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On Two Efficient Numerical Schemes for Nonlinear Burgers’ Type Equations

Year 2022, Volume: 12 Issue: 1, 70 - 87, 30.06.2022

Bilge Inan

https://doi.org/10.37094/adyujsci.1041652

Abstract

In this work, we investigate two finite difference schemes to solve for nonlinear Burgers’ type equations. In the first stage, we define the numerical methods to solve the equations. Secondly, numerical solutions are obtained and compared with the exact solutions. In comparison with other defined results in the literature, it is deduced in a conclusive way that the methods are reliable and convenient alternative methods for solving for nonlinear Burgers’ type equations.

Keywords

The Burgers’ type equations The generalized Burgers-Huxley equation Finite difference method Logarithmic finite difference method

References

  • Referans1 Wang, X. Y., Zhu Z. S., Lu, Y. K., Solitary wave solutions of the generalized Burgers-Huxley equation, J. Phys. A: Math. Gen., 23, 271-274, 1990.
  • Referans2 Hashim, I., Noorani, M. S. M., Batiha, B., A note on the Adomian decomposition method for the generalized Huxley Equation, Appl. Math. Comput., 181, 1439-1445, 2006.
  • Referans3 Sari, M., Gürarslan, G., Zeytinoglu, A., High-order finite difference schemes for numerical solutions of the generalized Burgers-Huxley equation, Numer. Methods Partial Differential Eq., 27, 1313-1326, 2010.
  • Referans4 Hemida, K., Mohamed, M. S., Numerical simulation of the generalized Huxley equation by homotopy analysis method, J. Appl. Func. Analysis, 5, 344-350, 2010.
  • Referans5 Hemida, K., Mohamed, M. S., Application of homotopy analysis method to fractional order generalized Huxley equation, J. Appl. Analysis, 7, 367-372, 2012.
  • Referans6 Hashemi, S. H., Daniali, H. R. M., Ganji, D. D., Numerical simulation of the generalized Huxley equation by He's homotopy perturbation method, Appl. Math. Comput., 192, 157-161, 2007.
  • Referans7 Batiha, B., Noorani, M. S. M., Hashim, I., Numerical simulation of the generalized Huxley equation by He's variational iteration method, Appl. Math. Comput., 186, 1322-1325, 2007.
  • Referans8 İnan, B., A new numerical scheme for the generalized Huxley equation, Bulletin of Math. Sci. Appl., 16, 105-111, 2016.
  • Referans9 İnan, B., Finite difference methods for the generalized Huxley and Burgers-Huxley equations, Kuwait J. Sci., 44, 20-27, 2017.
  • Referans10 El Morsy, S. A., El-Azab, M. S., Logarithmic finite difference method applied to KdVB equation, American Academic&Scholarly Research Journal, 4, 2, 2012.
  • Referans11 Srivastava, V. K., Awasthi, M. K., Singh, S., An implicit logarithmic finite-difference technique for two dimensional coupled viscous Burgers' equation, AIP Advances, 3, 122105, 2013.
  • Referans12 Srivastava, V. K., Tamsir, M., Awasthi, M. K., Singh, S., One dimensional coupled viscous Burgers' equation and its numerical solution by an implicit logarithmic finite-difference method, AIP Advances, 4, 037119, 2014.
  • Referans13 Srivastava, V. K., Tamsir, M., Rashidi, M. M., Analytic and numeric computation of two dimensional unsteady nonlinear coupled viscous generalized Burgers' equation, Asia Pacific J. Engineering Sci. Tech., 2, 23-35, 2016.
  • Referans14 Çelikten, G., Göksu, A., Yagub, G., Explicit logarithmic finite difference schemes for numerical solution of Burgers equation, European Int. J. Sci. Tech., 6, 57-67, 2017.
