Standart ve Standart Olmayan Theta Metotlarının Bazı Uygulamaları ve Sonuçları

Abstract

Bu çalışmada Adi Diferansiyel Denklemlerin nümerik çözüm metotlarından biri olan Theta Metodu araştırıldı. Mickens’in ortaya koyduğu Standart Olmayan Sonlu Fark Metotlarının bakış açısıyla bazı teoremler ve uygulamalar verildi. Bununla beraber, Standart ve Standart Olmayan Thata Metotları diğer klasik metotlar ile karşılaştırılarak güvenilirlikleri test edildi.

Keywords

• Theta Metodu,Diferansiyel Denklemler,Nümerik Metotlar

Some Results and Applications of Standard and Non-standard Theta Methods

Abstract

In this paper, The Theta Method which is one way of the numerical solution method of ordinary differential equations was investigated. Some theorems and applications were given from the point of non-standard finite difference methods discovered by Mickens. Moreover, the performance of  the Standard and Non-standard Theta method was tested by comparing with other classical numerical methods.

Keywords

• Theta Method,Differential Equations,Numerical Methods

References

1. Mickens, R. E., 1994. Nonstandart Finite Difference Models of Differential Equations, World Scientific Publishing Company Inc. River Edge, New Jersey.
2. Mickens, R.E., 2007. Calculation of Denominator Functions for Non-standard Finite Difference Schemes for Differential Equations Satisfying a Positivity Condition. Numerical Methods for Partial Differential Equations, 23, pp. 672-691.
3. Kama, P.,2009, Non-standard Finite Difference Methods in Dynamical Systems. University of Pretoria, Philosophie Doctor, Pretoria, p. 49.
4. Lubuma, J.M., A. Roux, 2003, An Improved Theta Method for Systems of Ordinary Differential Equations, Journal of Difference Equations and Applications, pp. 1023-1035.
5. Roux, A., 2002. Fourier Series and Spectral-Finite Difference Methods for the General Linear Diffusion Equation, 28 January 2002, Pretoria, p. 45.
6. Suli, E., 2010. Numerical Solution of Ordinary Differential Equations, 12 August 2010, pp. 21-53.
7. Farago, I., 2013. Convergence and Stability Constants of the Theta Method, Conference Applications of Mathematics, 186, pp. 42-51.
8. Arslan, D., Ongun, M. Y., Turhan, İ., 2011. Standart Olmayan Sonlu Fark Yönteminin Fuzzy Diferansiyel Denklemlere Uygulanması, III. Ulusal Konya Ereğli Kemal Akman Meslek Yüksekokulu Tebliğ Günleri, Sayı 3, No:1.

9. Ongun,M.Y., Turhan,I.,2013, A numerical comparison for a discrete HIV infection of $CD4^{+}T$-cells model derived from non-standard numerical scheme, Journal of Applied Mathematics, Special Issue:Iterative Methods for Nonlinear Equations or Systems and Their Applications (IMNES), Vol.2013, Article ID 375094, 9 pages doi:10.1155/2012/375094
10. Ongun,M.Y.,Arslan,D.,Garrappa,R., 2013,Non-standard Finite Difference Schemes for fractional order Brusselator system, Advances in Difference Equations,doi:10.1186/1687-1847-2013-102.
11. Barclay G. J., Griffiths D. F., Higham D.J., 2000, Theta Method Dynamics, LMS J. Comput. Math. 3, pp. 27–43.