Numerical Solution of Second-Order Ordinary Differential Equations by Improved Runge-Kutta Nystrom Method

In this paper we developed the Improved Runge-Kutta Nystrom (IRKN) method for solving second order ordinary differential equations. The methods are two step in nature and require lower number of function evaluations per step compared with the existing Runge-Kutta Nystrom (RKN) methods. Therefore, the methods are computationally more efficient at achieving the higher order of local accuracy. Algebraic order conditions of the method are obtained and the third and fourth order method are derived with two and three stages respectively. The numerical results are given to illustrate the efficiency of the proposed method compared to the existing RKN methods.

• Faranak Rabiei
• Fudziah Ismail
• S. Norazak
• Saeid Emadi

• Improved Runge-Kutta Nystrom method
• Two step method
• Second-order ordinary differential equations
• Order conditions

[1] P. Phohomsiri, F. E. Udwadia, Acceleration of Runge-Kutta integeration
schemes. Discrete Dynamics in Nature and Society. 2: 307-314
[2] F.E. Udwadia, A. Farahani, Accelerated Runge-Kutta methods. Discrete
Dynamics in Nature and Society. doi:10.1155/2008/790619
[3] F. Rabiei, F. Ismail, Third-order Improved Runge-Kutta method for solving ordinary differential equation. International Journal of Applied Physics and Mathematics (IJAPM). 2011 Vol.1(3): 191-194 ISSN:2010-
362X
[4] F. Rabiei, F. Ismail, M.Sulieman, Improved Runge-Kutta method for solving ordinary differential equation. Sains Malaysiana, Submitted (2011)
[5] F. Rabiei, F. Ismail, Fifth-order Improved Runge-Kutta method for solving
ordinary differential equation. Australian Journal of Basic and Applied
Sciencs, 6(3), pp 97-105, (2012)
[6] J. R. Dormand, Numerical Method for Differential Equations (A Computational
Approach). CRC Prees. Inc (1996)
[7] H. Van de Vyver, A Runge-Kutta-Nystrom pair for the numerical integration
of perturbed oscillators, Computer Physics Communications, vol.
167, no. 2, pp. 129142, 2005.
[8] E. Stiefel and D. G. Bettis, Stabilization of Cowells method, Numerische
Mathematik, vol. 13(2), pp. 154175, (1969).
[9] P.J. van der Houwen, B.P. Sommeijer, Explicit RungeKutta(Nystrom)
methods with reduced phase errors for computing oscillating solutions,
SIAM , Numerical Analysis. Vol 24, pp 595617,(1987)
[10] A. Garca, P. Martn, and A. B. Gonzalez, New methods for oscillatory problems based on classical codes, Applied Numerical Mathematics, vol.
42(13), pp. 141157, (2002).