  • Referans15 Hashim, I., Noorani, M. S. M., Said Al-Hadidi, M. R., Solving the generalized Burgers-Huxley Equation using the Adomian decomposition method, Math. Comput. Model., 43, 1404-1411, 2006.
  • Referans16 Javidi, M., A numerical solution of the generalized Burger's-Huxley equation by pseudospectral method and Darvishi's preconditioning, Appl. Math. Comput., 175, 1619-1628, 2006.
  • Referans17 Javidi, M., A numerical solution of the generalized Burger's-Huxley equation by spectral collocation method, Appl. Math. Comput., 178, 338-344, 2006.
  • Referans18 Darvishi, M. T., Kheybari, S., Khani, F., Spectral collocation method and Darvishi's preconditionings to solve the generalized Burgers-Huxley equation, Commun. Nonlinear Sci. Numer. Simul., 13, 2091-2103, 2008.
  • Referans19 Batiha, B., Noorani, M. S. M., Hashim, I., Application of variational iteration method to the generalized Burgers-Huxley equation, Chaos Soliton Fract., 36, 660-663, 2008.
  • Referans20 Sari, M., Gürarslan, G., Numerical solutions of the generalized Burgers-Huxley equation by a differential quadrature method, Math. Probl. Eng., doi: 10.1155/2009/370765, 2009.
  • Referans21 Khattak, A. J., A computational meshless method for the generalized Burger's-Huxley equation, Appl. Math. Model., 33, 3218-3729, 2009.
  • Referans22 Javidi, M., Golbabai, A., A new domain decomposition algorithm for generalized Burger's-Huxley equation based on Chebyshev polynomials and preconditioning, Chaos Soliton Fract., 39, 849-857, 2009.
  • Referans23 Biazar, J., Mohammadi, F., Application of differential transform method to the generalized Burgers-Huxley equation, Appl. Appl. Math., 5, 1726-1740, 2010.
  • Referans24 Bratsos, A. G., A fourth order improved numerical scheme for the generalized Burgers-Huxley equation, American J. Comput. Math., 1, 152-158, 2011.
  • Referans25 Çelik, İ., Haar wavelet method for solving generalized Burgers-Huxley equation, Arab J. Math. Sci., 18, 25-37, 2012.
  • Referans26 El-Kady, M., El-Sayed, S. M., Fathy, H. E., Development of Galerkin method for solving the generalized Burger's Huxley equation, Math. Probl. Eng., doi: 10.1155/2013/165492, 2013.
  • Referans27 Al-Rozbayani, A. M., Discrete Adomian decomposition method for solving Burger's-Huxley Equation, Int. J. Contemp. Math. Sci., 8, 623-631, 2013.
  • Referans28 Ismail, H. N. A., Raslan, K., Rabboh, A. A. A., Adomian decomposition method for Burger's-Huxley and Burger's-Fisher equations, Appl. Math. Comput., 159, 291-301, 2004.
  • Referans29 Macías-Díaz, J. E., On the numerical and structural properties of a logarithmic scheme for diffusion-reaction equations, Appl. Num. Math., 140, 104-114, 2019.
  • Referans30 İnan, B., A logarithmic finite difference technique for numerical solution of the generalized Huxley equation, Proceedings of 7th International Eurasian Conference on Mathematical Sciences and Applications, Kyiv, Ukraine, pp. 100-101, 2018.
  • Referans31 İnan, B., High accuracy numerical solutions by logarithmic finite difference method for the generalized Burgers-Huxley equation, Proceedings of 2nd International Conference on Mathematical and Related Sciences, Antalya, Turkey, pp.29, 2019.
  • Referans32 Macías-Díaz, J. E., İnan, B., Structural and numerical analysis of an implicit logarithmic scheme for diffusion equations with nonlinear reaction, Int. J. Modern Physics C, 30, 9, 1950065, 2019.
  • Referans33 Macías-Díaz, J. E., Hendy, A. S., On the stability and convergence of an implicit logarithmic scheme for diffusion equations with nonlinear reaction, J. Math. Chemistry, 58, 735–74, 2020.

