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Discrete Dynamics in Nature and Society
Volume 2014 (2014), Article ID 401696, 15 pages
http://dx.doi.org/10.1155/2014/401696
Research Article

Optimization Solution of Troesch’s and Bratu’s Problems of Ordinary Type Using Novel Continuous Genetic Algorithm

1Department of Mechatronics Engineering, Faculty of Engineering, The University of Jordan, Amman 11942, Jordan
2Department of Mathematics, Faculty of Science, Al-Balqa’ Applied University, Salt 19117, Jordan
3Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, Jordan
4Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received 23 August 2013; Accepted 16 December 2013; Published 9 February 2014

Academic Editor: Stepan Agop Tersian

Copyright © 2014 Zaer Abo-Hammour 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.

Abstract

A new kind of optimization technique, namely, continuous genetic algorithm, is presented in this paper for numerically approximating the solutions of Troesch’s and Bratu’s problems. The underlying idea of the method is to convert the two differential problems into discrete versions by replacing each of the second derivatives by an appropriate difference quotient approximation. The new method has the following characteristics. First, it should not resort to more advanced mathematical tools; that is, the algorithm should be simple to understand and implement and should be thus easily accepted in the mathematical and physical application fields. Second, the algorithm is of global nature in terms of the solutions obtained as well as its ability to solve other mathematical and physical problems. Third, the proposed methodology has an implicit parallel nature which points to its implementation on parallel machines. The algorithm is tested on different versions of Troesch’s and Bratu’s problems. Experimental results show that the proposed algorithm is effective, straightforward, and simple.