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
Volume 6, Issue 6, December 2017, Pages: 243-253
Received: Sep. 26, 2017;
Accepted: Oct. 8, 2017;
Published: Nov. 8, 2017
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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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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.
Differential Equation, Initial Value Problem, Error, Butcher’s Method, Runge-Kutta Method
To cite this article
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), Applied and Computational Mathematics.
Vol. 6, No. 6,
2017, pp. 243-253.
Copyright © 2017 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/
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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.
Butcher, J. C. (1995). On fifth order Runge-Kutta methods. BIT Numerical Mathematics, 35 (2), 202-209.
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.
Butcher, J. C. (1964). On Runge-Kutta processes of high order. Journal of the Australian Mathematical Society, 4 (02), 179-194.
Butcher, J. C. (1996). A history of Runge-Kutta methods. Applied numerical mathematics, 20 (3), 247-260.
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.
Goeken, D., & Johnson, O. (2000). Runge–Kutta with higher order derivative approximations. Applied numerical mathematics, 34 (2-3), 207-218.
Lambert, J. D. (1973). Computational methods in ordinary differential equations. Wiley, New York.
Hall, G. and Watt, J. M. (1976) Modern Numerical Methods for Ordinary Differential Equations. Oxford University Press, Oxford.
Mathews, J. H. (2005) Numerical Methods for Mathematics, Science and Engineering. Prentice-Hall, India.
Gerald, C. F. and Wheatley, P. O. (2002) Applied Numerical Analysis. Pearson Education, India.
Burden, R. L. and Faires, J. D. (2002) Numerical Analysis. Bangalore, India.
Sastry, S. S. (2000) Introductory Methods of Numerical Analysis. Prentice-Hall, India.
Balagurusamy, E. (2006) Numerical Methods. Tata McGraw-Hill, New Delhi.
Hossain, Md. S., Bhattacharjee, P. K. and Hossain, Md. E. (2013) Numerical Analysis. Titas Publications, Dhaka.