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

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.