Sawada-Kotera Denkleminin Nümerik Yöntemlerle Çözümü ve Çözümlerin Karşılaştırılması

Year 2019, , 256 - 268, 30.11.2019

Zekeriya Özkan , Ramazan Uyhan

https://doi.org/10.29233/sdufeffd.568179

Cited By: 1

Abstract

Hesaplama
biliminde bilgisayarın etkili ve verimli kullanılması ile beraber kısmi türevli
diferansiyel denklemleri çözebilmek için kaynaklarda çok çeşitli metotlar sunulmuştur.
Bu metotların bazıları analitik metot çözümü bulurken bazıları algoritma
tabanlı yaklaşık çözümü bulan metotlardır. Bu çalışmada, Sawada-Kotera denklemi
çizgiler metodu ve radyal baz fonksiyonları yardımı ile ağsız çizgiler metodu
kullanılarak çözülmüştür. Elde edilen sonuçlar üzerinden de bu iki metot karşılaştırılmıştır. 

Keywords

Çizgiler Yöntemi, Ağsız Yöntemler, Radyal Baz Fonksiyonları

Solution of Sawada-Kotera Equation with Numerical Methods and Comparison of Solutions

Abstract

Effective and fruitful use
of computers in computing science, along with part of the literature, many techniques
to obtain the solution of differantial equations is presented. Some of these
methods, while achieving analytical technique and some of them algorithm based
solution techniques that approximate the solution.   In this study, Sawada-Kotera equation is
solved using the method of lines and meshless method of lines using radial
basis functions. These two methods are compared on the obtained results. 

Keywords

Method of Lines, Meshless Method, Radial Basis Functions

References

• [1] L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers (3th ed.). Edinburg: Birkhauser, 2005.
• [2] İ. Çağlar, “Bazı özel kısmi türevli diferansiyel denklemlerin gezen dalga çözümleri,” Yüksek Lisans Tezi, Fen Bilimleri Enstitüsü, Selçuk Üniversitesi, Konya, Türkiye, 2012.
• [3] A.R. Mitchell, D.F. Griffiths, The Finite Difference Method in Partial Equations. Chichester: John Wiley & Sons, 1980.
• [4] L. Demkowicz, J.T. Oden, W. Rachowicz and O. Hardy, “Toward A universal h-p adaptive finite element strategy, part 1: constrained approximation and data structure”, Comput. Methods Appl. Mech. Eng., 77 (1), 79-112, 1989.
• [5] G.R. Liu, Meshfree Methods: Moving Beyond the Finite Element Method. Boca Raton: CRC Press, 2003.
• [6] S. Çalışkan, “Eleman bağımsız Galerkin ve yerel Petrov Galerkin ağsız yöntemlerinin bir boyutlu mühendislik problemlerine uygulaması,” Yüksek Lisans Tezi, Fen Bilimleri Enstitüsü, Karadeniz Teknik Üniversitesi, Trabzon, Türkiye, 2006.
• [7] R. Pregla, Analysis of Electromagnetic Fields and Waves: The Method of Lines. West Sussex: John Wiley & Sons, 2008.
• [8] W.E. Schiesser, The Numerical Method of Lines: Integration of Partial Differential Equations. San Diego: Academic Press, 1991.
• [9] G.H. Meyer, The Time-Discrete Method of Lines for Options and Bonds A PDE Approach. Singapore: World Scientific Publishing, 2015.
• [10] R.A. Gingold and J.J. Monaghan, “Smoothed particle hidrodinamics: theory and application to non-spherical stars,” Mon. Not. R. Astr. Soc., 181, 375-389, 1977.
• [11] W. Chen, Z. Fu and C.S. Chen, Recent Advances in Radial Basis Function Collocation Methods. New York: Springer, 2014.
• [12] Q. Shen, “A meshless method of lines for the numerical solution of KdV equation using radial basis functions,” Eng. Anal. Bound. Elem., 33, 1171-1180, 2009.
• [13] W.E. Schiesser and G.W. Griffiths, A Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab. Cambridge: Cambridge University Press, 2009.
• [14] W.E. Schiesser and G.W. Griffiths, Traveling Wave Analysis of Partial Differantial Equations. San Diego: Academis Press, 2012.
• [15] F. Durmuş, “Kısmi türevli diferansiyel denklemlerin nümerik çözümü için method of lines yöntemi,” Yüksek Lisans Tezi, Fen Bilimleri Enstitüsü, Selçuk Üniversitesi, Konya, Türkiye, 2015.
• [16] G.R. Liu and Y.T Gu, An Introduction to Meshfree Methods and Their Programming. Dordrecht: Springer, 2005.
• [17] V.P. Nguyen, T. Rabczuk, S. Bordas and M. Duflot, “Meshless methods: a review and computer ımplementation aspects,”, Math. Comput.Simulat., 79, 763-813, 2008.
• [18] N. Bibi, “Meshless method of lines for numerical solutions of nonlinear time dependent partial differential equations,” PhD Thesis, Ghulam Ishaq Khan of Engineering Sciences and Technology, Swabi, Pakistan, 2011.
• [19] D. Kaya and S.M. El-Sayed, “On a Generalized fifth order KdV equations,” Phys. Lett. A., 310, 44-51, 2003.
• [20] C. Köroğlu, “Üstel matris fonksiyonları yardımıyla amerikan opsiyon probleminin çizgiler yöntemi ile çözümü,” Doktora Tezi, Fen Bilimleri Enstitüsü, Ege Üniversitesi, İzmir, Türkiye, 2002.
• [21] F. Tchier, Mustafa İnc, Bülent Kılıç and Ali Akgül, “On soliton structures of generalized resonance equation with time dependent coefficients,” Optik, 128, 218-223, 2017.
• [22] A. Akgül, Mustafa İnç and Esra Karataş, “Reproducing kernel functions for difference equations,” Discrete Contin. Dyn. Syst. Ser., 8(6), 1055-1064, 2015.
• [23] A. Akgül, A. Kılıçman, and Mustafa İnç, “Improved (G '/G)-expansion method for the space and time fractional foam drainage and KdV equations,” Abstr. Appl. Anal., (Article ID: 414353), 7 pages, 2013.
• [24] A. Akgül, Y. Khanb, E. Karataş Akgül, D. Baleanu and M. M. Al Qurashi, “Solutions of nonlinear systems by reproducing kernel method,” J. Nonlinear Sci. Appl., 10, 4408-4417, 2017.
• [25] A. Akgül, “New reproducing kernel functions,” Math. Probl. Eng., (Article ID:158134), 10 pages, 2015.
• [26] M. İnç and A. Akgül, “Approximate solutions for MHD squeezing fluid flow by a novel method,” Bound. Value Probl., 2014 (Article ID:18), 18 pages, 2014.
• [27] M.İnç, B. Kılıç, E. Karataş and A. Akgül, “Solitary wave solutions for the Sawada-Kotera equation,” J. Adv. Phys., 6 (2), 288-293, 2017.