Year 2022, Volume: 12 Issue: 1, 70 - 87, 30.06.2022

Bilge Inan

https://doi.org/10.37094/adyujsci.1041652

Abstract

References

  • Referans1 Wang, X. Y., Zhu Z. S., Lu, Y. K., Solitary wave solutions of the generalized Burgers-Huxley equation, J. Phys. A: Math. Gen., 23, 271-274, 1990.
  • Referans2 Hashim, I., Noorani, M. S. M., Batiha, B., A note on the Adomian decomposition method for the generalized Huxley Equation, Appl. Math. Comput., 181, 1439-1445, 2006.
  • Referans3 Sari, M., Gürarslan, G., Zeytinoglu, A., High-order finite difference schemes for numerical solutions of the generalized Burgers-Huxley equation, Numer. Methods Partial Differential Eq., 27, 1313-1326, 2010.
  • Referans4 Hemida, K., Mohamed, M. S., Numerical simulation of the generalized Huxley equation by homotopy analysis method, J. Appl. Func. Analysis, 5, 344-350, 2010.
  • Referans5 Hemida, K., Mohamed, M. S., Application of homotopy analysis method to fractional order generalized Huxley equation, J. Appl. Analysis, 7, 367-372, 2012.
  • Referans6 Hashemi, S. H., Daniali, H. R. M., Ganji, D. D., Numerical simulation of the generalized Huxley equation by He's homotopy perturbation method, Appl. Math. Comput., 192, 157-161, 2007.
  • Referans7 Batiha, B., Noorani, M. S. M., Hashim, I., Numerical simulation of the generalized Huxley equation by He's variational iteration method, Appl. Math. Comput., 186, 1322-1325, 2007.
  • Referans8 İnan, B., A new numerical scheme for the generalized Huxley equation, Bulletin of Math. Sci. Appl., 16, 105-111, 2016.
  • Referans9 İnan, B., Finite difference methods for the generalized Huxley and Burgers-Huxley equations, Kuwait J. Sci., 44, 20-27, 2017.
  • Referans10 El Morsy, S. A., El-Azab, M. S., Logarithmic finite difference method applied to KdVB equation, American Academic&Scholarly Research Journal, 4, 2, 2012.
  • Referans11 Srivastava, V. K., Awasthi, M. K., Singh, S., An implicit logarithmic finite-difference technique for two dimensional coupled viscous Burgers' equation, AIP Advances, 3, 122105, 2013.
  • Referans12 Srivastava, V. K., Tamsir, M., Awasthi, M. K., Singh, S., One dimensional coupled viscous Burgers' equation and its numerical solution by an implicit logarithmic finite-difference method, AIP Advances, 4, 037119, 2014.
  • Referans13 Srivastava, V. K., Tamsir, M., Rashidi, M. M., Analytic and numeric computation of two dimensional unsteady nonlinear coupled viscous generalized Burgers' equation, Asia Pacific J. Engineering Sci. Tech., 2, 23-35, 2016.
  • Referans14 Çelikten, G., Göksu, A., Yagub, G., Explicit logarithmic finite difference schemes for numerical solution of Burgers equation, European Int. J. Sci. Tech., 6, 57-67, 2017.
  • Referans15 Hashim, I., Noorani, M. S. M., Said Al-Hadidi, M. R., Solving the generalized Burgers-Huxley Equation using the Adomian decomposition method, Math. Comput. Model., 43, 1404-1411, 2006.
  • Referans16 Javidi, M., A numerical solution of the generalized Burger's-Huxley equation by pseudospectral method and Darvishi's preconditioning, Appl. Math. Comput., 175, 1619-1628, 2006.
  • Referans17 Javidi, M., A numerical solution of the generalized Burger's-Huxley equation by spectral collocation method, Appl. Math. Comput., 178, 338-344, 2006.
  • Referans18 Darvishi, M. T., Kheybari, S., Khani, F., Spectral collocation method and Darvishi's preconditionings to solve the generalized Burgers-Huxley equation, Commun. Nonlinear Sci. Numer. Simul., 13, 2091-2103, 2008.
  • Referans19 Batiha, B., Noorani, M. S. M., Hashim, I., Application of variational iteration method to the generalized Burgers-Huxley equation, Chaos Soliton Fract., 36, 660-663, 2008.
  • Referans20 Sari, M., Gürarslan, G., Numerical solutions of the generalized Burgers-Huxley equation by a differential quadrature method, Math. Probl. Eng., doi: 10.1155/2009/370765, 2009.
  • Referans21 Khattak, A. J., A computational meshless method for the generalized Burger's-Huxley equation, Appl. Math. Model., 33, 3218-3729, 2009.
  • Referans22 Javidi, M., Golbabai, A., A new domain decomposition algorithm for generalized Burger's-Huxley equation based on Chebyshev polynomials and preconditioning, Chaos Soliton Fract., 39, 849-857, 2009.
  • Referans23 Biazar, J., Mohammadi, F., Application of differential transform method to the generalized Burgers-Huxley equation, Appl. Appl. Math., 5, 1726-1740, 2010.
  • Referans24 Bratsos, A. G., A fourth order improved numerical scheme for the generalized Burgers-Huxley equation, American J. Comput. Math., 1, 152-158, 2011.
  • Referans25 Çelik, İ., Haar wavelet method for solving generalized Burgers-Huxley equation, Arab J. Math. Sci., 18, 25-37, 2012.
  • Referans26 El-Kady, M., El-Sayed, S. M., Fathy, H. E., Development of Galerkin method for solving the generalized Burger's Huxley equation, Math. Probl. Eng., doi: 10.1155/2013/165492, 2013.
  • Referans27 Al-Rozbayani, A. M., Discrete Adomian decomposition method for solving Burger's-Huxley Equation, Int. J. Contemp. Math. Sci., 8, 623-631, 2013.
  • Referans28 Ismail, H. N. A., Raslan, K., Rabboh, A. A. A., Adomian decomposition method for Burger's-Huxley and Burger's-Fisher equations, Appl. Math. Comput., 159, 291-301, 2004.
  • Referans29 Macías-Díaz, J. E., On the numerical and structural properties of a logarithmic scheme for diffusion-reaction equations, Appl. Num. Math., 140, 104-114, 2019.
  • Referans30 İnan, B., A logarithmic finite difference technique for numerical solution of the generalized Huxley equation, Proceedings of 7th International Eurasian Conference on Mathematical Sciences and Applications, Kyiv, Ukraine, pp. 100-101, 2018.
  • Referans31 İnan, B., High accuracy numerical solutions by logarithmic finite difference method for the generalized Burgers-Huxley equation, Proceedings of 2nd International Conference on Mathematical and Related Sciences, Antalya, Turkey, pp.29, 2019.
  • Referans32 Macías-Díaz, J. E., İnan, B., Structural and numerical analysis of an implicit logarithmic scheme for diffusion equations with nonlinear reaction, Int. J. Modern Physics C, 30, 9, 1950065, 2019.
  • Referans33 Macías-Díaz, J. E., Hendy, A. S., On the stability and convergence of an implicit logarithmic scheme for diffusion equations with nonlinear reaction, J. Math. Chemistry, 58, 735–74, 2020.
There are 33 citations in total.

