A CUBIC HERMITE COLLOCATION APPROACH FOR SOLVING THE KORTEWEG–DE VRIES–BURGERS EQUATION

Year 2025, Volume: 26 Issue: 4, 445 - 459, 25.12.2025

Murat Arı

https://doi.org/10.18038/estubtda.1743984

Abstract

This study presents a robust numerical approach for solving the nonlinear Korteweg–de Vries–Burgers (KdVB) equation using the Cubic Hermite Collocation Method. The method employs piecewise cubic Hermite basis functions, which ensure both high-order accuracy and smooth derivative continuity across element boundaries. These features make the method particularly suitable for problems involving sharp gradients or smooth solution profiles. The proposed scheme is rigorously tested on a set of benchmark problems to demonstrate its effectiveness in accurately capturing the complex interplay between the dispersive and dissipative behavior inherent to the KdVB equation. Numerical results which are given with L_2 and L_∞ error norms exhibit excellent agreement with known analytical or previously published numerical solutions, and confirm the method’s stability, efficiency, and reliability. In addition, graphical representations of the numerical solutions are provided to visually illustrate the method’s performance. Due to its flexibility, accuracy, and ease of implementation, the Cubic Hermite Collocation Method proves to be a promising and efficient alternative for the numerical solution of nonlinear PDEs with mixed physical effects.

Keywords

Korteweg–de Vries–Burgers equation , Cubic Hermite collocation method , Legendre roots , Chebyshev roots

