Type Nonlinear Equations, in Numerical Solutions of Realistic Nonlinear Phenomena. 2020;Springer:107-123.
Singh A, Dahiya S, Singh S. A fourth-order B-spline collocation method for nonlinear Burgers–Fisher equation. Mathematical Sciences. 2020;14(1):75-85.
Mohanty RK, Sharma S. A high-resolution method based on off-step non-polynomial spline approximations for the solution of Burgers-Fisher and coupled nonlinear Burgers’ equations. Engineering Computations; 2020.
Wasim I, Abbas M, Amin M. Hybrid B-spline collocation method for solving the generalized Burgers-Fisher and Burgers-Huxley equations. Mathematical Problems in Engineering; 2018.
Yadav OP, Jiwari R. Finite element analysis and approximation of Burgers’‐Fisher equation. Numerical Methods for Partial Differential Equations. 2017;33(5):1652-1677.
Triki H, Wazwaz AM. Trial equation method for solving the generalized Fisher equation with variable coefficients. Physics Letters A. 2016;380(13):1260-1262.
Chandraker V, Awasthi A, Jayaraj S. Numerical treatment of Burger-Fisher equation. Procedia Technology. 2016;25:1217-1225.
Zhang L, Wang L, Ding, X. Exact finite difference scheme and nonstandard finite difference scheme for Burgers and Burgers-Fisher equations. Journal of Applied Mathematics; 2014.
Zhang R, Yu X, Zhao G. The local discontinuous Galerkin method for Burger’s–Huxley and Burger’s–Fisher equations. Applied Mathematics and Computation. 2012;218(17):8773-8778.
Kocacoban D, et al. A better approximation to the solution of Burger-Fisher equation. in Proceedings of the World Congress on Engineering; 2011.
Zhang J, Yan G. A lattice Boltzmann model for the Burgers–Fisher equation. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2010;20(2):023129.
Sari M, Gürarslan G, Dağ İ. A compact finite difference method for the solution of the generalized Burgers–Fisher equation. Numerical Methods for Partial Differential Equations: An International Journal. 2010;26(1):125-134.
Rashidi M, Ganji D, Dinarvand S. Explicit analytical solutions of the generalized Burger and Burger–Fisher equations by homotopy perturbation method. Numerical Methods for Partial Differential Equations: An International Journal. 2009;25(2):409-417.
Khattak AJ. A computational meshless method for the generalized Burger’s–Huxley equation. Applied Mathematical Modelling. 2009;33(9):3718-3729.
Golbabai A, Javidi M. A spectral domain decomposition approach for the generalized Burger’s–Fisher equation. Chaos, Solitons & Fractals. 2009;39(1):385-392.
Ismail HN, Raslan K, Abd Rabboh AA. Adomian decomposition method for Burger's–Huxley and Burger's–Fisher equations. Applied mathematics and computation. 2004;159(1):291-301.
Javidi M. Spectral collocation method for the solution of the generalized Burger–Fisher equation. Applied Mathematics and Computation. 2006;174(1):345-352.
Wazwaz AM. The tanh method for generalized forms of nonlinear heat conduction and Burgers–Fisher equations. Applied Mathematics and Computation. 2005;169(1):321-338.
Ismail HN, Abd Rabboh AA. A restrictive Padé approximation for the solution of the generalized Fisher and Burger–Fisher equations. Applied Mathematics and Computation. 2004;154(1):203-210.
Kaya D, El-Sayed SM. A numerical simulation and explicit solutions of the generalized Burgers–Fisher equation. Applied Mathematics and computation. 2004;152(2):403-413.
Mickens R, Gumel A. Construction and analysis of a non-standard finite difference scheme for the Burgers-Fisher equation. Journal of sound and vibration. 2002;257(4):791-797.
Knoll DA, Keyes DE. Jacobian-free Newton–Krylov methods: A survey of approaches and applications. Journal of Computational Physics. 2004;193(2):357-397.
Heyouni M. Newton Generalized Hessenberg method for solving nonlinear systems of equations. Numerical Algorithms. 1999;21(1-4):225-246.
Mousseau VA, Knoll DA, Rider WJ. A Multigrid Newton-Krylov Solver for Non-linear Systems. Lecture Notes in Computational Science and Engineering. 2000;14:200-206.
Brown PN, Saad Y. Hybrid Krylov methods for nonlinear systems of equations. SIAM Journal on Scientific and Statistical Computing. 1990;11(3):450-481.
Chan TF, Jackson KR. Nonlinearly preconditioned Krylov subspace methods for discrete Newton algorithms. SIAM Journal on Scientific and Statistical Computing.1984;5(3):533-542.
Hajinezhad H, et al. Numerical solution and convergence analysis of steam injection in heavy oil reservoirs. Computational Geosciences. 2018;22(6): 1433-1444.
Chen BK, et al. Exp-function method for solving the Burgers-Fisher equation with variable coefficients. arXiv preprint arXiv:1004.1815; 2010.
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