A Comparative Study on Fourth Order and Butcher’s Fifth Order Runge-Kutta Methods with Third Order Initial Value Problem (IVP)

Md. Babul Hossain; Md. Jahangir Hossain; Md. Musa Miah; Md. Shah Alam

doi:doi:10.11648/j.acm.20170606.12

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A Comparative Study on Fourth Order and Butcher’s Fifth Order Runge-Kutta Methods with Third Order Initial Value Problem (IVP)

Md. Babul Hossain, Md. Jahangir Hossain, Md. Musa Miah, Md. Shah Alam

Published in Applied and Computational Mathematics (Volume 6, Issue 6)

Received: 26 September 2017 Accepted: 8 October 2017 Published: 8 November 2017

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Abstract

In this paper, Butcher’s fifth order Runge-Kutta (RK5) and fourth order Runge-Kutta (RK4) methods have been employed to solve the Initial Value Problems (IVP) involving third order Ordinary Differential Equations (ODE). These two proposed methods are quite proficient and practically well suited for solving engineering problems based on such problems. To obtain the accuracy of the numerical outcome for this study, we have compared the approximate results with the exact results and found a good agreement between the exact and approximate solutions. In addition, to achieve more accuracy in the solution, the step size needs to be very small. Moreover, the error terms have been analyzed for these two methods and also compared by an appropriate example.

Published in	Applied and Computational Mathematics (Volume 6, Issue 6)
DOI	10.11648/j.acm.20170606.12
Page(s)	243-253
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), 2017. Published by Science Publishing Group

Keywords

Differential Equation, Initial Value Problem, Error, Butcher’s Method, Runge-Kutta Method

References

[1]	Rabiei, F., & Ismail, F. (2012). Fifth-order Improved Runge-Kutta method for solving ordinary differential equation. Australian Journal of Basic and Applied Sciences, 6 (3), 97-105.
[2]	Butcher, J. C. (1995). On fifth order Runge-Kutta methods. BIT Numerical Mathematics, 35 (2), 202-209.
[3]	Islam, M. A. (2015). A Comparative Study on Numerical Solutions of Initial Value Problems (IVP) for Ordinary Differential Equations (ODE) with Euler and Runge-kutta Methods. American Journal of Computational Mathematics, 5 (03), 393.
[4]	Butcher, J. C. (1964). On Runge-Kutta processes of high order. Journal of the Australian Mathematical Society, 4 (02), 179-194.
[5]	Butcher, J. C. (1996). A history of Runge-Kutta methods. Applied numerical mathematics, 20 (3), 247-260.
[6]	Islam, M. A. (2015). Accuracy Analysis of Numerical solutions of Initial Value Problems (IVP) for Ordinary Differential Equations (ODE). IOSR Journal of Mathematics, 11, 18-23.
[7]	Goeken, D., & Johnson, O. (2000). Runge–Kutta with higher order derivative approximations. Applied numerical mathematics, 34 (2-3), 207-218.
[8]	Lambert, J. D. (1973). Computational methods in ordinary differential equations. Wiley, New York.
[9]	Hall, G. and Watt, J. M. (1976) Modern Numerical Methods for Ordinary Differential Equations. Oxford University Press, Oxford.
[10]	Mathews, J. H. (2005) Numerical Methods for Mathematics, Science and Engineering. Prentice-Hall, India.
[11]	Gerald, C. F. and Wheatley, P. O. (2002) Applied Numerical Analysis. Pearson Education, India.
[12]	Burden, R. L. and Faires, J. D. (2002) Numerical Analysis. Bangalore, India.
[13]	Sastry, S. S. (2000) Introductory Methods of Numerical Analysis. Prentice-Hall, India.
[14]	Balagurusamy, E. (2006) Numerical Methods. Tata McGraw-Hill, New Delhi.
[15]	Hossain, Md. S., Bhattacharjee, P. K. and Hossain, Md. E. (2013) Numerical Analysis. Titas Publications, Dhaka.

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Md. Babul Hossain, Md. Jahangir Hossain, Md. Musa Miah, Md. Shah Alam. (2017). A Comparative Study on Fourth Order and Butcher’s Fifth Order Runge-Kutta Methods with Third Order Initial Value Problem (IVP). Applied and Computational Mathematics, 6(6), 243-253. https://doi.org/10.11648/j.acm.20170606.12

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ACS Style

Md. Babul Hossain; Md. Jahangir Hossain; Md. Musa Miah; Md. Shah Alam. A Comparative Study on Fourth Order and Butcher’s Fifth Order Runge-Kutta Methods with Third Order Initial Value Problem (IVP). Appl. Comput. Math. 2017, 6(6), 243-253. doi: 10.11648/j.acm.20170606.12

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AMA Style

Md. Babul Hossain, Md. Jahangir Hossain, Md. Musa Miah, Md. Shah Alam. A Comparative Study on Fourth Order and Butcher’s Fifth Order Runge-Kutta Methods with Third Order Initial Value Problem (IVP). Appl Comput Math. 2017;6(6):243-253. doi: 10.11648/j.acm.20170606.12

