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), 2019. Published by Science Publishing Group |
Ordinary Differential Equation, Runge-Kutta, Stability Region, Java Programming
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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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