( ) ( ) ( ) u f uu gu hx ′′ ′ + + = ( ) () () () 3 5 0 u x lu x mu x nu x ′′ + + + = World Applied Sciences Journal 28 (5): 636-643, 2013 ISSN 1818-4952 © IDOSI Publications, 2013 DOI: 10.5829/idosi.wasj.2013.28.05.1645 Corresponding Author: Suheel Abdullah Malik, Department of Electronic Engineering, Faculty of Engineering and Technology, International Islamic University, Islamabad, Pakistan. Tel: +92-333-5178315, Fax: +92-51-9258025. 636 Numerical Solution of Lienard Equation Using Hybrid Heuristic Computation Suheel Abdullah Malik, Ijaz Mansoor Qureshi, Muhammad Amir and Ihsanul Haq 1 2 1 1 Department of Electronic Engineering, Faculty of Engineering and Technology, 1 International Islamic University, Islamabad, Pakistan Department of Electrical Engineering, Air University, Islamabad, Pakistan 2 Abstract: In this paper, a hybrid heuristic computing technique, stochastic in nature, is used for obtaining an approximate numerical solution of the Lienard equation. The proposed technique converts the nonlinear differential equation into an equivalent global error minimization problem. A trial solution is developed using a fitness function with unknown adaptable parameters. The memetic computation or hybrid genetic algorithms (HGAs) combining genetic algorithm (GA) with interior point algorithm (IPA), active set algorithm (ASA) and pattern search (PS) is used to solve the minimization problem and to obtain the unknown adaptable parameters. The accuracy and efficacy of the proposed technique is illustrated by considering the Lienard’s equation with two special cases. Comparison of numerical results is made with the exact solution and two important deterministic standard methods, including variational iteration method (VIM) and differential transform method (DTM). The comparison of numerical results validate the effectiveness and viability of the suggested technique. The results obtained by the proposed method are found to be in excellent agreement with the exact solution. Key words: Lienard equation Memetic computation Hybrid genetic algorithms (HGAs) Interior point algorithm (IPA) Active set algorithm (ASA). INTRODUCTION force and external force respectively). Moreover it is used Nonlinear problems appearing in many physical when taking different choices of f(u), g(u) and h(x). For phenomena, engineering and scientific applications are example, the choices f(u) = (u -1), g(u) = u and h(x) = 0 modeled with nonlinear differential equations. Since most lead (1) to the well-known Van der Pol equation of of the nonlinear differential equations are difficult to be nonlinear electronic oscillator [1-5]. Some nonlinear solved using analytical techniques, these problems must evolution equations such as Burgers-KdV equation can be tackled using approximate analytical and numerical also be transformed to (1) [5]. Therefore the study of (1) methods. Many approximate analytical and numerical is of great importance. methods like finite difference method, differential In the general case, it is commonly believed that it is transform method (DTM), homotopy perturbation very difficult to find the exact solution of (1) by usual method (HPM), adomian decomposition method (ADM), ways [1-5], the following special case was investigated by variational iteration method (VIM) etc. have been vastly authors in [1-7] and references there in. used for solving nonlinear differential equations. In this paper, we consider the Lienard equation [1-5] (2) (1) where l, m and n are real coefficients. Which is regarded as a generalization of damped Many methods including differential transform pendulum equation or damped spring-mass system (where method (DTM) [1], variational iteration method (VIM) f(u), g(u) and h(x) represent the damping force, restoring [2], variational homotopy perturbation method as nonlinear models in many physically significant fields 2