In this paper, the classical fourth-order Runge-Kutta methodis presented for solving the first-order ordinary differential equation. First, the given solution domain is discretizedby using a uniform discretization grid point. Next by applyingthe forward difference method, we discretized the given ordinary differential equation. And formulating a difference equation. Then using this difference equation, the given first-order ordinary differential equation is solved by using the classicalfourth-order Runge-Kutta method at each specified grid point. To validate the applicability of the proposed method, two model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supportedthe theoretical and mathematical statementsand the accuracy of the solution is obtained. The accuracy of the present methodhas been shown in the sense ofmaximumabsolute error and the local behavior of the solution is captured exactly. Numerical and exact solutions have been presented in tables and graphs and the corresponding maximumabsolute errorisalso presented in tables and graphs. The present method approximates the exact solution very well and it is quite efficient and practically well suitedfor solving first-order ordinary differential equations. The numerical result presented in tables and graphsindicates that the approximate solution is in good agreement with the exact solution. Hence the proposed method is accruable to solve ordinary differential equations.
| Published in | International Journal of Systems Science and Applied Mathematics (Volume 6, Issue 1) |
| DOI | 10.11648/j.ijssam.20210601.11 |
| Page(s) | 1-8 |
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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. |
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Copyright © The Author(s), 2024. Published by Science Publishing Group |
Ordinary Differential Equation, Runge-Kutta Method, Initial Value Problem, Boundary Value Problem, Stability and Convergent Analysis, Maximum Absolute Error
| [1] | Ray, S. (2018). Numerical analysis with algorithms and programming. CRC Press. |
| [2] | Francis B. Hildebrand (1987). Introduction to numerical analysis, Second edition |
| [3] | Sastry, S. S. (2006). Introductory method of numerical analysis, Fourth-edition. |
| [4] | Walter Gautschi (2012) Numerical Analysis Second Edition |
| [5] | Butcher, J. C., & Goodwin, N. (2008). Numerical methods for ordinary differential equations (Vol. 2). New York: Wiley. |
| [6] | Md. Amirul Islam. "A Comparative Study on numerical Solutions of Initial Value Problems (IVP) for Ordinary Differential Equations (ODE) with Euler and RungeKutta Methods”, America Journal of Computational Mathematics, |
| [7] | Kadalbajoo, M. K. "Geometric mesh FDM for selfadjoint singular perturbation boundary value problems", Applied Mathematics and Computation, |
| [8] | Le Veque, R. J. (2007). Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems. Society for Industrial and Applied Mathematics. |
| [9] | Kaw, A. (2009). Runge-Kutta 4th Order Method for Ordinary Differential Equations. Ordinary Differential Equations, 08-04. |
| [10] | Ralston, A. (1962). Runge-Kutta methods with minimum error bounds. Mathematics of computation, 16 (80), 431-437. |
| [11] | Hull, T. E., Enright, W. H., Fellen, B. M., & Sedgwick, A. E. (1972). Comparing numerical methods for ordinary differential equations. SIAM Journal on Numerical Analysis, 9 (4), 603-637. |
| [12] | Mungkasi, S., & Christian, A. (2017, January). Runge-Kutta and rational block methods for solving initial value problems. In Journal of Physics: Conference Series (Vol. 795, No. 1, p. 012040). IOP Publishing. |
| [13] | Butcher, J. C. (1966). On the convergence of numerical solutions to ordinary differential equations. Mathematics of Computation, 20 (93), 1-10. |
| [14] | Griffiths, D. F., Sweby, P. K., & Yee, H. C. (1992). On spurious asymptotic numerical solutions of explicit Runge-Kutta methods. IMA Journal of numerical analysis, 12 (3), 319-338. |
| [15] | Islam, M. A. (2015). Accurate solutions of initial value problems for ordinary differential equations with the fourth-order RungeKutta method. Journal of Mathematics Research, 7 (3), 41. |
| [16] | Griffiths, D. F.; Swaby, P. K.; and Yee, Helen C. (1992)., On spurious asymptotic numerical solutions of explicit Runge Kutta methods NASA Publications. 244. |
| [17] | Habtamu Garoma Debela and Masho Jima Kabeto (2017) Numerical solution of fourth-order ordinary differential equations using fifth-order Runge – Kuttamethod. Asian Journal of Science and Technology Vol. 08, Issue, 02, pp. 4332-4339. |
