This paper is in the context of the numerical resolution of ordinary differential equations. Most equations are unsolved in the analytic aspect. The goal is to find among the existing methods, the best method of numerical resolution. Also to facilitate the implementation of methods by introducing a calculation software. To do this, we use the Runge-Kutta method which is one of the best methods of numerical resolutions. That is why a family of Runge-Kutta methods of order 7 is presented. This family depends on the parameter b8 and contains the well known method of Butcher [8] (b8 =77/1440). To obtain convincing results, we compare methods according to the values of b8 with those of Butcher. The stability region is also studied to essentially perceive the numerical behavior that manifests itself when the steps of discretization tend to 0. The study shows that the stability region of this method does not depend on the coefficient b8. To get the values of b8, Java programming is used. Finally, to facilitate the implementation of the resolution, very simple software for numerical resolution of the ordinary differential equations is given. This software is designed for all students, also for all those who have no basis in numerical analysis and java programming to be able to find a solution approached with error estimate to an ordinary differential equation.
| Published in | Mathematics and Computer Science (Volume 4, Issue 3) |
| DOI | 10.11648/j.mcs.20190403.12 |
| Page(s) | 68-75 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Ordinary Differential Equation, Runge-Kutta, Stability Region, Java Programming
| [1] | Najmuddin Ahamad, Shiv Sharan “Study of Numerical Solution of Fourth Order Ordinary Differential Equations by Fifth Order Runge-Kutta Method”. IJSRSET. 6 (1). 2019. |
| [2] | Robert I. McLachlan et al. “Butcher Series. A story of rooted trees and numerical methods for evolution equations”. Asia Pacific Mathematics Newsletter. 2017. |
| [3] | Gashu Gadisa Kiltu, Habtamu Garoma. “Comparison of Higher Order Taylor’s Method and Runge-Kutta Methods for Solving First Order Differential Equations”. Journal of Computer and Mathematical Sciences. 8 (1). 2017. |
| [4] | Trésor Kanyiki. “Contribution of a Runge-Kutta Order 4 Method in a Dynamic of Mechanical System”. ISTE Open Science. 2018. |
| [5] | F. A. Fawzi et al. “An Embedded 6 (5) pair of Explicit Runge-Kutta Method for Periodic IVPs”. Far East Journal of Mathematical Sciences. Vol 100. Issue 11. P 1841-1857 (2016). |
| [6] | Séka Hippolyte, Kouassi Assui Richard, “A New Eighth Order Runge-Kutta Family Method”, Journal of Mathematics Research. 11 (2), 2019. |
| [7] | Abbas Fadhil Abbas Al-Shimmary, “Solving initial value using Runge-Kutta 6th order method”, ARPN Journal of Engineering and Applied Sciences. 12 (12), 2017. |
| [8] | J. C. Butcher “Numericals Methods for Ordinary Differential Equations”. Second Edition, 2008. |
| [9] | Mohammed S. Mechee et al. “On the rehabilitee stability of direct explicit Runge-Kutta integrators”. Global Journal of Pure and Applied Mathematics 12 (4). 2016, pp. 3959-3975. |
| [10] | Z. W. Sun et al. “Stability of the fourth order Runge-Kutta method for time-dependent partial differential equations”. Annals of Mathematical sciences and Application. 2 (2). 2017. |
| [11] | Rachid Ait-Haddou “New Stability Results for Explicit Runge-Kutta Methods”. arXiv: 1804.09896. (2018). |
| [12] | Kim Gaik Taya, et al., “The fourth Order Runge-Kutta Spreadsheet Calculator Using VBA Programing for ordinary differential equations”, 4th World Congress on Technical and Vocational Education and Training (WoCTVET), 5th–6th November, 2014, Malaysia. |
| [13] | Sergey Khashin, “List of some known Runge-Kutta methods.” http://math.ivanovo.ac.ru/dalgebra/Khashin/rk/sh_rk.html |
| [14] | Kasim Hussain, et al., “Runge-Kutta Type Methods for Directly Solving Special Fourth-, Order Ordinary Differential Equations.” Problems in Engineering Volume 2015. |
| [15] | Md. Babul Hossain, Md. Jahangir Hossain Md. Musa. Miah and Md. Shah Alam. A Comparative Study on Fourth Order and Butchers Fifth Order Runge-Kutta Methods with Third Order Initial Value Problem (IVP). Applied and Computational Mathematics. December 2015 Volume 2015. |
| [16] | Md. Babul Hossain, et al., “A Comparative Study on Fourth Order and Butchers Fifth Order Runge-Kutta Methods with Third Order Initial Value Problem (IVP).” |
| [17] | ROSETTACODE. ORG. https://rosettacode.org/wiki/Runge-Kutta method. |
| [18] | Stephan Houben, Stability regions of Runge-Kutta methods. Eindhoven University of Technology. February 19, 2002. |
| [19] | Jackiewicz, Zdzislaw, General Linear Methods for Ordinary Differential Equations. 482 p., 2009. isbn 978-0-470-40855-1. doi 10.1002/9780470522165. publisher John Wiley & Sons, Inc. |
| [20] | M. Calvo and J. I. Montijano and L. Randez, A new embedded pair of Runge-Kutta formulas of orders 5 and 6. Comput. Math. Appl. Vol. 20, No. 1, 1990. Issn 0898-1221. 15-24 p. doi http://dx.doi.org/10.1016/0898-1221(90)90064-Q, Pergamon Press, Inc., Tarrytown, NY, USA. |
| [21] | C. R. Cassity, The complete solution of the fifth order Runge-Kutta equations, SIAM J. Numer. Anal., vol. 6, 1969, 432-436 p. |
| [22] | T. Feagin, A tenth-order Runge-Kutta method with error estimate, Proceedings of the IAENG Conf. on Scientific Computing, 2007, Hong Kong. |
