%0 Journal Article
%T A New Stochastic Technique for Painlev¨¦ Equation-I Using Neural Network Optimized with Swarm Intelligence
%A Muhammad Asif Zahoor Raja
%A Junaid Ali Khan
%A Siraj-ul-Islam Ahmad
%A Ijaz Mansoor Qureshi
%J Computational Intelligence and Neuroscience
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/721867
%X A methodology for solution of Painlev¨¦ 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. 1. Introduction The history of Painlev¨¦ equations is more than one century old. These are six second-order nonlinear irreducible equations that define new transcendental functions known as Painlev¨¦ Transcendents. These functions describe different physical processes and have been extensively used in both pure [1] and applied mathematics [2], along with theoretical physics [3]. For instance, Painlev¨¦ Transcendents are used in solutions to Korteweg-de Vries (KdV), cylindrical KdV and Bussinesq equations [4, 5], bifurcations in nonlinear non-integral models [6], matrix models of quantum gravity with continuous limits [7, 8], among others. The history, importance, and applications of Painlev¨¦ transcendents can be seen elsewhere [9, 10]. In recent publications, Painlev¨¦ equation-I (PE-I) is used in Tronqu¨¦e [11] and hyperasymptotic [12] solutions, as well as in modeling the viscous shocks in Hele-shaw flow and Stokes phenomena [13]. Beside this, many researchers have employed PE-I in diverse fields of applied science and engineering [14¨C22]. In this article, our investigation is confined to find out solution of initial value problem (IVP) of nonlinear second-order PE-I, written in the following form: In the present study, the strength of feed forward artificial neural networks (ANNs) is exploited for approximate mathematical model of PE-I. The real strength of such model to solve the differential equations can be achieved by using modern stochastic solvers for optimization of weights based on particle swarm optimization (PSO) technique hybrid with local search
%U http://www.hindawi.com/journals/cin/2012/721867/