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Computational Intelligence and Neuroscience
Volume 2012, Article ID 721867, 10 pages
Research Article

A New Stochastic Technique for Painlevé Equation-I Using Neural Network Optimized with Swarm Intelligence

1Department of Electronic Engineering, International Islamic University, Islamabad, Pakistan
2Center for Computational Intelligence, P.O. Box 2300, Islamabad, Pakistan
3Department of Electrical Engineering, COMSATS Institute of Information Technology, Attock, Pakistan
4Pakistan Institute of Engineering and Applied Science, Nilore, Pakistan

Received 6 March 2012; Revised 9 April 2012; Accepted 13 April 2012

Academic Editor: Christian W. Dawson

Copyright © 2012 Muhammad Asif Zahoor Raja 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.


A methodology for solution of Painlevé equation-I is presented using computational intelligence technique based on neural networks and particle swarm optimization hybridized with active set algorithm. The mathematical model of the equation is developed with the help of linear combination of feed-forward artificial neural networks that define the unsupervised error of the model. This error is minimized subject to the availability of appropriate weights of the networks. The learning of the weights is carried out using particle swarm optimization algorithm used as a tool for viable global search method, hybridized with active set algorithm for rapid local convergence. The accuracy, convergence rate, and computational complexity of the scheme are analyzed based on large number of independents runs and their comprehensive statistical analysis. The comparative studies of the results obtained are made with MATHEMATICA solutions, as well as, with variational iteration method and homotopy perturbation method.