| [23] | T. Feagin, High-Order Explicit Runge-Kutta Methods Using m-Symmetry. Neural, Parallel & Scientific Computations. Vol 20. Issue 3/4, p 437 (2012). |
| [24] | E. Hairer, S. P. Norsett, G. Wanner, isbn 3-540-56670-8. 528 p., 2Ed. Springer-Verlag, Solving ordinary differential equations I. Nonstiff Problems, 2000. |
| [25] | Hairer, Ernst; Norsett, Syvert Paul; Wanner, Gerhard, Solving ordinary differential equations I. Nonstiff Problems, Springer-Verlag, isbn 978-3-540-56670-0, 2008. |
| [26] | Sindhuja Reddy Velgala. Stability analysis of the 4th order Runge-Kutta method in application to colloidal particle interactions. Thesis University of Illinois at Urbana-Champaign, 2014. |
| [27] | M. Z. Liu, M. H. Song, Z. W. Yang, Stability of RungeKutta methods in the numerical solution of equation u’(t) = au(t) + a0u([t]). Volume 166, Issue 2, 15 April 2004, Pages 361-370. |
APA Style
Hippolyte Séka, Assui Richard Kouassi. (2019). A New Seventh Order Runge-kutta Family: Comparison with the Method of Butcher and Presentation of a Calculation Software. Mathematics and Computer Science, 4(3), 68-75. https://doi.org/10.11648/j.mcs.20190403.12
ACS Style
Hippolyte Séka; Assui Richard Kouassi. A New Seventh Order Runge-kutta Family: Comparison with the Method of Butcher and Presentation of a Calculation Software. Math. Comput. Sci. 2019, 4(3), 68-75. doi: 10.11648/j.mcs.20190403.12
AMA Style
Hippolyte Séka, Assui Richard Kouassi. A New Seventh Order Runge-kutta Family: Comparison with the Method of Butcher and Presentation of a Calculation Software. Math Comput Sci. 2019;4(3):68-75. doi: 10.11648/j.mcs.20190403.12
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Download@article{10.11648/j.mcs.20190403.12,
author = {Hippolyte Séka and Assui Richard Kouassi},
title = {A New Seventh Order Runge-kutta Family: Comparison with the Method of Butcher and Presentation of a Calculation Software},
journal = {Mathematics and Computer Science},
volume = {4},
number = {3},
pages = {68-75},
doi = {10.11648/j.mcs.20190403.12},
url = {https://doi.org/10.11648/j.mcs.20190403.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20190403.12},
abstract = {This paper is in the context of the numerical resolution of ordinary differential equations. Most equations are unsolved in the analytic aspect. The goal is to find among the existing methods, the best method of numerical resolution. Also to facilitate the implementation of methods by introducing a calculation software. To do this, we use the Runge-Kutta method which is one of the best methods of numerical resolutions. That is why a family of Runge-Kutta methods of order 7 is presented. This family depends on the parameter b8 and contains the well known method of Butcher [8] (b8 =77/1440). To obtain convincing results, we compare methods according to the values of b8 with those of Butcher. The stability region is also studied to essentially perceive the numerical behavior that manifests itself when the steps of discretization tend to 0. The study shows that the stability region of this method does not depend on the coefficient b8. To get the values of b8, Java programming is used. Finally, to facilitate the implementation of the resolution, very simple software for numerical resolution of the ordinary differential equations is given. This software is designed for all students, also for all those who have no basis in numerical analysis and java programming to be able to find a solution approached with error estimate to an ordinary differential equation.},
year = {2019}
}
TY - JOUR T1 - A New Seventh Order Runge-kutta Family: Comparison with the Method of Butcher and Presentation of a Calculation Software AU - Hippolyte Séka AU - Assui Richard Kouassi Y1 - 2019/10/14 PY - 2019 N1 - https://doi.org/10.11648/j.mcs.20190403.12 DO - 10.11648/j.mcs.20190403.12 T2 - Mathematics and Computer Science JF - Mathematics and Computer Science JO - Mathematics and Computer Science SP - 68 EP - 75 PB - Science Publishing Group SN - 2575-6028 UR - https://doi.org/10.11648/j.mcs.20190403.12 AB - This paper is in the context of the numerical resolution of ordinary differential equations. Most equations are unsolved in the analytic aspect. The goal is to find among the existing methods, the best method of numerical resolution. Also to facilitate the implementation of methods by introducing a calculation software. To do this, we use the Runge-Kutta method which is one of the best methods of numerical resolutions. That is why a family of Runge-Kutta methods of order 7 is presented. This family depends on the parameter b8 and contains the well known method of Butcher [8] (b8 =77/1440). To obtain convincing results, we compare methods according to the values of b8 with those of Butcher. The stability region is also studied to essentially perceive the numerical behavior that manifests itself when the steps of discretization tend to 0. The study shows that the stability region of this method does not depend on the coefficient b8. To get the values of b8, Java programming is used. Finally, to facilitate the implementation of the resolution, very simple software for numerical resolution of the ordinary differential equations is given. This software is designed for all students, also for all those who have no basis in numerical analysis and java programming to be able to find a solution approached with error estimate to an ordinary differential equation. VL - 4 IS - 3 ER -
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