Abstract
A numerical approach to approximate the nonlinear Burgers’ equation solution is presented in this article. By temporally discretizing the problem using the Crank-Nicolson scheme, we present a discrete formulation along with approximate solutions at each time step. Next, we express these approximations in terms of the Chelyshkov polynomial basis. Then, Newton’s method is utilized to solve the resulting nonlinear system of equations to find the values of unknowns. Three numerical examples are provided to assess the accuracy and efficiency of the suggested method. The proposed approach demonstrates more accurate numerical results than those existing in the literature.
Similar content being viewed by others
Data availability
No data were used to support this study.
References
Whitham, G.B.: Lectures on Wave Propagation. Narosa Pub House, New Delhi (1979)
Kevorkian, J.: Partial Differential Equations: Analytical Solution Techniques. Brooks/Cole Pub, Pacific Grove (1990)
Burgers, J.M.: The Nonlinear Diffusion Equation. Reidel, Dordrecht (1974)
Quarteroni, A., Sacco, R., Saleri, F.: Méthodes Numériques: Algorithmes, analyse et applications. Springer, Berlin (2008)
Benia, Y., Khaled, S.B.: Existence of solutions to burgers equations in Domains that can be transformed into Rectangles. Electron. J. Differ. Eqn. 2016(157), 1–13 (2016)
Biazar, J., Aminikhah, H.: Exact and numerical solutions for non-linear Burger’s equation by VIM. Math. Comput. Model. 49(7), 1394–1400 (2009)
Asaithambi, A.: Numerical solution of the Burgers’ equation by automatic differentiation. Appl. Math. Comput. 216(9), 2700–2708 (2010)
Umar M., Helil N., Rahman K.: Fourth-order finite difference approach for numerical solution of Burgers equation. In: International Conference on Multimedia Information Networking and Security. pp 603–607 (2010 ).
Mittal, R.C., Jain, R.K.: Numerical solutions of nonlinear Burgers’ equation with modified cubic B-splines collocation method. Appl. Math. Comput. 218(15), 7839–7855 (2012)
Kutluay, S., Esen, A., Dag, I.: Numerical solutions of the Burgers’ equation by the least-squares quadratic B-spline finite element method. J. Comput. Appl. Math. 167(1), 21–33 (2004)
Uçar, Y., Yağmurlu, M., Çelikkaya, İ: Numerical solution of Burger’s Type equation using finite element collocation method with strange splitting. Math. Sci. Appl. 8(1), 29–45 (2020)
Zhao, J., Li, H., Fang, Z., Bai, X.: Numerical solution of Burgers’ equation based on mixed finite volume element methods. Discret. Dyn. Nat. Soc. p 13 (2020)
Al-Shaher, O.I., Mechee, M.S.: A study of Legendre polynomials approximation for solving initial value problems. J. Phys: Conf. Ser. 1897(1), 012058 (2021)
Alotaibi A.M., El-Moneam M.A., Badr Badr S.: The solutions of Legendre’s and Chebyshev’s differential equations by using the differential transform method. Math. Probl. Eng. p 39 (2022)
Laouar, Z., Arar, N., Talaat, A.: Efficient spectral Legendre Galerkin approach for the advection diffusion equation with constant and variable coefficients under mixed Robin boundary conditions. Adv. Theory Nonlinear Anal. Appl. 7(1), 133–147 (2023)
Venkatesh, S.G., Ayyaswamy, S.K., Balachandar, Raja S.: Legendre approximation solution for a class of higher-order Volterra integro-differential equations. Ain Shams Engineering Journal 3(4), 417–422 (2012)
Laouar Z., Arar N., Ben Makhlouf A.: Spectral collocation method for handling integral and integrodifferential equations of n-th order via certain combinations of shifted Legendre polynomials. Math. Probl. Eng. p 10 (2022)
Talaei Y., Noeiaghdam S., Hosseinzadeh H.: Numerical solution of fractional Fredholm integro-differential equations by spectral method with fractional basis functions. arXiv preprint arXiv:2209.10912 (2022).
Duangpan, A., Boonklurb, R., Treeyaprasert, A.: Finite integration method with shifted Chebyshev polynomials for solving time-fractional Burgers’ equations. Mathematics 7(12), 1201 (2019)
Sadri, K., Hosseini, K., Baleanu, D., et al.: Bivariate Chebyshev polynomials of the fifth kind for variable-order time-fractional partial integro-differential equations with weakly singular kernel. Adv. Difference Equ. 348, 1–26 (2021)
Öztürk, Y., Mutlu, U.: Numerical solution of fractional differential equations using fractional Chebyshev polynomials. Asian-Eur. J. Math. 15(3), 2250048 (2022)
Laouar Z., Arar N., Ben Makhlouf A.: Theoretical and numerical study for Volterra-Fredholm fractional integro-differential equations based on Chebyshev polynomials of the third kind. Complexity, p 13 (2023)
Talaei, Y.: Chelyshkov collocation approach for solving linear weakly singular Volterra integral equations. J. Appl. Math. Comput. 60(1–2), 201–222 (2019)
Oğuz, C., Sezer, M.: Chelyshkov collocation method for a class of mixed functional integro-differential equations. Appl. Math. Comput. 259, 943–954 (2015)
Chelyshkov, V.S.: Alternative orthogonal polynomials and quadratures. Electron. Trans. Numer. Anal. 25, 17–26 (2006)
Rasty, M., Hadizadeh, M.: A product integration approach based on new orthogonal polynomials for nonlinear weakly singular integral equations. Acta Appl. Math. 109, 861–873 (2010)
Mohammadi, F., Hassani, H.: Numerical solution of time-fractional telegraph equation by using a new class of orthogonal polynomials. Bol. Soc. Paran. Mat. 40, 1–13 (2022)
Talaei, Y., Asgari, M.: An operational matrix based on Chelyshkov polynomials for solving multi-order fractional differential equations. Neural Comput. Appl. 28, 1–7 (2017)
Ngo, H.T.B., Razzaghi, M., Vo, T.N.: Fractional-order Chelyshkov wavelet method for solving variable-order fractional differential equations and an application in variable-order fractional relaxation system. Numer. Algorithms 92, 1571–1588 (2022)
Hamid, M., Usman, M., Haq, R.U., Wang, W.: A Chelyshkov polynomial based algorithm to analyze the transport dynamics and anomalous diffusion in fractional model. Phys. A 551, 124227 (2020)
Gulsu, M., Ozis, T.: Numerical solutions of Burger’s equation with restrictive Taylor approximation. Appl. Math. Comput. 171, 1192–1200 (2005)
Bahadir, A.R., Saglam, M.: A mixed finite difference a boundary element approach to one-dimensional Burgers’ equation. Appl. Math. Comput. 160, 8663–8673 (2005)
Funding
The authors have not disclosed any funding.
Author information
Authors and Affiliations
Contributions
This work was carried out as a collaboration between all the authors.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no potential conflict of interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Arar, N., Deghdough, B., Dekkiche, S. et al. Numerical Solution of the Burgers’ Equation Using Chelyshkov Polynomials. Int. J. Appl. Comput. Math 10, 33 (2024). https://doi.org/10.1007/s40819-023-01663-8
Accepted:
Published:
DOI: https://doi.org/10.1007/s40819-023-01663-8