An Application of Trigonometric Quintic B-Spline Collocation Method for Sawada-Kotera Equation

Year 2022, Volume: 12 Issue: 2, 269 - 282, 30.12.2022

Hatice Karabenli , Alaattin Esen , Murat Yağmurlu

https://doi.org/10.37094/adyujsci.1156498

Abstract

In this paper, we deal with the numerical solution of Sawada-Kotera (SK) equation classified as the type of fifth order Korteweg-de Vries (gfKdV) equation. In the first step of our study consisting of several steps, nonlinear model problem is split into the system with the coupled new equations by using the transformation w_xxx=v. In the second step, to get rid of the nonlinearity of the problem, Rubin-Graves type linearization is used. After these applications, the approximate solutions are obtained by using the trigonometric quintic B-Spline collocation method. The efficiency and accuracy of the present method is demonstrated with the tables and graphs. As it is seen in the tables given with the error norms L_2 and L_∞ for different time and space steps, the present method is more accurate for the larger element numbers and smaller time steps.

Keywords

Sawada-Kotera Equation, Collocation Finite Element Method, Trigonometric Quintic B-Spline, Rubin-Graves Type Linearization

Sawada-Kotera Denklemi için Trigonometrik Beşli Baz Fonksiyonları Kollokasyon Yönteminin Bir Uygulaması

Abstract

Bu çalışmada, beşinci dereceden Korteweg-de Vries (gfKdV) denklemlerinin türü olarak sınıflandırılan Sawada-Kotera (SK) denkleminin nümerik çözümü ele alınmaktadır. Birkaç adımdan oluşan çalışmamızın ilk adımında, lineer olmayan model problem w_xxx=v dönüşümü kullanılarak iki yeni denklem sistemine ayrıştırılmıştır. İkinci adımda, problemin lineer olmama durumundan kurtulmak için Rubin-Graves tipi lineerleştirme kullanılmıştır. Bu uygulamalardan sonra trigonometrik beşli B-Spline kollokasyon yöntemi kullanılarak yaklaşık çözümler elde edilmiştir. Mevcut yöntemin etkinliği ve doğruluğu tablolar ve grafiklerle gösterilmiştir. Farklı zaman ve konum adımı için L_2 ve L_∞ hata normları ile verilen tablolardan görüldüğü üzere, mevcut yöntem daha büyük eleman sayıları ve daha küçük zaman adımları için yüksek doğruluktadır.

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