Comparison of Numerical Methods for System of First Order Ordinary Differential Equations

Jemal Demsie Abraha

doi:doi:10.11648/j.pamj.20200902.11

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Comparison of Numerical Methods for System of First Order Ordinary Differential Equations

Jemal Demsie Abraha

Received: 17 January 2020 Accepted: 26 February 2020 Published: 14 April 2020

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Abstract

In this paper three numerical methods are discussed to find the approximate solutions of a systems of first order ordinary differential equations. Those are Classical Runge-Kutta method, Modified Euler method and Euler method. For each methods formulas are developed for n systems of ordinary differential equations. The formulas explained by these methods are demonstrated by examples to identify the most accurate numerical methods. By comparing the analytical solution of the dependent variables with the approximate solution, absolute errors are calculated. The resulting value indicates that classical fourth order Runge-Kutta method offers most closet values with the computed analytical values. Finally from the results the classical fourth order is more efficient method to find the approximate solutions of the systems of ordinary differential equations.

DOI	10.11648/j.pamj.20200902.11
Published in	Pure and Applied Mathematics Journal (Volume 9, Issue 2, April 2020)
Page(s)	32-36
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

Keywords

Euler Method, Modified Euler Method, Runge-Kutta Method, System of Ordinary Differential Equations

References

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[2]	Richard L., J.Burden, F.Douglas. Numerical Analysis. Richard Stratton, Canada, ninth edition, 2011.
[3]	Steven C. Chapra. Applied Numerical Methods with MATLAB for Engineers and Scientists. HcGraw-Hill, New York, NY10020, third edition, 2012.
[4]	N. Shawagfer, D. Kaya (2004) Comparing Numerical Methods for the Solutions of Systems of Ordinary Differential Equations.Elsevier, Applied Mathematics Letters 17, pp. 323-328.
[5]	S. R. K. Iyengar and R. K. Jain. Numerical Methods. New Age International (P) Limited. New Delhi-110002, 2009.
[6]	Joe D. Hoffman. Numerical Methods for Engineers and Scientists. Mc Graw-Hill,Inc(New York, 1992) second edition.
[7]	Md. Jahangir Hossain, Md. Shah Alam, Md. Babul Hossain (2017). A stdy on the Numerical Solutions of Second Order Initial Value Problems(IVP) for Ordinary Differential Equations with Fourth Order and Butcher’s Fifth Order Runge-Kutta Mthods. American Journal of Computational and Applied Mathematics, 7 (5), pp. 129- 137.
[8]	A. Iserles. A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, Cambridge, United Kingdom, 1996.
[9]	S. E. Fadugba and T. E. Olaosebikan (2018). Comparative Study of Class of One-Step Methods for Numerical Solution of Some Initial Value Problems in Ordinary Differential Equations. Research Journal of Mathematics and Computer Science, 2: 9, pp. 1-11.
[10]	Gadamsetty Revathi (2017). Numerical Solution of Ordinary Differential equations and Applications, International Journal of Management and Applied Science, ISSN: 2394-7926 Vol 3, pp. 1-5.
[11]	C. Senthilnathan(2018). Anumerical Solutions of Initial Value Problems (IVP) for Ordinary Differential Equations (ODE) with Euler and Higher Order of Runge Kutta Methods Using Matlab. International Journal of Engineering Science Invention (IJESI) ISSN (online): 2319-6734, ISSN (print): 2319-6726 Volume 7, pp. 25- 31.
[12]	Murad Hossen, Zain Ahmed, Rashadul Kabir, Zakir Hossan (2019). Acomparative Investigation on Numerical Solution of Initial Value Problem by Using Modiffied Euler Method and Runge Kutta Method. ISOR Journal of Mathematics (IOSR-JM)e-ISSN: 2278-5728, P-ISSN: 2319-765X. Volume 15, pp. 40-45.
[13]	J. C. Butcher. The Numerical Analysis of Ordinary Differential Equations: Runge Kutta and General Linear Methods. John Wiley, New York, NY, 1987.
[14]	John H. Mathews, Kurtis D. Fink. Numerical Methods Using Matlab, prentice Hall, Upper Saddle River, NJ 07458, third edition, 1999.
[15]	Stephen J. Chapman. MATLAB Programming for Engineers, Thomson Learning, 2004.

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Jemal Demsie Abraha. (2020). Comparison of Numerical Methods for System of First Order Ordinary Differential Equations. Pure and Applied Mathematics Journal, 9(2), 32-36. https://doi.org/10.11648/j.pamj.20200902.11

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ACS Style

Jemal Demsie Abraha. Comparison of Numerical Methods for System of First Order Ordinary Differential Equations. Pure Appl. Math. J. 2020, 9(2), 32-36. doi: 10.11648/j.pamj.20200902.11

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AMA Style

Jemal Demsie Abraha. Comparison of Numerical Methods for System of First Order Ordinary Differential Equations. Pure Appl Math J. 2020;9(2):32-36. doi: 10.11648/j.pamj.20200902.11