Details

Primary Language English
Subjects Applied Mathematics
Journal Section Mathematics
Authors

Bilge Inan 0000-0002-6339-5172

Publication Date June 30, 2022
Submission Date December 24, 2021
Acceptance Date May 13, 2022
Published in Issue Year 2022 Volume: 12 Issue: 1

Cite

APA Inan, B. (2022). On Two Efficient Numerical Schemes for Nonlinear Burgers’ Type Equations. Adıyaman University Journal of Science, 12(1), 70-87. https://doi.org/10.37094/adyujsci.1041652
AMA Inan B. On Two Efficient Numerical Schemes for Nonlinear Burgers’ Type Equations. ADYU J SCI. June 2022;12(1):70-87. doi:10.37094/adyujsci.1041652
Chicago Inan, Bilge. “On Two Efficient Numerical Schemes for Nonlinear Burgers’ Type Equations”. Adıyaman University Journal of Science 12, no. 1 (June 2022): 70-87. https://doi.org/10.37094/adyujsci.1041652.
EndNote Inan B (June 1, 2022) On Two Efficient Numerical Schemes for Nonlinear Burgers’ Type Equations. Adıyaman University Journal of Science 12 1 70–87.
IEEE B. Inan, “On Two Efficient Numerical Schemes for Nonlinear Burgers’ Type Equations”, ADYU J SCI, vol. 12, no. 1, pp. 70–87, 2022, doi: 10.37094/adyujsci.1041652.
ISNAD Inan, Bilge. “On Two Efficient Numerical Schemes for Nonlinear Burgers’ Type Equations”. Adıyaman University Journal of Science 12/1 (June 2022), 70-87. https://doi.org/10.37094/adyujsci.1041652.
JAMA Inan B. On Two Efficient Numerical Schemes for Nonlinear Burgers’ Type Equations. ADYU J SCI. 2022;12:70–87.
MLA Inan, Bilge. “On Two Efficient Numerical Schemes for Nonlinear Burgers’ Type Equations”. Adıyaman University Journal of Science, vol. 12, no. 1, 2022, pp. 70-87, doi:10.37094/adyujsci.1041652.
Vancouver Inan B. On Two Efficient Numerical Schemes for Nonlinear Burgers’ Type Equations. ADYU J SCI. 2022;12(1):70-87.
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