The essence of this paper is to analyze the stability and implementation of a newly derived explicit fourth-stage fourth-order Runge-Kutta method. Efforts will be made to carry out a comparative analysis with an existing classical fourth stage fourth order explicit Runge Kutta method. The implementation on initial-value problems revealed that the method compared favorably well with the existing classical fourth stage fourth order explicit Runge Kutta method. The stability analysis revealed that the method is absolute stable, and capable in handling initial value problems in ordinary differential equations. The Taylor series expansion was carried out on the general explicit fourth stage fourth order Runge Kutta scheme, and then, parameters and coefficients were varied with the expansion to generate a set of linear / nonlinear equations which were resolved to generate the method. This approach has shown that when parameters are properly varied, and all equations in the set, whether lineaer / nonlinear, it will definitely give birth to a method that will improve results.
| Published in | International Journal of Systems Science and Applied Mathematics (Volume 7, Issue 3) |
| DOI | 10.11648/j.ijssam.20220703.12 |
| Page(s) | 52-59 |
| 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), 2022. Published by Science Publishing Group |
Stability, Linear and Non-linear Equations, Taylor Series, Parameters, Initial-Value Problems, Implementation
| [1] | Agbeboh; G. U (2013) “On the Stability Analysis of a Geometric 4th order Runge–Kutta Formula”. (Mathematical Theory and Modeling ISSN 2224 – 5804 (Paper) ISSN 2225 – 0522 (Online) Vol. 3, (4)) www.iiste.org.the international institute for science, technology and education, (IISTE). |
| [2] | Butcher, J. C., (1987);” The Numerical Analysis of Ordinary Differential Equations, Runge-Kutta and General linear methods”, John Wiley & Sons. |
| [3] | Barletti, L, Brugnano, L, and Yifa, T., (2020): “Spectrally accurate space time solution of manakov systems”, Mathematics, J. Comput. Appl. Math 2020. |
| [4] | Brugnano, L, Gurion, G, and Yadian, S., (2019): “Energy conserving Hamiltian Boundary value methods for numerical solution of korteweg de vries equations”, Mathematics, J Compt. Apll. Math. 2019. |
| [5] | Butcher J. C., (2000). “Numerical methods for ordinary differential equations in the 20th century”, journal of computational and applied mathematics 125 (2000) 1-29. |
| [6] | Butcher J. C., (2003); “Numerical methods for ordinary differential equations”, wiley, chichester. |
| [7] | Butcher J. C., (2008); “Numerical methods for ordinary differential equations (2nd ed.)”, John wiley and sons ltd. |
| [8] | Butcher J. C., (2009);” Trees and Numerical methods for ordinary differential equations”, Numerical Algorithms (Springer online). |
| [9] | Butcher J. C., (2009), “On the fifth and sixth order explicit Runge-Kutta methods”. Order conditions and order Barries, Canadian applied Mathematics quarterly volume 17, numbers pg 433-445. |
| [10] | Gear, C. W., (1971). “The automatic integration of ordinary differential equations” Comm. ACM., 14, 176-179. |
| [11] | Gianluca, F, Lavermero, F, and Vespri, V., (2021): “A new frame work for polynomial approximation to differential equations”, Mathematics, Computer Science, 2021. |
| [12] | Gianluca, F, Lavermero, F, and Vespri, V., (2021): “A new frame work for approximating differential equations”, Mathematics, Computer Science, 2021. |
| [13] | Van der Houwen, P. J., Sommeijer, B. P., (2015); “Runge-Kutta projection methods with low dispersion and dissipation errors”. Advances in computational methods, 41: 231-251. |
APA Style
Esekhaigbe Aigbedion Christopher, Okodugha Edward. (2022). Stability Analysis and Implementation of a New Fully Explicit Fourth-Stage Fourth-Order Runge-Kutta Method. International Journal of Systems Science and Applied Mathematics, 7(3), 52-59. https://doi.org/10.11648/j.ijssam.20220703.12
ACS Style
Esekhaigbe Aigbedion Christopher; Okodugha Edward. Stability Analysis and Implementation of a New Fully Explicit Fourth-Stage Fourth-Order Runge-Kutta Method. Int. J. Syst. Sci. Appl. Math. 2022, 7(3), 52-59. doi: 10.11648/j.ijssam.20220703.12
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Download@article{10.11648/j.ijssam.20220703.12,
author = {Esekhaigbe Aigbedion Christopher and Okodugha Edward},
title = {Stability Analysis and Implementation of a New Fully Explicit Fourth-Stage Fourth-Order Runge-Kutta Method},
journal = {International Journal of Systems Science and Applied Mathematics},
volume = {7},
number = {3},
pages = {52-59},
doi = {10.11648/j.ijssam.20220703.12},
url = {https://doi.org/10.11648/j.ijssam.20220703.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20220703.12},
abstract = {The essence of this paper is to analyze the stability and implementation of a newly derived explicit fourth-stage fourth-order Runge-Kutta method. Efforts will be made to carry out a comparative analysis with an existing classical fourth stage fourth order explicit Runge Kutta method. The implementation on initial-value problems revealed that the method compared favorably well with the existing classical fourth stage fourth order explicit Runge Kutta method. The stability analysis revealed that the method is absolute stable, and capable in handling initial value problems in ordinary differential equations. The Taylor series expansion was carried out on the general explicit fourth stage fourth order Runge Kutta scheme, and then, parameters and coefficients were varied with the expansion to generate a set of linear / nonlinear equations which were resolved to generate the method. This approach has shown that when parameters are properly varied, and all equations in the set, whether lineaer / nonlinear, it will definitely give birth to a method that will improve results.},
year = {2022}
}
TY - JOUR T1 - Stability Analysis and Implementation of a New Fully Explicit Fourth-Stage Fourth-Order Runge-Kutta Method AU - Esekhaigbe Aigbedion Christopher AU - Okodugha Edward Y1 - 2022/08/31 PY - 2022 N1 - https://doi.org/10.11648/j.ijssam.20220703.12 DO - 10.11648/j.ijssam.20220703.12 T2 - International Journal of Systems Science and Applied Mathematics JF - International Journal of Systems Science and Applied Mathematics JO - International Journal of Systems Science and Applied Mathematics SP - 52 EP - 59 PB - Science Publishing Group SN - 2575-5803 UR - https://doi.org/10.11648/j.ijssam.20220703.12 AB - The essence of this paper is to analyze the stability and implementation of a newly derived explicit fourth-stage fourth-order Runge-Kutta method. Efforts will be made to carry out a comparative analysis with an existing classical fourth stage fourth order explicit Runge Kutta method. The implementation on initial-value problems revealed that the method compared favorably well with the existing classical fourth stage fourth order explicit Runge Kutta method. The stability analysis revealed that the method is absolute stable, and capable in handling initial value problems in ordinary differential equations. The Taylor series expansion was carried out on the general explicit fourth stage fourth order Runge Kutta scheme, and then, parameters and coefficients were varied with the expansion to generate a set of linear / nonlinear equations which were resolved to generate the method. This approach has shown that when parameters are properly varied, and all equations in the set, whether lineaer / nonlinear, it will definitely give birth to a method that will improve results. VL - 7 IS - 3 ER -
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