A Numerical Solution of the Generalized Burgers-Huxley Equation

Yıl 2022, , 75 - 84, 28.02.2022

Gonca Çelikten , Adem Cankurt

https://doi.org/10.35414/akufemubid.1006385

Öz

In this study numerical solutions of the generalized Burgers-Huxley equation are obtained utilizing a new approach: The Crank Nicolson logarithmic finite difference method (CN-LFDM). The effectiveness of the suggested method is demonstrated by a numerical example for various parameter cases. Presented tables demonstrate that the obtained results are in excellent agreement with the exact solutions and better than numerical results acquired by other methods in the literature. The method was analyzed with the von-Neumann stability analysis method and it was shown that the method was unconditionally stable.

Anahtar Kelimeler

Crank Nicolson logarithmic finite difference method, Generalized Burgers-Huxley equation, von Neumann Stability analysis.

Genelleştirilmiş Burgers-Huxley Denkleminin Bir Sayısal Çözümü

Öz

Bu çalışmada, genelleştirilmiş Burgers-Huxley denkleminin sayısal çözümleri yeni bir yaklaşım kullanılarak elde edilmiştir: Crank Nicolson logaritmik sonlu farklar yöntemi (CN-LSFY). Önerilen yöntemin etkinliği, çeşitli parametre durumları için sayısal bir örnekle gösterilmiştir. Sunulan tablolar, elde edilen sonuçların tam çözümlerle mükemmel bir uyum içinde olduğunu ve literatürdeki diğer yöntemlerle elde edilen sayısal sonuçlardan daha iyi olduğunu göstermektedir. Yöntem, von-Neumann kararlılık analizi yöntemi ile analiz edilmiş ve yöntemin koşulsuz kararlı olduğu gösterilmiştir.

Anahtar Kelimeler

Crank Nicolson logaritmik sonlu fark yöntemi, Genelleştirilmiş Burgers-Huxley denklemi, von Neumann kararlılık analizi.

Kaynakça

• Batiha, B., Noorani, M.S.M., Hashim, I., 2008. Application of variational iteration method to the generalized Burgers-Huxley equation. Chaos, Solitons & Fractals, 36, 660-663.
• Biazar, J., Mohammadi, F., 2010. Application of differential transform method to the generalized Burgers-Huxley equation. Applications and Applied Mathematics: An International Journal, 5 (10) , 1726-1740.
• Bratsos, A. G., 2011. A fourth order improved numerical scheme for the generalized Burgers-Huxley equation. American Journal of Computational Mathematics, 1 , 152-158.
• Çelik, I, 2012. Haar wavelet method for solving generalized Burgers-Huxley equation. Arab Journal of Mathematical Sciences, 18, 25-37.
• Çelik, I., 2016. Chebyshev Wavelet collocation method for solving generalized Burgers–Huxley equation. Mathematical Methods in the Applied Sciences, 39, 366–377.
• Darvishi, M.T., Kheybari, S., Khani, F., 2008. Spectral collocation method and Darvishi’s preconditionings to solve the generalized Burgers-Huxley equation. Communications in Nonlinear Science and Numerical Simulation, 13, 2091-2103.
• Deng, X., 2008. Travelling wave solutions for the generalized Burgers–Huxley equation. Applied Mathematics and Computation, 204, 733–737.
• Duan, Y., Kong, L., Zhang, R., 2012. A lattice Boltzmann model for the generalized Burgers–Huxley equation. Physica A, 391, 625–632.
• El-Kady, M., El-Sayed, S.M., Fathy, H.E., 2013. Development of Galerkin method for solving the generalized Burger’s Huxley equation. Mathematical Problems in Engineering, 2013, 1-9.
• Hashim, I., Noorani, M.S.M., Al-Hadidi, M.R.S., 2006. Solving the generalized Burgers-Huxley Equation using the Adomian decomposition method. Mathematical and Computer Modelling, 43, 1404-1411.
• Hilal, N., Injrou, S., Karroum, R., 2020. Exponential finite difference methods for solving Newell–Whitehead–Segel equation. Arabian Journal of Mathematics, 9, 367-379.
• Inan, B., 2017. Finite difference methods for the generalized Huxley and Burgers-Huxley equations, Kuwait Journal of Science, 44 (3), 20-27.
• Inan, B., Bahadir, A.R., 2015. Numerical solutions of the generalized Burgers-Huxley equation by implicit exponential finite difference method. Journal of Applied Mathematics, Statistics and Informatics, 11 (2), 57-67.
• Javidi, M., 2006. A numerical solution of the generalized Burger’s-Huxley equation by pseudospectral method and Darvishi’s preconditioning. Applied Mathematics and Computations, 175, 1619-1628.
• Javidi, M., 2006. A numerical solution of the generalized Burger’s-Huxley equation by spectral collocation method. Applied Mathematics and Computations, 178, 338-344.
• Javidi, M., Golbabai, A., 2009. A new domain decomposition algorithm for generalized Burger’s-Huxley equation based on Chebyshev polynomials and preconditioning. Chaos, Solitons & Fractals, 39(2), 849-857.
• Khan, A.,, Mohan, M. T., Ruiz-Baier, R., 2021. Conforming, nonconforming and DG methods for the stationary generalized Burgers- Huxley equation. Journal of Scientific Computing, 3, 1-21.
• Khattak, A.J., 2009. A computational meshless method for the generalized Burger’s-Huxley equation. Applied Mathematical Modelling, 33, 218-3729.
• Loyinmi, A.C., Akinfe, T.K., 2020. An algorithm for solving the Burgers–Huxley equation using the Elzaki transform. SN Applied Sciences, 2 (7), 1-17.
• Mittal, R.C., Tripathi, A., 2015. Numerical solutions of generalized Burgers–Fisher and generalized Burgers–Huxley equations using collocation of cubic B-splines. International Journal of Computer Mathematics, 92 (5), 1053–1077.
• Mohan, M.T., Khan, A., 2021. On the generalızed Burgers-Huxley equation: existence, uniqueness, regularity, global attractors and numerical studies. Discrete and Continuous Dynamical Systems Series B, 26 (7), 3943-3988.
• Sari, M., Gürarslan, G., 2009. Numerical solutions of the generalized Burgers-Huxley equation by a differential quadrature method. Mathematical Problems in Engineering, 2009, 1-11.
• Satsuma, S., 1987. Topics in Soliton Theory and Exactly Solvable Nonlinear Equations, Ablowitz, M., Fuchssteiner, B., Kruskal, M., World Scientific, 1-354.
• Singh, B.K., Arora, G., Singh, M.K., 2016. A numerical scheme for the generalized Burgers–Huxley equation. Journal of the Egyptian Mathematical Society, 24, 629–637.
• Tomasiello, S., 2010. Numerical solutions of the Burgers–Huxley equation by the IDQ method. International Journal of Computer Mathematics, 87 (1) , 129-140.
• Wazwaz A-M., 2005. Travelling wave solutions of generalized forms of Burgers, Burgers–KdV and Burgers–Huxley equations. Applied Mathematics and Computation, 169, 639–656.