## Application of the collocation method with B-splines to the GEW equation

Halil Zeybek and S. Battal Gazi Karakoç

### Abstract

In this paper, the generalized equal width (GEW) wave equation is solved numerically by using a quintic B-spline collocation algorithm with two different linearization techniques. Also, a linear stability analysis of the numerical scheme based on the von Neumann method is investigated. The numerical algorithm is applied to three test problems consisting of a single solitary wave, the interaction of two solitary waves, and a Maxwellian initial condition. In order to determine the performance of the numerical method, we compute the error in the $L_{2}$- and $L_{\infty }$- norms and in the invariants $I_1$, $I_2$, and $I_3$ of the GEW equation. These calculations are compared with earlier studies. Afterwards, the motion of solitary waves according to different parameters is designed.

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### Key words

GEW equation, finite element method, quintic B-spline, soliton, solitary waves

### AMS subject classifications

41A15, 65N30, 76B25

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