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Abstract and Applied Analysis
Volume 2015 (2015), Article ID 594685, 10 pages
http://dx.doi.org/10.1155/2015/594685
Research Article

Generalized Runge-Kutta Method with respect to the Non-Newtonian Calculus

1Department of Mathematics, Faculty of Sciences, Gazi University, 06100 Ankara, Turkey
2Department of Mathematics, Faculty of Sciences and Arts, Bozok University, 66200 Yozgat, Turkey
3Department of Mathematics, Faculty of Sciences and Arts, Batman University, 72060 Batman, Turkey

Received 1 August 2014; Revised 28 October 2014; Accepted 12 November 2014

Academic Editor: Allaberen Ashyralyev

Copyright © 2015 Uğur Kadak and Muharrem Özlük. 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.

Abstract

Theory and applications of non-Newtonian calculus have been evolving rapidly over the recent years. As numerical methods have a wide range of applications in science and engineering, the idea of the design of such numerical methods based on non-Newtonian calculus is self-evident. In this paper, the well-known Runge-Kutta method for ordinary differential equations is developed in the frameworks of non-Newtonian calculus given in generalized form and then tested for different generating functions. The efficiency of the proposed non-Newtonian Euler and Runge-Kutta methods is exposed by examples, and the results are compared with the exact solutions.