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@article{10.11648/j.acm.20170606.12,
  author = {Md. Babul Hossain and Md. Jahangir Hossain and Md. Musa Miah and Md. Shah Alam},
  title = {A Comparative Study on Fourth Order and Butcher’s Fifth Order Runge-Kutta Methods with Third Order Initial Value Problem (IVP)},
  journal = {Applied and Computational Mathematics},
  volume = {6},
  number = {6},
  pages = {243-253},
  doi = {10.11648/j.acm.20170606.12},
  url = {https://doi.org/10.11648/j.acm.20170606.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20170606.12},
  abstract = {In this paper, Butcher’s fifth order Runge-Kutta (RK5) and fourth order Runge-Kutta (RK4) methods have been employed to solve the Initial Value Problems (IVP) involving third order Ordinary Differential Equations (ODE). These two proposed methods are quite proficient and practically well suited for solving engineering problems based on such problems. To obtain the accuracy of the numerical outcome for this study, we have compared the approximate results with the exact results and found a good agreement between the exact and approximate solutions. In addition, to achieve more accuracy in the solution, the step size needs to be very small. Moreover, the error terms have been analyzed for these two methods and also compared by an appropriate example.},
 year = {2017}
}

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TY  - JOUR
T1  - A Comparative Study on Fourth Order and Butcher’s Fifth Order Runge-Kutta Methods with Third Order Initial Value Problem (IVP)
AU  - Md. Babul Hossain
AU  - Md. Jahangir Hossain
AU  - Md. Musa Miah
AU  - Md. Shah Alam
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AB  - In this paper, Butcher’s fifth order Runge-Kutta (RK5) and fourth order Runge-Kutta (RK4) methods have been employed to solve the Initial Value Problems (IVP) involving third order Ordinary Differential Equations (ODE). These two proposed methods are quite proficient and practically well suited for solving engineering problems based on such problems. To obtain the accuracy of the numerical outcome for this study, we have compared the approximate results with the exact results and found a good agreement between the exact and approximate solutions. In addition, to achieve more accuracy in the solution, the step size needs to be very small. Moreover, the error terms have been analyzed for these two methods and also compared by an appropriate example.
VL  - 6
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Author Information

Md. Babul Hossain

Department of Mathematics, Mawlana Bhashani Science and Technology University, Tangail-1902, Bangladesh
Md. Jahangir Hossain

Department of Basic Science, World University of Bangladesh, Dhaka, Bangladesh
Md. Musa Miah

Department of Mathematics, Mawlana Bhashani Science and Technology University, Tangail-1902, Bangladesh
Md. Shah Alam

Department of Mathematics, Mawlana Bhashani Science and Technology University, Tangail-1902, Bangladesh

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Plain Text BibTeX RIS

APA Style

Md. Babul Hossain, Md. Jahangir Hossain, Md. Musa Miah, Md. Shah Alam. (2017). A Comparative Study on Fourth Order and Butcher’s Fifth Order Runge-Kutta Methods with Third Order Initial Value Problem (IVP). Applied and Computational Mathematics, 6(6), 243-253. https://doi.org/10.11648/j.acm.20170606.12

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ACS Style

Md. Babul Hossain; Md. Jahangir Hossain; Md. Musa Miah; Md. Shah Alam. A Comparative Study on Fourth Order and Butcher’s Fifth Order Runge-Kutta Methods with Third Order Initial Value Problem (IVP). Appl. Comput. Math. 2017, 6(6), 243-253. doi: 10.11648/j.acm.20170606.12

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AMA Style

Md. Babul Hossain, Md. Jahangir Hossain, Md. Musa Miah, Md. Shah Alam. A Comparative Study on Fourth Order and Butcher’s Fifth Order Runge-Kutta Methods with Third Order Initial Value Problem (IVP). Appl Comput Math. 2017;6(6):243-253. doi: 10.11648/j.acm.20170606.12

Copy | Download

@article{10.11648/j.acm.20170606.12,
  author = {Md. Babul Hossain and Md. Jahangir Hossain and Md. Musa Miah and Md. Shah Alam},
  title = {A Comparative Study on Fourth Order and Butcher’s Fifth Order Runge-Kutta Methods with Third Order Initial Value Problem (IVP)},
  journal = {Applied and Computational Mathematics},
  volume = {6},
  number = {6},
  pages = {243-253},
  doi = {10.11648/j.acm.20170606.12},
  url = {https://doi.org/10.11648/j.acm.20170606.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20170606.12},
  abstract = {In this paper, Butcher’s fifth order Runge-Kutta (RK5) and fourth order Runge-Kutta (RK4) methods have been employed to solve the Initial Value Problems (IVP) involving third order Ordinary Differential Equations (ODE). These two proposed methods are quite proficient and practically well suited for solving engineering problems based on such problems. To obtain the accuracy of the numerical outcome for this study, we have compared the approximate results with the exact results and found a good agreement between the exact and approximate solutions. In addition, to achieve more accuracy in the solution, the step size needs to be very small. Moreover, the error terms have been analyzed for these two methods and also compared by an appropriate example.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - A Comparative Study on Fourth Order and Butcher’s Fifth Order Runge-Kutta Methods with Third Order Initial Value Problem (IVP)
AU  - Md. Babul Hossain
AU  - Md. Jahangir Hossain
AU  - Md. Musa Miah
AU  - Md. Shah Alam
Y1  - 2017/11/08
PY  - 2017
N1  - https://doi.org/10.11648/j.acm.20170606.12
DO  - 10.11648/j.acm.20170606.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 243
EP  - 253
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20170606.12
AB  - In this paper, Butcher’s fifth order Runge-Kutta (RK5) and fourth order Runge-Kutta (RK4) methods have been employed to solve the Initial Value Problems (IVP) involving third order Ordinary Differential Equations (ODE). These two proposed methods are quite proficient and practically well suited for solving engineering problems based on such problems. To obtain the accuracy of the numerical outcome for this study, we have compared the approximate results with the exact results and found a good agreement between the exact and approximate solutions. In addition, to achieve more accuracy in the solution, the step size needs to be very small. Moreover, the error terms have been analyzed for these two methods and also compared by an appropriate example.
VL  - 6
IS  - 6
ER  -

Copy | Download