World Applied Sciences Journal 27 (12): 1605-1613, 2013 ISSN 1818-4952 © IDOSI Publications, 2013 DOI: 10.5829/idosi.wasj.2013.27.12.2648 Corresponding Author: Z.A. Zaidi, Department of Mathematics, COMSATS Institute of Information Technology, University Road, Post code 22060, Abbottabad, NWFP, Pakistan 1605 Application of the Optimal Homotopy Asymptotic Method for Fins with Variable temperature Surface Heat Flux Z.A. Zaidi and A. Shahzad Department of Mathematics, COMSATS Institute of Information Technology, University Road, Post code 22060, Abbottabad, NWFP, Pakistan Submitted: Mar 28, 2013; Accepted: May 7, 2013; Published: Dec 22, 2013 Abstract: In this paper, Optimal Homotopy Asymptotic Method (OHAM) is employed to obtain the temperature distribution, efficiency and heat transfer of rectangular fin with temperature dependent heat transfer coefficient i.e. (power-law function of temperature). OHAM provides the optimal analytical solution in the form of an infinite series. Obtained results of OHAM and numerical results for different values of heat transfer mode (n =-1, 0.1, 1, 2, 3, 4, 5) are compared graphically. Optimal values of constants for efficiencies and heat transfer are tabulated for different values of conductive convective parameter M = 0.01, 0.4, 0.8, 1.2, 1.6, 2. It is observed that the results of OHAM are simple, effective and easy. Key words: Optimal homotopy asymptotic method • heat transfer • fins INTRODUCTION The aim of this article is to present an approximate solution of the nonlinear differential equation for steady-state one dimensional temperature distribution in rectangular fin by a well-known Optimal Homotopy Asymptotic Method (OHAM). Nonlinearity in the differential equation is due to the assumption of the variable local heat transfer coefficient. The local heat transfer coefficient is assumed to be a power-law function of temperature. Consequently, the exact analytic solution of the differential is not possible. In recent decades, numerical methods become more and more important due to their good means of analyzing the equations, but in parallel it is also believed that the approximate solutions are also useful and practical. Perturbation method is one of the well-known method that produces solutions for small parameters. Aziz [1] presented the approximate solution for convective fin with internal heat generation and temperature dependent thermal conductivity by perturbation method. He observed the accuracy of the method by comparing the results with numeric results and maximum error was found to be 2%. Due to the presence of small parameter, perturbation method cannot explain the behaviour of physical problem for large values parameters. Singular differential equation of order two considered by Mohyud-Din et al. [2] using He's polynomial. Homotopy Perturbation Method introduced by He [3-5] to avoid the presence of small parameter. Noor et al . [6, 7] and Mohyud-Din et al . [8] investigated the solution of higher-order linear boundary value equations by using variational iteration method and homotopy perturbation method by taking He's polynomial. As an application of engineering and mathematical physics, Mohyud-Din et al . [9] uses He's polynomial for the solution of seventh-order KdV equation for the traveling waves. Also Mohyud-Din et al . [10] obtained analytical soliton solution for kaup- kupershmidt equation without direct transformation. To solve the nonlinear partial differential equations Asif et al . [11] proposed the method by combining the homotopy analysis method and Laplace decomposition method. Recently, Herisanu et al . [12] introduced Optimal Homotopy Asymptotic Method (OHAM) while studying the nonlinear dynamic behaviour of an electric machine exhibiting nonlinear vibration. This method is flexible than Homotopy Analysis Method (HAM) due to its built in convergence criteria. Marinca et al. [13-15] investigated the solution of nonlinear heat transfer problems and fluid flow problems with OHAM. They compared OHAM results with numeric results and found good agreement of OHAM solutions. Idrees et al . [16, 17] solved fourth and sixth order boundary value problem with OHAM. Iqbal et al. [18] investigated the effectiveness of the OHAM in solving the heat transfer flow of a third grade fluid between two heated parallel plates. They represented a comparative analysis between OHAM and numerical solution by finite element method and found that the method of OHAM is simple, precise, effective and easy to use.