Numerical solution of a coupled Korteweg–de Vries equations by collocation method

Abstract

A numerical method for solving the coupled Korteweg-de Vries (CKdV) equation based on the collocation method with quintic B-spline finite elements is set up to simulate the solution of CKdV equation. Invariants and error norms are studied wherever possible to determine the conservation properties of the algorithm. Simulation of single soliton, interaction of two solitons, and birth of solitons are presented. A linear stability analysis shows the scheme to be unconditionally stable.

Numerical simulations of the improved Boussinesq equation

“…By using the approximation (11) and quintic B-splines (10), nodal values U and their first and second derivatives U x and U xx at the knots x m are obtained in terms of the element parameters as follows:…”

A numerical treatment based on Haar wavelets for coupled KdV equation

“…For the periodic initial boundary value problem of the cKdV system a finite difference scheme produced by Wazwaz [15]. By using collocation method and quintic splines Ismail [16] solved cKdV system. A quadratic B-spline Galerkin approach applied by Kutluay and Ucar [17] for solving cKdV system.…”

Finite difference method with different high order approximations for solving complex equation

“…Such as the Korteweg-de Vries (KdV) equation [3,4,5,6] and the nonlinear Schrodinger equation has been solved by [7,8]. Numerical solution of coupled partial differential equations, as an example, the coupled nonlinear Schrodinger equation admits soliton solution and it has many applications in communication, this system has been solved numerically by Ismail [9,10,11,12] and the coupled Korteweg-de Vries equation has been solved numerically [13,14,15,16]. The complex nonlinear partial differential equations have been solved in [17,18,19,20,21].…”