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.

References

• Nagashima, H., Experiment on Solitary Waves in the Nonlinear Transmission Line Discribed by the Equation u_t +uu_ξ-u_ξξξξξ =0, Journal of the Physical Society of Japan, 47 (4), 1387-1388, 1979.
• Kawahara, T., Oscillatory solitary waves in dispersive media, Journal of the Physical Society of Japan, 33 (1), 260-264, 1972.
• Wazwaz, A.M., Analytic Study on the Generalized KdV Equation: New Solution and Periodic Solutions, Elsevier, Amsterdam, 2006.
• Bakodah, H.O., Modified Adomian Decomposition Method for the Generalized Fifth Order KdV Equations, American Journal of Computational Mathematics , 3, 53-58, 2013.
• Kaya, D., The use of Adomian Decomposition Methods for Solving a Specific Nonlinear Partial Differential Equations, Bulletin of the Belgian Mathematical Society Simon Stevin, 9 (3), 343-349, 2002.
• Wazwaz, A.M., Partial Differential Equations and Solitary Waves Theory, Higher Education Press, Beijing, Springer-Verlag, Berlin, 2009. He, J.H., Wu, X.H., Construction of solitary solution and compacton-like solution by variational iteration method, Chaos, Solitons and Fractals, 29 (1), 108-113, 2006.
• Odibat, Z., Momani, S., Application of variational iteration method to nonlinear differential equations of fractional order, International Journal of Nonlinear Science and Numerical Simulation, 7 (1), 27-34, 2006.
• Biazar, J., Hosseini, K., Gholamin, P., Homotopy Perturbation Method for Solving KdV and Sawada-Kotera Equations, Journal of Applied Mathematics, Islamic Azad University, 6, 2009.
• He, J.H., Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons and Fractals, 26 (3), 695-700, 2005.
• Khan, Y., An effective modification of the Laplace decomposition method for nonlinear equations, International Journal of Nonlinear Sciences and Numerical Simulation, 10 (11-12), 1373-1376, 2009.
• Handibag, S., Karande, B.D., Existence the solutions of some fifth-order kdv equation by laplace decomposition method, American Journal of Computational Mathematics, 3, 80-85, 2013.
• Djidjeli, K., Price, W.G., Twizell, E.H., Wang, Y., Numerical Methods for the Soltution of the Third and Fifth-Order Dispersive Korteweg-De Vries Equations, Journal of Computational and Applied Mathematics, 58 (3), 307-336, 1995.
• Sawada, K., Kotera, T., A method for finding N-soliton solutions of the KdV equation and KdV-like equation, Progress of Theoretical Physics, 51 (5), 1355-1367, 1974.
• Prenter, P.M., Splines and Variational Methods, New York, John Wiley, 1975.
• Schoenberg, I.J., On trigonometric spline interpolation, Journal of Mathematics and Mechanics, 13, 795, 1964.
• Yagmurlu, N.M., Tasbozan, O., Ucar, Y., Esen, A., Numerical solutions of the Combined KdV-mKdV Equation by a Quintic B-spline Collocation Method, Applied Mathematics & Information Sciences Letters, 4 (1), 19-24, 2016.
• Hepson, O.E., Numerical solutions of the Gardner equation via trigonometric quintic B-spline collocation method, Sakarya University Journal of Science, 22 (6), 1576–1584, 2018.
• Rubin, S.G., Graves, R.A., A Cubic Spline Approximation for Problems in Fluid Mechanics, NASA, Washington, District of Columbia, October, 1975.