Numerical methods for solving Ordinary Differential Equations differ in accuracy, performance, and applicability. This paper presents a comparative study of numerical methods, mainly Euler’s method, the Runge-Kutta method of order 4th & 6th and the Adams-Bashforth-Moulton method for solving initial value problems in ordinary differential equations. Our aim in this paper is to show that which method gives better accuracy for the initial value problem in numerical methods. Comparisons are made among the direct method, Euler’s method, Runge-Kutta fourth and sixth order and the Adams-Bashforth-Moulton method for solving the initial value problem. The comparisons with error analysis are also shown in the graphical and tabular form. MATHEMATICA 5.2 software is used for programming code and solving the particular problems numerically. It is found that the calculated results for a particular problem using the Runge-Kutta fourth order give good agreement with the exact solution, whereas the Runge-Kutta sixth order defers slightly for a particular problem. Approximate solution using the Adams-Bashforth method with error estimation is also investigated. Moreover, we are also investigated of the Euler methods, the Runge-Kutta methods of order 4th & 6th and the Adams-Bashforth method for solving a particular initial value problem. Finally, it is found that the Adams-Bashforth method gives a better approximation result among the others mentioned methods for solving initial value problems in ordinary differential equations.
| Published in | American Journal of Applied Mathematics (Volume 11, Issue 6) |
| DOI | 10.11648/j.ajam.20231106.12 |
| Page(s) | 106-118 |
| 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), 2023. Published by Science Publishing Group |
Euler’s Method (EM), Runge-Kutta Method of Order Four (RK-4), Runge-Kutta Method of Order Six (RK-6), Adamsh-Bashforth Moulton Method (ABMM)
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APA Style
Sumon, M. M. I., Nurulhoque, M. (2023). A Comparative Study of Numerical Methods for Solving Initial Value Problems (IVP) of Ordinary Differential Equations (ODE). American Journal of Applied Mathematics, 11(6), 106-118. https://doi.org/10.11648/j.ajam.20231106.12
ACS Style
Sumon, M. M. I.; Nurulhoque, M. A Comparative Study of Numerical Methods for Solving Initial Value Problems (IVP) of Ordinary Differential Equations (ODE). Am. J. Appl. Math. 2023, 11(6), 106-118. doi: 10.11648/j.ajam.20231106.12
AMA Style
Sumon MMI, Nurulhoque M. A Comparative Study of Numerical Methods for Solving Initial Value Problems (IVP) of Ordinary Differential Equations (ODE). Am J Appl Math. 2023;11(6):106-118. doi: 10.11648/j.ajam.20231106.12
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Download@article{10.11648/j.ajam.20231106.12,
author = {Md. Monirul Islam Sumon and Md. Nurulhoque},
title = {A Comparative Study of Numerical Methods for Solving Initial Value Problems (IVP) of Ordinary Differential Equations (ODE)},
journal = {American Journal of Applied Mathematics},
volume = {11},
number = {6},
pages = {106-118},
doi = {10.11648/j.ajam.20231106.12},
url = {https://doi.org/10.11648/j.ajam.20231106.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20231106.12},
abstract = {Numerical methods for solving Ordinary Differential Equations differ in accuracy, performance, and applicability. This paper presents a comparative study of numerical methods, mainly Euler’s method, the Runge-Kutta method of order 4th & 6th and the Adams-Bashforth-Moulton method for solving initial value problems in ordinary differential equations. Our aim in this paper is to show that which method gives better accuracy for the initial value problem in numerical methods. Comparisons are made among the direct method, Euler’s method, Runge-Kutta fourth and sixth order and the Adams-Bashforth-Moulton method for solving the initial value problem. The comparisons with error analysis are also shown in the graphical and tabular form. MATHEMATICA 5.2 software is used for programming code and solving the particular problems numerically. It is found that the calculated results for a particular problem using the Runge-Kutta fourth order give good agreement with the exact solution, whereas the Runge-Kutta sixth order defers slightly for a particular problem. Approximate solution using the Adams-Bashforth method with error estimation is also investigated. Moreover, we are also investigated of the Euler methods, the Runge-Kutta methods of order 4th & 6th and the Adams-Bashforth method for solving a particular initial value problem. Finally, it is found that the Adams-Bashforth method gives a better approximation result among the others mentioned methods for solving initial value problems in ordinary differential equations.
},
year = {2023}
}
TY - JOUR T1 - A Comparative Study of Numerical Methods for Solving Initial Value Problems (IVP) of Ordinary Differential Equations (ODE) AU - Md. Monirul Islam Sumon AU - Md. Nurulhoque Y1 - 2023/11/21 PY - 2023 N1 - https://doi.org/10.11648/j.ajam.20231106.12 DO - 10.11648/j.ajam.20231106.12 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 106 EP - 118 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20231106.12 AB - Numerical methods for solving Ordinary Differential Equations differ in accuracy, performance, and applicability. This paper presents a comparative study of numerical methods, mainly Euler’s method, the Runge-Kutta method of order 4th & 6th and the Adams-Bashforth-Moulton method for solving initial value problems in ordinary differential equations. Our aim in this paper is to show that which method gives better accuracy for the initial value problem in numerical methods. Comparisons are made among the direct method, Euler’s method, Runge-Kutta fourth and sixth order and the Adams-Bashforth-Moulton method for solving the initial value problem. The comparisons with error analysis are also shown in the graphical and tabular form. MATHEMATICA 5.2 software is used for programming code and solving the particular problems numerically. It is found that the calculated results for a particular problem using the Runge-Kutta fourth order give good agreement with the exact solution, whereas the Runge-Kutta sixth order defers slightly for a particular problem. Approximate solution using the Adams-Bashforth method with error estimation is also investigated. Moreover, we are also investigated of the Euler methods, the Runge-Kutta methods of order 4th & 6th and the Adams-Bashforth method for solving a particular initial value problem. Finally, it is found that the Adams-Bashforth method gives a better approximation result among the others mentioned methods for solving initial value problems in ordinary differential equations. VL - 11 IS - 6 ER -
Department of Business Administration, Z. H. Sikder University of Science and Technology, Shariatpur, Bangladesh
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