Numerical solutions of the modified KdV Equation with collocation method

Seydi Battal Gazi Karakoc

doi:10.26637/MJM0604/0020

Abstract

In this article, numerical solutions of the modified Korteweg-de Vries (MKdV) equation have been obtained by a numerical technique attributed on collocation method using quintic B-spline finite elements. The suggested numerical scheme is controlled by applying three test problems involving single solitary wave, interaction of two and three solitary waves. To check the performance of the newly applied method, the error norms, $L_2$ and $L_{\infty}$, as well as the three lowest invariants, $I_1, I_2$ and $I_3$, have been calculated. The acquired numerical results are compared with some of those available in the literature. Linear stability analysis of the algorithm is also examined.

Keywords:

Modified Korteweg-de Vries equation, finite element method, collocation, quintic B-spline, soliton.

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

Seydi Battal Gazi Karakoc Department of Mathematics, Faculty of Science and Art, Nevsehir Haci Bektas Veli University, Nevsehir, 50300, Turkey.

Pages: 835-842
Date Published: 01-10-2018
Vol. 6 No. 04 (2018): Malaya Journal of Matematik (MJM)

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary wave, Philosophical Magazine. $39(1895), 422-443$.

N. J. Zabusky, A synergetic approach to problem of nonlinear dispersive wave propagation and interaction, in: $W$. Ames (Ed.), Proc. Symp. Nonlinear Partial Diff. Equations, Academic Press. (1967) 223-258.

B. Fornberg and G. B. Whitham, A numerical and theoretical study of certain nonlinear wavephenomena, Philos. Trans. Roy. Soc. 289(1978), 373-404.

N. J. Zabusky and M. D. Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett. 15(6)(1965), 240-243.

C. S Gardner, J. M. Green, M. D. Kruskal and R. M. Miura, Method for solving Korteweg- de Vries equation, Phys. Rev. 19(1967), 1095-1097.

$mathrm{K}$. Goda, On instability of some finite difference schemes for Korteweg-de Vries equation, J.Phys. Soc. Japan. 39(1975), 229-236.

A. C. Vliengenthart, On finite difference methods for the Korteweg-de Vries equation, J. Eng. Math. 5(1971), $137-155$.

A. A. Soliman, Collocation solution of the Korteweg-De Vries equation using septic splines, Int. J. Comput. Math. $81(2004), 325-331$

D. Irk, İ. Dağ and B. Saka, A small time solutions for the Korteweg-de Vries equation using spline approximation, Appl. Math. Comput. 173(2)(2006), 834-846.

A. Canıvar, M. Sarı and I. Dağ, A Taylor-Galerkin finite element method for the KdV equation using cubic Bsplines, Physica B. 405(2010), 3376-3383.

A. Korkmaz, Numerical algorithms for solutions of Korteweg-de Vries Equation, Numerical Methods for Partial Differential Equations. 26(6)(2010), 1504-1521.

Ö. Ersoy and I. Dağ, Cubic B-Spline Algorithm for Korteweg-de Vries Equation, Advances in Numerical Analysis, 2015(2015), 1-8.

E. N. Aksan and A. Ozdes, Numerical solution of Korteweg-de Vries equation by Galerkin B-spline finite element method, Applied Mathematics and Computation, $175(2006), 1256-1265$.

D. Irk, Quintic B-spline Galerkin method for the KdV equation, Anadolu University Journal of Science and Technology B- Theoritical Sciences. 5(2)(2017), 111-119.

B. Saka, Cosine expansion-based differential quadrature method for numerical solution of the $mathrm{KdV}$ equation, Chaos Soliton Fract. 40(2009),2181-2190.

S. Kutluay, A. R. Bahadır and A. Ozdes, A small time solutions for the Korteweg-de Vriesequation, Appl. Math. Comput. 107(2000), 203-210.

D. Kaya, An application for the higher order modified KdV equation by decomposition method, Commun. in Nonlinear Science and Num. Simul. 10(2005), 693-702.

A. Biswas and K. R .Raslan, Numerical simulation of the modified Korteweg-de Vries Equation, Physics of Wave Phenomena. 19(2)(2011), 142-147.

K. R. Raslan and H. A. Baghdady, A finite difference scheme for the modified Korteweg-de Vries equation, General Mathematics Notes. 27(1)(2015), 101-113.

K. R. Raslan and H. A. Baghdady, New algorithm for solving the modi ed Korteweg-de Vries (mKdV) equation, International Journal of Research and Reviews in Applied Sciences. $18(1)(2014), 59-64$.

A. M. Wazwaz, A variety of $(3+1)$-dimensional mKdV equations derived by using the $mathrm{mKdV}$ recursion operator, Computers and Fluids. 93(10)(2014), 41-45.

A. M. Wazwaz, New (3+1)-dimensional nonlinear evolution equations with $mathrm{mKdV}$ equation constituting its main part: multiple soliton solutions, Chaos, Solitons and Fractals. 76(2015), 93-97.

T. Ak, S. B. G. Karakoc and A. Biswas, A New Approach for Numerical Solution of Modified Korteweg-de Vries Equation, Iran J Sci Technol Trans Sci. 41(2017), 11091121.

T. Ak, S. B. G. Karakoc and A. Biswas, Application of Petrov-Galerkin nite element method to shallow water waves model: Modified Korteweg-de Vries equation, Scientia Iranica B. 24(3)(2017), 1148-1159.

Prenter P. M. Splines and Variational Methods. John Wiley & Sons, New York, NY.USA, (1975).

R. M. Miura, C. S. Gardner and M. D. Kruskal, Kortewegde Vries equation and generalizations. II. Existence of conservation laws and constants of motion, J. Math. Phys. $9(8)(1968), 1204-1212$.