J. Basic. Appl. Sci. Res., 3(7)10-16, 2013 © 2013, TextRoad Publication ISSN 2090-4304 Journal of Basic and Applied Scientific Research www.textroad.com *Corresponding Author: Suheel Abdullah Malik, Department of Electronic Engineering, Faculty of Engineering & Technology, International Islamic University Islamabad, Pakistan, Email: suheel.abdullah@iiu.edu.pk Numerical Solution to Troesch’s Problem Using Hybrid Heuristic Computing Suheel Abdullah Malik a , Ijaz Mansoor Qureshi b , Muhammad Zubair a , Muhammad Amir a a Department of Electronic Engineering, Faculty of Engineering and Technology, International Islamic University, Islamabad, Pakistan b Department of Electrical Engineering, Air University, Islamabad, Pakistan ABSTRACT In this paper an evolutionary computational technique, which is stochastic in nature, is applied for the first time to the numerical solution of the Troesch’s problem. An approximate mathematical model employing the linear combinations of log sigmoid basis functions is deduced. A fitness function (FF), containing unknown adjustable parameters and representing the trial solution along with the initial conditions, is developed. The Genetic algorithm (GA) as global optimizer is hybridized with local optimizers, such as Pattern Search (PS) and Interior Point algorithm (IPA), for achieving the unknown adjustable parameters. Comparisons of the numerical results obtained for three special cases of the Troesch’s problem are made with some important deterministic classical approximation methods, as well as, the exact solutions. The numerical results are found to be in sharp agreement with the exact solutions. KEYWORDS: Troesch’s Problem; Boundary Value Problem; Genetic Algorithm (GA), Interior point Algorithm (IPA), Pattern Search (PS), Evolutionary Computing (EC) 1. INTRODUCTION Nonlinear ordinary differential equations (ODEs) frequently arise in a wide variety of problems in applied science and engineering. These problems are by and large formulated as initial and/or boundary value problems. Traditionally these nonlinear boundary value problems (BVPs) are solved by the approximate analytical and numerical methods such as homotopy perturbation method (HPM) [1], homotopy analysis method (HAM) [2], variational iteration method (VIM) [3], adomian decomposition method (ADM) [4], shooting method [5] etc. This paper is devoted primarily to the investigation of the Troesch’s problem [1-8 ] ݑ ′′ ( ݔ) = ߣsinh( ߣݑ( ݔ) ) 0 ≤ ݔ≤ 1 (1) with the boundary conditions ݑ(0) = 0,  ݑ(1) = 1 Troesch’s problem appears in the investigation of the confinement of a plasma column by radiation pressure and applied physics [1-9]. This BVP was formulated and solved by Weibel [10]. Troesch’s obtained the numerical solution of this problem by the shooting method [11]. The Troesch’s problem has been extensively studied and various methods have been proposed for its numerical solution. Hassan et al. [2] applied homotopy analysis method (HAM) to the Troesch’s problem for its numerical solution. Mirmoradi et al. [1] found the solution of Troesch’s problem using homotopy perturbation method (HPM). Deeba et al. [4] employed decomposition based method (ADM) for the approximate solution of this problem. Recently Geng and Cui [8] proposed a method based on the combination of ADM and reproducing kernel method (RKM) for the numerical solution of the Troesch’s problem. Several other authors have obtained numerical solution of the Troesch’s problem by utilizing various approximate methods such as modified homotopy perturbation method (MHPM) [7], He’s polynomials [6], B-spline collocation [9], and simple shooting method [5]. Although a rich variety of approximate numerical techniques have been proposed for handling nonlinear ODEs, an incredible amount of research work is still carried out in this direction. Besides the traditional numerical techniques, stochastic solvers based on evolutionary computing and neural networks (NN) have been successfully applied to nonlinear ODEs. The efficiency and reliability of these stochastic solvers has been demonstrated by several authors [12-15]. Khan et al. [12] used a neural network (NN) model optimized by hybrid Particle Swarm Optimization (PSO) for the solution of nonlinear ODEs including the Wessinger’s equation. Behrang et al. [13] solved a nonlinear differential equation arising from similarity solution of inverted cone embedded in porous medium employing a PSO based neural network (NN). Malik et al. [14] obtained the numerical solution of Duffing van der pol equation using heuristic computing technique. Zahoor et al. [15] applied hybrid evolutionary computing approach for the solution of fractional order Riccati differential equation. The main goal of this study is to obtain the approximate numerical solution of the Troesch’s problem (1) using hybrid heuristic computing approach. Genetic algorithm (GA), pattern search (PS), interior point algorithm (IPA), and two hybrid schemes called as GA-PS (hybridization of GA and PS), and GA-IPA (hybridization of GA and IPA) have been utilized in this study. The efficiency and reliability of the proposed method are demonstrated by solving the Troesch’s problem for three special cases of the constant ߣ (0.5,1,  10) . Comparisons of the numerical results are made with the exact solutions and the classical approximate techniques. 10