A Collocation method for the KdV-Kawahara equation by trigonometric quintic B-spline basis

B. Karaagac   ; A. Esen   ; K. M. Owolabi   ; E. Pindza

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 26 (2023) no. 3, pp. 263-276

Résumé

In this paper, an efficient numerical method which is a collocation method is considered in order to obtain numerical solutions of the KdV-Kawahara equation. The numerical method is based on a finite element formulation and a spline interpolation by trigonometric quintic B-spline basis. Firstly, the KdV-Kawahara equation is split into a coupled equation via an auxiliary variable as $v=u_{xxx}$. Subsequently, a collocation method is applied to the coupled equation together with the forward difference and the Cranck-Nicolson formula. This application leads us to obtain an algebraic equation system in terms of time variables and trigonometric quintic B-spline basis. In order to measure the error between numerical solutions and exact ones, the error norms $L_2$ and $L_\infty$. are calculated successfully. The results are illustrated by means of two numerical examples with their graphical representations and comparisons with other methods.

Bibliographie

[1] Ali A.H.A., Gardner G.A., and Gardner L.R.T., “A collocation solution for Burgers' equation using cubic B-spline finite elements”, Computer Methods in Applied Mechanics and Engineering, 100:3 (1992), 325–337 | DOI | MR | Zbl

[2] Kutluay S., Ucar Y., and Yagmurlu N.M., “Numerical solutions of the modified Burgers equation by a cubic B-spline collocation method”, Bulletin of the Malaysian Mathematical Sciences Society, 39:4 (2016), 1603–1614 | DOI | MR | Zbl

[3] Hepson O.E. and Dag I., “Finite element method for Schnakenberg model”, Mathematical Methods in Engineering, Springer, Cham, 2019, 41–51 | DOI | MR

[4] Mittal R.C. and Rohila R., “A fourth order cubic B-spline collocation method for the numerical study of the RLW and MRLW equations”, Wave Motion, 80 (2018), 47–68 | DOI | MR | Zbl

[5] Yagmurlu N.M., Karaagac B., and Kutluay S., “Numerical solutions of Rosenau-RLW equation using Galerkin cubic B-spline finite element method”, American J. of Computational and Applied Mathematics, 7:1 (2017), 1–10 | MR

[6] Ali K., Raslan K.R., and Hadhoud A.R., “Numerical studies of non-local hyperbolic partial differential equations using collocation methods”, Computational Methods for Differential Equations, 6:3 (2018), 326–338 | MR | Zbl

[7] Dag I., Irk D., Ka?maz O., Adar N., “Trigonometric B-spline collocation algorithm for solving the RLW equation”, Applied and Computational Mathematics, 15:1 (2016), 96–105 | MR | Zbl

[8] Ömer O., Esen A., and Bulut F., “A Haar wavelet collocation method for coupled nonlinear Schrödinger-KdV equations”, Intern. J. of Modern Physics C, 27:09 (2016), 1–16 | MR

[9] Rohila R. and Mittal R.C., “Numerical study of reaction diffusion Fisher's equation by fourth order cubic B-spline collocation method”, Mathematical Sciences, 12 (2018), 79–89 | DOI | MR | Zbl

[10] Yağmurlu N.M., Taşbozan O., Uçar Y., Esen A., “Numerical solutions of the combined KdV-mKdV equation by a quintic B-spline collocation method”, Appl. Math. Inf. Sci. Lett., 4:1 (2016), 19–24 | MR

[11] Owolabi K.M., “Robust IMEX schemes for solving two-dimensional reaction-diffusion models”, Intern. J. of Nonlinear Sciences and Numerical Simulation, 16 (2015), 271–284 | DOI | MR | Zbl

[12] Kassam A.K., Trefethen L.N., “Fourth-order time-stepping for stiff PDEs”, SIAM J. Sci. Comput., 26 (2005), 1214–1233 | DOI | MR | Zbl

[13] Owolabi K.M., Second or fourth-order finite difference operators, which one is most effective?, Intern. J. of Applied Mathematics and Statistics, 1:3 (2014), 044–054

[14] Hochbruck M., Ostermann A., “Exponential multistep methods of Adams-type”, BIT Numer. Math., 51 (2011), 889–908 | DOI | MR | Zbl

[15] Owolabi K.M., “Mathematical study of two-variable systems with adaptive numerical methods”, Numerical Analysis and Applications, 9 (2016), 218–230 | DOI | MR | Zbl

[16] Pindza E. and Owolabi K.M., “Fourier spectral method for higher order space fractional reaction-diffusion equations”, Communications in Nonlinear Science and Numerical Simulation, 40 (2016), 112–128 | DOI | MR

[17] Ceballos J.C., Sepulveda M., and Villagran O.P.V., “The Korteweg-de Vries-Kawahara equation in a bounded domain and some numerical results”, Applied Mathematics and Computation, 190:1 (2007), 912–936 | DOI | MR | Zbl

[18] Koley U., “Finite difference schemes for the Korteweg-de Vries-Kawahara equation”, Intern. J. of Numerical Analysis and Modeling, 13:3 (2016), 344–367 | MR | Zbl

[19] Koley U., Convergence of numerical schemes for the Korteweg-de Vries-Kawahara equation, 2011, arXiv: 1106.1781v1

[20] Sepulveda M., Paulo O., and Villagran O.P.V., “Numerical methods for a transport equation perturbed by dispersive terms of 3rd and 5th order”, Scientia. Series A. Mathematical Sciences. New Series, 13 (2006), 13–21 | MR | Zbl

[21] Alam M.P., Kumar D., and Khan A., “Trigonometric quintic B-spline collocation method for singularly perturbed turning point boundary value problems”, Intern. J. of Computer Mathematics, 98:5 (2021), 1029–1048 | DOI | MR | Zbl

[22] Chertock A. and Levy D., “A particle method for the KdV equation”, J. of Scientific Computing, 17 (2002), 491–499 | DOI | MR | Zbl

[23] Owolabi K.M. and Patidar K.C., “Robust numerical simulation of nonlinear stiff higher-order PDEs”, Transylvanian Review, 24:6 (2016), 509–526 | MR