A Collocation method for the KdV-Kawahara equation by trigonometric quintic B-spline basis
Bcrat Karaagac1, Alaattin Esen2, Kolade Malthew Owolabi3, Edson Pindza4,5
1Department of Mathematics Education, Adiyaman University, Adiyaman, Turkey
2Department of Mathematics, Inonu University, Malatya, Turkey
3Department of Mathematical Sciences, Federal University of Technology Akure, Akure, Nigeri
4Department of Mathematics and Applied Mathematics University of Pretoria, Pretoria West, South Africa
5Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria West, South Africa
Keywords: KdV-Kawahara equation, collocation method, quintic trigonometric B-spline basis, stability
Abstract
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 υ=uxxx. 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 L2 and L∞. are calculated successfully. The results are illustrated by means of two numerical examples with their graphical representations and comparisons with other methods.
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