| [18] | Shampine L. F. (1984). Stability of explicit Runge-Kutta methods Camp & Mark with Appi. 7. Vol. IO. No. 6. pp 419-432. 1. |
APA Style
Kedir Aliyi Koroche. (2021). Numerical Solution of First Order Ordinary Differential Equation by Using Runge-Kutta Method. International Journal of Systems Science and Applied Mathematics, 6(1), 1-8. https://doi.org/10.11648/j.ijssam.20210601.11
ACS Style
Kedir Aliyi Koroche. Numerical Solution of First Order Ordinary Differential Equation by Using Runge-Kutta Method. Int. J. Syst. Sci. Appl. Math. 2021, 6(1), 1-8. doi: 10.11648/j.ijssam.20210601.11
AMA Style
Kedir Aliyi Koroche. Numerical Solution of First Order Ordinary Differential Equation by Using Runge-Kutta Method. Int J Syst Sci Appl Math. 2021;6(1):1-8. doi: 10.11648/j.ijssam.20210601.11
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Download@article{10.11648/j.ijssam.20210601.11,
author = {Kedir Aliyi Koroche},
title = {Numerical Solution of First Order Ordinary Differential Equation by Using Runge-Kutta Method},
journal = {International Journal of Systems Science and Applied Mathematics},
volume = {6},
number = {1},
pages = {1-8},
doi = {10.11648/j.ijssam.20210601.11},
url = {https://doi.org/10.11648/j.ijssam.20210601.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20210601.11},
abstract = {In this paper, the classical fourth-order Runge-Kutta methodis presented for solving the first-order ordinary differential equation. First, the given solution domain is discretizedby using a uniform discretization grid point. Next by applyingthe forward difference method, we discretized the given ordinary differential equation. And formulating a difference equation. Then using this difference equation, the given first-order ordinary differential equation is solved by using the classicalfourth-order Runge-Kutta method at each specified grid point. To validate the applicability of the proposed method, two model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supportedthe theoretical and mathematical statementsand the accuracy of the solution is obtained. The accuracy of the present methodhas been shown in the sense ofmaximumabsolute error and the local behavior of the solution is captured exactly. Numerical and exact solutions have been presented in tables and graphs and the corresponding maximumabsolute errorisalso presented in tables and graphs. The present method approximates the exact solution very well and it is quite efficient and practically well suitedfor solving first-order ordinary differential equations. The numerical result presented in tables and graphsindicates that the approximate solution is in good agreement with the exact solution. Hence the proposed method is accruable to solve ordinary differential equations.},
year = {2021}
}
TY - JOUR T1 - Numerical Solution of First Order Ordinary Differential Equation by Using Runge-Kutta Method AU - Kedir Aliyi Koroche Y1 - 2021/03/10 PY - 2021 N1 - https://doi.org/10.11648/j.ijssam.20210601.11 DO - 10.11648/j.ijssam.20210601.11 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 - 1 EP - 8 PB - Science Publishing Group SN - 2575-5803 UR - https://doi.org/10.11648/j.ijssam.20210601.11 AB - In this paper, the classical fourth-order Runge-Kutta methodis presented for solving the first-order ordinary differential equation. First, the given solution domain is discretizedby using a uniform discretization grid point. Next by applyingthe forward difference method, we discretized the given ordinary differential equation. And formulating a difference equation. Then using this difference equation, the given first-order ordinary differential equation is solved by using the classicalfourth-order Runge-Kutta method at each specified grid point. To validate the applicability of the proposed method, two model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supportedthe theoretical and mathematical statementsand the accuracy of the solution is obtained. The accuracy of the present methodhas been shown in the sense ofmaximumabsolute error and the local behavior of the solution is captured exactly. Numerical and exact solutions have been presented in tables and graphs and the corresponding maximumabsolute errorisalso presented in tables and graphs. The present method approximates the exact solution very well and it is quite efficient and practically well suitedfor solving first-order ordinary differential equations. The numerical result presented in tables and graphsindicates that the approximate solution is in good agreement with the exact solution. Hence the proposed method is accruable to solve ordinary differential equations. VL - 6 IS - 1 ER -
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