Birinci Dereceden Diferansiyel Denklemlerin Çözümü için Yüksek Dereceli Runge-Kutta Yöntemlerinin Karşılaştırmalı Araştırması

Abstract

Birinci dereceden adi diferansiyel denklemlerin (ODE’ler) çözümünde, bu makale çeşitli yüksek dereceli Runge-Kutta yöntemlerini kapsamlı bir şekilde karşılaştırmaktadır. Araştırmanın ana amacı, çeşitli Runge-Kutta şemalarının etkinliğini, doğruluğunu ve uygulanabilirliğini gözden geçirmek ve bunların sayısal yaklaşımlarda kullanımını vurgulamaktır. Çalışma, beşinci dereceli, altı aşamalı Runge-Kutta (RK56), altıncı dereceli, yedi aşamalı Runge-Kutta (RK67) ve yedinci dereceli, dokuz aşamalı Runge-Kutta (RK79) gibi geleneksel yaklaşımları araştırmakta olup, bu yöntemlerin bireysel avantaj ve dezavantajlarına dair kapsamlı bir anlayış sunmayı amaçlamaktadır. Akademisyenler ve uygulayıcıların, problemin özelliklerine göre en uygun yaklaşımı seçmelerine yardımcı olmak için teorik temeller ve gerçek dünya uygulamaları kullanılarak karşılaştırmalı ölçütler oluşturulmuştur. Dayanıklılık değerlendirmeleri ve hassasiyet analizleri, bu tekniklerin farklı bağlamlardaki esnekliğini aydınlatarak karşılaştırma araştırmasını tamamlamaktadır. Bu çalışmanın sonuçları, yüksek dereceli Runge-Kutta yöntemlerinin nasıl çalıştığına dair önemli yeni anlayışlar sunmakta ve bu yöntemlerin çeşitli bilim ve mühendislik alanlarında birinci dereceden diferansiyel problemleri çözmek için uygulanması konusunda kapsamlı bir kılavuz sağlamaktadır. Üç yüksek dereceli Runge-Kutta algoritmasının incelenmesi, RK56’nın birinci dereceden ODE’leri çözmede daha etkili olduğunu ortaya koymaktadır.

Keywords

• Runge-Kutta tekniği
• adi diferansiyel denklemler
• sayısal entegrasyon
• hata analizi
• hesaplamalı karşılaştırma.

An Investigative Comparison of Higher-Order Runge-Kutta Techniques for Resolving First-Order Differential Equations

Abstract

In the context of solving first-order ordinary differential equations (ODEs), this paper thoroughly compares various higher-order Runge-Kutta methods. Reviewing the effectiveness, precision, and practicality of several Runge-Kutta schemes and highlighting their usage in numerical approximation is the main goal of the research. The study explores traditional approaches, including the fifth-order, six-stage Runge-Kutta (RK56), the sixth-order, seven-stage Runge-Kutta (RK67), and the seventh-order, nine-stage Runge-Kutta (RK79), with the goal of offering a comprehensive comprehension of their individual advantages and disadvantages. In order to help academics and practitioners choose the best approach based on the features of the problem, comparative benchmarks are constructed, utilizing both theoretical underpinnings and real-world implementations. Robustness evaluations and sensitivity analysis complement the comparison research by illuminating how flexible these techniques are in various context. The results of this study provide important new understandings of how higher-order Runge-Kutta methods function and provide a thorough manual for applying them to solve first-order differential problems in a variety of scientific and engineering fields. The study’s examination of three higher order Runge-Kutta algorithms reveals that the RK56 is more effective at solving first order ODEs.

Keywords

• Runge-Kutta Technique
• Ordinary Differential Equations
• Numerical Integration
• Error Analysis
• Computational Comparison