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@article{10.11648/j.pamj.20200902.11,
  author = {Jemal Demsie Abraha},
  title = {Comparison of Numerical Methods for System of First Order Ordinary Differential Equations},
  journal = {Pure and Applied Mathematics Journal},
  volume = {9},
  number = {2},
  pages = {32-36},
  doi = {10.11648/j.pamj.20200902.11},
  url = {https://doi.org/10.11648/j.pamj.20200902.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20200902.11},
  abstract = {In this paper three numerical methods are discussed to find the approximate solutions of a systems of first order ordinary differential equations. Those are Classical Runge-Kutta method, Modified Euler method and Euler method. For each methods formulas are developed for n systems of ordinary differential equations. The formulas explained by these methods are demonstrated by examples to identify the most accurate numerical methods. By comparing the analytical solution of the dependent variables with the approximate solution, absolute errors are calculated. The resulting value indicates that classical fourth order Runge-Kutta method offers most closet values with the computed analytical values. Finally from the results the classical fourth order is more efficient method to find the approximate solutions of the systems of ordinary differential equations.},
 year = {2020}
}

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TY  - JOUR
T1  - Comparison of Numerical Methods for System of First Order Ordinary Differential Equations
AU  - Jemal Demsie Abraha
Y1  - 2020/04/14
PY  - 2020
N1  - https://doi.org/10.11648/j.pamj.20200902.11
DO  - 10.11648/j.pamj.20200902.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 32
EP  - 36
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20200902.11
AB  - In this paper three numerical methods are discussed to find the approximate solutions of a systems of first order ordinary differential equations. Those are Classical Runge-Kutta method, Modified Euler method and Euler method. For each methods formulas are developed for n systems of ordinary differential equations. The formulas explained by these methods are demonstrated by examples to identify the most accurate numerical methods. By comparing the analytical solution of the dependent variables with the approximate solution, absolute errors are calculated. The resulting value indicates that classical fourth order Runge-Kutta method offers most closet values with the computed analytical values. Finally from the results the classical fourth order is more efficient method to find the approximate solutions of the systems of ordinary differential equations.
VL  - 9
IS  - 2
ER  -

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Author Information

Jemal Demsie Abraha

Departiment of Mthematics, Wolaita Sodo University, Wolaita Sodo, Ethiopia

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APA Style

Jemal Demsie Abraha. (2020). Comparison of Numerical Methods for System of First Order Ordinary Differential Equations. Pure and Applied Mathematics Journal, 9(2), 32-36. https://doi.org/10.11648/j.pamj.20200902.11

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ACS Style

Jemal Demsie Abraha. Comparison of Numerical Methods for System of First Order Ordinary Differential Equations. Pure Appl. Math. J. 2020, 9(2), 32-36. doi: 10.11648/j.pamj.20200902.11

Copy | Download

AMA Style

Jemal Demsie Abraha. Comparison of Numerical Methods for System of First Order Ordinary Differential Equations. Pure Appl Math J. 2020;9(2):32-36. doi: 10.11648/j.pamj.20200902.11

Copy | Download

@article{10.11648/j.pamj.20200902.11,
  author = {Jemal Demsie Abraha},
  title = {Comparison of Numerical Methods for System of First Order Ordinary Differential Equations},
  journal = {Pure and Applied Mathematics Journal},
  volume = {9},
  number = {2},
  pages = {32-36},
  doi = {10.11648/j.pamj.20200902.11},
  url = {https://doi.org/10.11648/j.pamj.20200902.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20200902.11},
  abstract = {In this paper three numerical methods are discussed to find the approximate solutions of a systems of first order ordinary differential equations. Those are Classical Runge-Kutta method, Modified Euler method and Euler method. For each methods formulas are developed for n systems of ordinary differential equations. The formulas explained by these methods are demonstrated by examples to identify the most accurate numerical methods. By comparing the analytical solution of the dependent variables with the approximate solution, absolute errors are calculated. The resulting value indicates that classical fourth order Runge-Kutta method offers most closet values with the computed analytical values. Finally from the results the classical fourth order is more efficient method to find the approximate solutions of the systems of ordinary differential equations.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Comparison of Numerical Methods for System of First Order Ordinary Differential Equations
AU  - Jemal Demsie Abraha
Y1  - 2020/04/14
PY  - 2020
N1  - https://doi.org/10.11648/j.pamj.20200902.11
DO  - 10.11648/j.pamj.20200902.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 32
EP  - 36
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20200902.11
AB  - In this paper three numerical methods are discussed to find the approximate solutions of a systems of first order ordinary differential equations. Those are Classical Runge-Kutta method, Modified Euler method and Euler method. For each methods formulas are developed for n systems of ordinary differential equations. The formulas explained by these methods are demonstrated by examples to identify the most accurate numerical methods. By comparing the analytical solution of the dependent variables with the approximate solution, absolute errors are calculated. The resulting value indicates that classical fourth order Runge-Kutta method offers most closet values with the computed analytical values. Finally from the results the classical fourth order is more efficient method to find the approximate solutions of the systems of ordinary differential equations.
VL  - 9
IS  - 2
ER  -

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