KdV DENKLEMİ İÇİN KUİNTİK B-SPLİNE GALERKİN METODU

Öz

Korteweg de Vries (KdV) denklemi, Crank Nicolson parçalanması ile birlikte kuintik B-spline şekil ve taban fonksiyonlarının kullanıldığı Galerkin sonlu elemanlar metoduyla yaklaşık olarak çözülmüştür. Bir solitonun yayılması ve iki solitonun çarpışmasını içeren iki klasik test problemi kullanılarak önerilen yöntemin doğruluğu kontrol edilmiştir.  Sonuç olarak önerilen yaklaşık yöntemin KdV denkleminin sayısal çözümü için faydalı bir yöntem olduğu görülmüştür.

Anahtar Kelimeler

• Soliton,B-spline

QUINTIC B-SPLINE GALERKIN METHOD FOR THE KdV EQUATION

Öz

The Korteweg de Vries (KdV) equation is solved numerically based on Crank Nicolson discretization and Galerkin finite element method using quintic B-splines as weight and element shape functions. Two classical test problems, including propagation of a single soliton and interaction of two solitons, are used to validate the proposed method. Finally, we conclude that the proposed numerical method is a useful approach for numerical solution of KdV equation.

Anahtar Kelimeler

• Soliton,B-spline

Kaynakça

1. [1] Korteweg DJ and de Vries G. On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves. Philosophical Magazine. 1895; 39 (240): 422–443.
2. [2] Zabusky NJ and Kruskal M D. Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States. Phys. Rev. Lett. 1965; 15 (6): 240–243.
3. [3] Soliman AA. Collocation solution of the Korteweg-De Vries equation using septic splines. Int. J. Comput. Math. 2004; 81: 325-331.
4. [4] Canıvar, A. Sarı M. and Dağ I. A Taylor-Galerkin finite element method for the KdV equation using cubic B-splines. Physica B Condensed Matter 2010; 405: 3376-3383.
5. [5] Ersoy Ö. and Dağ İ. The Exponential Cubic B-Spline Algorithm for Korteweg-de Vries Equation. Advances in Numerical Analysis 2015; 2015: 1-8.
6. [6] Saka B. Cosine expansion-based differential quadrature method for numerical solution of the KdV equation, Chaos, Solitons and Fractals 2009; 40: 2181-2190.
7. [7] Korkmaz A. Numerical Algorithms for solutions of Korteweg-de Vries Equation, Numer. Meth. Part. D. E. 2010; 26(6): 1504–1521.
8. [8] Irk D. Dağ İ. and Saka B. A small time solutions for the Korteweg–de Vries equation using spline approximation. Applied mathematics and computation 2006; 173 (2): 834-846.

9. [9] Crank J. and Nicolson P. A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type. Proc. Camb. Phil. Soc. 1947; 43: 50–67.
10. [10] Prenter PM. Splines and Variational Methods, Wiley 1975.