References

1. Lee KC, Senu N, Ahmadian A, Ibrahim SI & Baleanu D. Numerical Study of Third-Order Ordinary Differential Equations Using a New Class of Two Derivative Runge-Kutta Type Methods. Alex Eng J 2020; 59, 2449–2467.
2. Poornima S, and Nirmala T. Comparative Study of Runge-Kutta Methods of Solving Ordinary Differential Equations. Int J Res in Eng, Sci and Mgt 2020; .3: 557-559.
3. Jamali N. Analysis and Comparative Study of Numerical Methods to Solve Ordinary Differential Equation with Initial Value Problem. Int J Adv Res 2020; 7(5): 117-128.
4. Okeke AA, Hambagda BM, & Tumba P. Accuracy Study on Numerical Solutions of Initial Value Problems (IVP) in Ordinary Differential Equations. Int J Math and Stat Invention 2019. 7(2), 2321-4759.
5. Soliu AA. Comparative Study on Some Numerical Algorithms for First Order Ordinary Differential Equations. B. Tech, Federal University of Technology, Minna, Nigeria. 2023.
6. Mesa F, Devia-Narvaez DM, Correa-Velez G. Numerical Comparison by Different Methods (Second Order Runge Kutta Methods, Heun Method, fixed Point Method and Ralston Method) to Differential Equations with Initial Condition. Scientia et Technica 2020; 25(2): 299-305.
7. Smith J, & Johnson A. Comparative Analysis of Runge-Kutta Methods for Solving Ordinary Differential Equations. J Comput Math 2019. 45(2), 210-225.
8. Wang L, & Li HA. Review of Higher-Order Runge-Kutta Methods in Scientific Computing. Applied Numerical Analysis 2020; 35(4): 567-582.

9. Jones R, Brown M. Performance Evaluation of Runge-Kutta Techniques in Atmospheric Modeling. J Atmosph Sci 2018; 25(3): 410-425.
10. Garcia P, & Martinez E. Comparative Study of Runge-Kutta Methods for Solving Heat Transfer Equations in Engineering Applications. Heat Trans Eng 2017;, 33(1): 89-104.
11. Chen Y, & Zhang Q. A Survey of Runge-Kutta Methods for Solving Chemical Reaction Kinetics. Chem Eng. J. 2016; 40(2): 315-330.
12. Agbeboh GU, Adoghe LO, Ehiemua ME, Ononogbo BC. On the Derivation of a Sixth-Stage-Fifth-Order Runge-Kutta Method for Solving Initial Value Problems in Ordinary Differential Equations. American J Sci Eng Res. 2020; 3(5): 29-41.
13. Başhan A, Battal S, Karakoç G, and Geyikli T. Approximation of the KdVB Equation by the Quintic B-spline Differential Quadrature method. Kuwait J.Sci 2015; 42(2): 67-92.
14. Bashan A, Ucar Y, Yagmurlu NM, Esen A. An effective approach to numerical soliton solutions for the Schrodinger equation via modified cubic B-spline differential quadrature method. Article in Chaos Solitons & Fractals 2017; 100, 45–56.
15. Bashan A. An effective application of differential quadrature method based on modified cubic B-splines to numerical solutions of the kdV equation. Turk J Math 2018; 42: 373 – 394.
16. Bashan A, Ucar Y, Yagmurlu NM, Esen A. Numerical Solutions for the Fourth Order Extended Fisher-Kolmogorov Equation with High Accuracy by Differential Quadrature Method. Sigma J Eng & Nat Sci 2018; 9(3): 273-284.
17. Ucar Y, Yagmurlu NM, Bashan A. Numerical Solutions and Stability Analysis of Modified Burgers Equation via Modified Cubic B-Spline Differential Quadrature Methods. Sigma J Eng & Nat Sci 2019; 37 (1): 129-142.
18. Al-Shimmary AF. Solving initial value problem using Runge-Kutta 6th order method. ARPN J Eng Appl Sci 2017; 12(13): 3953-3961.
19. Trikkaliotis GD & Gousidou-Koutita MCh. Production of the Reduction Formula of Seventh Order Runge-Kutta Method with Step Size Control of an Ordinary Differential Equation. Appl Math 2022; 13, 325-337.
20. Tuba G. Some Approaches For Solving Mulplicative Second Order Linear Differential Equations with Variable Exponentials and Multiplicative Airy’s Equation. Turk J Sci & Tech 2023; 18(2): 301-309.
21. Audu KJ, Taiwo AR, Soliu AA. Assessment of Numerical Performance of Some Runge-Kutta Methods and New Iteration Method on First Order Differential Problems. Dutse J Pure & Appl Sci 2023; 9(4a): 58-70.
22. Arora G, Joshi V & Garki I. Developments in Runge–Kutta Method to Solve Ordinary Differential. Recent Advan Math Eng 2020; pp 193-202.
23. Hetmaniok E, Pleszczynski M. Comparison of the Selected Methods Used for Solving the Ordinary Differential Equations and Their Systems. Mathematics 2022; 10(3): 1-15.