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Mathematical Problems in Engineering
Volume 2015, Article ID 196595, 12 pages
Research Article

Two Embedded Pairs of Runge-Kutta Type Methods for Direct Solution of Special Fourth-Order Ordinary Differential Equations

1Department of Mathematics, Faculty of Science, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia
2Department of Mathematics, College of Science, Al-Mustansiriyah University, Baghdad, Iraq
3Institute for Mathematical Research, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia

Received 14 August 2015; Revised 3 November 2015; Accepted 8 November 2015

Academic Editor: Tarek Ahmed-Ali

Copyright © 2015 Kasim Hussain et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We present two pairs of embedded Runge-Kutta type methods for direct solution of fourth-order ordinary differential equations (ODEs) of the form denoted as RKFD methods. The first pair, which we will call RKFD5, has orders 5 and 4, and the second one has orders 6 and 5 and we will call it RKFD6. The techniques used in the derivation of the methods are that the higher order methods are very precise and the lower order methods give the best error estimate. Based on these pairs, we have developed variable step codes and we have used them to solve a set of special fourth-order problems. Numerical results show the robustness and the efficiency of the new RKFD pairs as compared with the well-known embedded Runge-Kutta pairs in the scientific literature after reducing the problems into a system of first-order ordinary differential equations (ODEs) and solving them.