Collocation method with quintic b-spline method for solving hirota-satsuma coupled KDV equation
DOI:
https://doi.org/10.14419/ijamr.v5i2.6138Published:
2016-05-17Keywords:
Collocation Method, Quintic B-Splines Method, Coupled KDV EquationAbstract
In the present paper, a numerical method is proposed for the numerical solution of a coupled system of KdV (CKdV) equation with appropriate initial and boundary conditions by using collocation method with quintic B-spline on the uniform mesh points. The method is shown to be unconditionally stable using von-Neumann technique. To test accuracy the error norms, are computed. Three invariants of motion are predestined to determine the preservation properties of the problem, and the numerical scheme leads to careful and active results. Furthermore, interaction of two and three solitary waves is shown. These results show that the technique introduced here is easy to apply. We make linearization for the nonlinear term.
References
[1] R. Hirota and J. Satsuma, “Soliton solutions of a coupled Korteweg-de Vries equation,†Physics Letters a, vol. 85, no. 8-9, pp. 407–408, 1981. http://dx.doi.org/10.1016/0375-9601(81)90423-0.
[2] A. A. Halim, S. P. Kshevetskii, and S. B. Leble, “Numerical integration of a coupled Korteweg-de Vries system,†Computers & Mathematics with Applications, vol. 45, no. 4-5, pp. 581–591, 2003. http://dx.doi.org/10.1016/S0898-1221(03)00018-X.
[3] A. A. Halim and S. B. Leble, “Analytical and numerical solution of a coupled KdV- MKdV System,†Chaos, Solitons and Fractals, vol. 19, no. 1, pp. 99–108, 2004. http://dx.doi.org/10.1016/S0960-0779(03)00085-7.
[4] M. S. Ismail, “Numerical solution of a coupled Korteweg-de Vries equations by collocation method,†Numerical Methods for Partial Differential Equations, vol. 25, no. 2, pp. 275–291, 2009. http://dx.doi.org/10.1002/num.20343.
[5] D. Kaya and I. E. Inan, “Exact and numerical traveling wave solutions for nonlinear coupled equations using symbolic computation,†Applied Mathematics and Computation, vol. 151, no. 3, pp. 775–787, 2004. http://dx.doi.org/10.1016/S0096-3003(03)00535-6.
[6] M. S. Ismail and H. A. Ashi, “A Numerical Solution for Hirota-Satsuma Coupled KdV Equation,†Abstract and Applied Analysis, Volume 2014, 9 pages, 2014.
[7] L. M. B. Assas, “Variational iteration method for solving coupled-KdV equations,†Chaos, Solitons and Fractals, vol. 38, no. 4, pp. 1225–1228, 2008. S. Abbasbandy, “The application of homotopy analysis method to solve a generalized http://dx.doi.org/10.1016/j.chaos.2007.02.012.
[8] Hirota-Satsuma coupled KdV equation,†Physics Letters A: General, Atomic and Solid State Physics, vol. 361, no. 6, pp. 478–483, 2007.
[9] A. Wazwaz, “The KdV equation,†in Handbook of Differential Equations: Evolutionary Equations. VOL. IV, Handb. Differ. Equ., pp. 485–568, Elsevier/North- Holland, Amsterdam, the Netherlands, 2008. http://dx.doi.org/10.1016/S1874-5717(08)00009-1.
[10] S. Kutluay and Y.Ucar, “A quadratic B-spline Galerkin approach for solving a coupled KdV equation,†Mathematical Modelling and Analysis, vol. 18, no. 1, pp.
[11] M. S. Ismail, “Numerical solution of coupled nonlinear Schr¨odinger equation by Galerkin method,†Mathematics and Computers in Simulation, vol. 78, no. 4, pp. 532– 547, 2008. http://dx.doi.org/10.1016/j.matcom.2007.07.003.
[12] M. S. Ismail and T. R. Taha, “A linearly implicit conservative scheme for the coupled nonlinear Schr¨odinger equation,†Mathematics and Computers in Simulation, vol. 74, no. 4-5, pp. 302–311, 2007. http://dx.doi.org/10.1016/j.matcom.2006.10.020.
[13] M. S. Ismail and S. Z. Alamri, “Highly accurate finite difference method for coupled nonlinear Schr¨odinger equation,†International Journal of Computer Mathematics, vol. 81, no. 3, pp. 333–351, 2004. http://dx.doi.org/10.1080/00207160410001661339.
[14] T. S. EL-Danaf, K. R. Raslan and Khalid K. Ali," collocation method with cubic B- Splines for solving the GRLW equation", Int. J. of Num. Meth. And Appl. Vol. 15 (1), pp. 39-59, 2016.
[15] Rubin SG, Graves RA. Cubic spline approximation for problems in fluid mechanics. Nasa TR R-436. Washington, DC; 1975.
[16] T. S. EL-Danaf, K. R. Raslan and Khalid K. Ali,"New Numerical treatment for the Generalized Regularized Long Wave Equation based on finite difference scheme", Int. J. of S. Comp. and Eng. (IJSCE)’, 4, 16-24, 2014.
License
Authors who publish with this journal agree to the following terms:
                          [1]           Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
                          [2]           Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
                          [3]           Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).