References

• [1] Su CH, Gardner CS. Derivation of the Kortewege–de Vries and Burgers’ equation, J. Math. Phys. 1969, 10, 536–539.
• [2] Soliman AA. The modified extended tanh-function method for solving Burgers-type equations. Physica A: Statistical Mechanics and its Applications, 2006, 361(2), 394-404.
• [3] Wang GW, Liu YT, Zhang YY. New exact solutions to the compound KdV-Burgers system with nonlinear terms of any order. Afrika Matematika, 2014, 25(2), 357-362.
• [4] Wazzan LA. More soliton solutions to the KdV and the KdV -Burgers, Proc. Pakistan Acad. Sci. 2007, 44(2):117-120.
• [5] Shu JJ. The proper analytical solution of the Korteweg-de Vries-Burgers equation. Journal of Physics A: Mathematical and General, 1987, 20(2), L49-L56.
• [6] Yuanxi X, Jiashi T. New solitary wave solutions to the KdV-Burgers equation. International Journal of Theoretical Physics, 2005, 44(3), 293-301.
• [7] Korkut SÖ, İmamoğlu KN. A reliable explicit method to approximate the general type of the kdv–burgers’ equation, Iranian Journal of Science and Technology, Transactions A: Science 2022, 1–11.
• [8] Ahmad H, Seadawy AR, Khan TA. Numerical solution of korteweg–de vries-burgers equation by the modified variationaliteration algorithm-ii arising in shallow water waves, Physica Scripta, 2020, 95 (4) 045210.
• [9] Benkhaldoun F, Seaid M. New finite-volume relaxation methods for the third-order differential equations, Commun. Comput. Phys, 2008, 4 (820-837) 3.
• [10] Abbasbandy S. The application of homotopy analysis method to solve a generalized hirota–satsuma coupled kdv equation, Physics Letters A, 2007, 361 (6) 478–483.
• [11] Bektas M, Inc M, Cherruault Y. Geometrical interpretation and approximate solution of non-linear kdv equation, Kybernetes 34 (7/8) (2005) 941–950.
• [12] Wang Q. Numerical solutions for fractional KdV–Burgers equation by Adomian decomposition method, Applied Mathematics and Computation, 2006, Volume 182, Issue 2, Pages 1048-1055.
• [13] Assas LMB. Approximate solutions for the generalized KdV–Burgers' equation by He's variational iteration method Physica Scripta, 2007, Volume 76, Number 2, 76 161
• [14] Karaagac B, Esen A, Owolabi KM, Pindza E. A collocation method for solving time fractional nonlinear Korteweg–de Vries–Burgers equation arising in shallow water waves. Int J Mod Phys C, 2023, 34(07):2350096.
• [15] El Sadat R, Ali MR. Application of the method of lines for solving the KdV–Burger equation. BISKA NTMSCT, 2017, 12:39–51.
• [16] Saka B, Dağ I. Quartic B-spline Galerkin approach to the numerical solution of the KdVB equation. Applied Mathematics and Computation, 2009, 215, 746–758.
• [17] Kudryashov NA. On ’new travelling wave solutions’ of the KdV and the KdV-Burgers equations. Communications in Nonlinear Science and Numerical Simulation, 2009, 14(5), 1891-1900.
• [18] Haq S, Islam S.U, Uddin M. A mesh-free method for the numerical solution of the KdV-Burgers equation. Applied Mathematical Modelling, 2009, 33(8), 3442-3449.
• [19] Mittal AK, Ganaie IA, Kukreja VK, Parumasur N, Singh P. Solution of diffusion–dispersion models using a computationally efficient technique of orthogonal collocation on finite elements with cubic Hermite as basis. Computer and Chemical Engineering, 2013, 58, 203–210.
• [20] Ganaie IA, Kukreja VK. Numerical solution of Burgers’ equation by cubic Hermite collocation method. Applied Mathematics and Computation, 2014, 237, 571–581.
• [21] Ganaie IA, Arora S, Kukreja VK. Cubic Hermite collocation solution of Kuramoto–Sivashinsky equation. International Journal of Computer Mathematics, 2016, 93(1), 223–235.
• [22] Arora S, Kaur I. Applications of Quintic Hermite collocation with time discretization to singularly perturbed problems. Applied Mathematics and Computation, 2018, 316, 409–421.
• [23] Arora S, Jain R, Kukreja VK. Solution of Benjamin-Bona-Mahony-Burgers equation using collocation method with quintic Hermite splines. Applied Numerical Mathematics, 2020, 154, 1–16.
• [24] Yousaf A, Abdeljawad T, Yaseen M, Abbas M. Novel cubic trigonometrical B-spline approach based on the Hermite formula for solving the convection–diffusion equation. Mathematical Problems in Engineering, 2020.
• [25] Kaur SP, Mittal AK, Kukreja VK, Kaundal A, Parumasur N, Singh P. Analysis of a linear and non-linear model for diffusion–dispersion phenomena of pulp washing by using quintic Hermite interpolation polynomials. Afrika Matematika, 2021, 32, 997–1019.
• [26] Kudryashov NA. Generalized Hermite polynomials for the Burgers hierarchy and point vortices. Chaos, Solitons and Fractals, 2021, 151, 111256.
• [27] Kumari A, Kukreja VK. Robust septic Hermite collocation technique for singularly perturbed generalized Hodgkin–Huxley equation. International Journal of Computer Mathematics, 2021.
• [28] Kumari A, Kukreja VK. Septic Hermite collocation method for the numerical solution of Benjamin–Bona–Mahony–Burgers equation. Journal of Difference Equations and Applications, 2021, 27, 1193–1217.
• [29] Kutluay S, Yağmurlu NM, Karakaş AS. A novel perspective for simulations of the modified equal-width wave equation by cubic Hermite B-spline collocation method. Wave Motion, 2024, 129, 103342.
• [30] Kutluay S, Yağmurlu M, Karakaş AS. A robust quintic Hermite collocation method for one-dimensional heat conduction equation. Journal of Mathematical Sciences and Modelling, 2024, 7(2), 82–89. https://doi.org/10.33187/jmsm.1475294
• [31] Kutluay S, Yağmurlu NM, Karakaş AS. A powerful robust cubic Hermite collocation method for the numerical calculations and simulations of the equal width wave equation. arXiv preprint, 2023, arXiv:2309.02439.
• [32] Kutluay S, Yağmurlu NM, Karakaş AS. A novel perspective for simulations of the modified equal-width wave equation by cubic Hermite B-spline collocation method. Wave Motion, 2024, 129, 103342.
• [33] Ganaie IA, Gupta B, Parumasur N, Singh P, Kukreja VK. Asymptotic convergence of cubic Hermite collocation method for parabolic partial differential equation. Applied Mathematics and Computation, 2013, 220, 560–567.
• [34] Hepson OE, Korkmaz A, Dağ İ. Extended b-spline collocation method for KdV-Burgers equation, TWMS J. App. Eng. Math. 2019, V.9, N.2, pp. 267-278.
• [35] Saka B, Dağ İ. Quartic B-spline collocation method to the numerical solutions of the Burgers’ equation. Chaos, Solitons & Fractals, 2007, 32(3), 1125–1137.
• [36] Dağ İ, Irk D, Kaçmaz O, Adar N. Trigonometric B-spline collocation algorithm for solving the RLW equation. Applied and Computational Mathematics, 2016, 15(1), 96–105.
• [37] Irk D. Sextic B-spline collocation method for the modified Burgers' equation. Kybernetes, 2009, 38(9), 1599–1620.
• [38] Korkmaz B, Dereli Y. Numerical solution of the Rosenau–KdV–RLW equation by using RBFs collocation method. International Journal of Modern Physics C, 2016, 27(10), 1650117.
• [39] Kaplan AG, Dereli Y. Numerical solutions of the GEW equation using MLS collocation method. International Journal of Modern Physics C, 2017, 28(1), 1750011.
• [40] Kaplan A, Dereli Y. Numerical solutions of the MRLW equation using moving least square collocation method. Communications Faculty of Sciences University of Ankara, Series A1: Mathematics and Statistics, 2017, 66(2), 349–361.
• [41] Arı M, Dereli Y. Numerical solutions of Boussinesq type equations by meshless methods. Estuscience - Se, 2024, 25(3), 471–484.
• [42] Arı M, Karaman B, Dereli Y. Comparison between two meshless methods for the Rosenau–Kawahara equation. International Journal of Applied Mathematics and Statistics, 2018, 57(4), 117–124.
• [43] Finlayson BA. Non Linear Analysis in Chemical Engineering, Mc-Graw Hill, New York, 1980.
• [44] Douglas J, Dupont T. A finite element collocation method for quasilinear parabolic equations, Math. Comput., 1973, 121, 17–28.
• [45] Prenter PM. Splines and Variational Methods, John Wiley, New York, 1975.
• [46] Kaya D. An application of the decomposition method for the KdVB equation, Appl. Math. Comput. 2004, 152, 279-288.