ISSN 1 746-7233, England, UK World Journal of Modelling and Simulation Vol. 5 (2009) No. 3, pp. 225-231 Application of homotopy perturbation method to non-homogeneous parabolic partial and non linear differential equations Hamid El Qarnia * Faculty of Sciences Semlalia, Physics Department, Fluid Mechanics and Energetic Laboratory, Cadi Ayyad University, P. O. Box 2390, Marrakech 40000, Morocco (Received August 11 2008, Accepted June 25 2009) Abstract. In this paper homotopy perturbation method (HPM) is employed to solve two kinds of differential equations: one dimensional non homogeneous parabolic partial differential equation and non linear differen- tial equation. Using the HPM, an exact analytical solution to non homogeneous parabolic partial differential equation and an approximate explicit solution for a non linear differential equation were obtained. The re- sults obtained by HPM for the non linear differential equation were compared with those results obtained by the exact analytical solution. The comparison shows a complete agreement between results and also shows that this new method may be applicable for solving engineering problem because it needs less computations efforts and is easier than others. Keywords: homotopy peturbation method, non linear differential equation, non homogeneous partial differ- ential equation 1 Introduction Most physical phenomena that occurred in nature such as heat transfer are governed by non-linear partial differential equations (NLPDE). To understand such phenomena, one must solve the corresponding NLPDE. Howeverv, most of them do not have exact analytical solutions. Therefore, these NLPDE should be solved using other methods such as numerical methods or semi-analytical method. Some investigators also proposed the combination of these two methods for obtaining the approximate solution to NLPDE. Another known method to solve the NLPDE is a perturbation method which is studied by several in- vestigators for solving some physical problems [9, 10] . Nevertheless, a perturbation method necessitates the existence of a small parameter, which limits its use for different applications. This limitation is overcome us- ing the homotopy perturbation method (HPM) which was first proposed by He [5–8] . Comparatively to classical methods, the HPM method, presents some advantages: obtaining explicit solution with high accuracy, minimal calculations without loss of physical verification. This method has found application in different fields of non linear equations such as fluid mechanics and heat transfer [1–4] . The objective of the present study is to implement the HPM for finding: (1) the exact analytical solution to one dimensional non-homogeneous parabolic partial differential equation with a variable coefficient and (2) the approximate solution of a non linear differential equation that governs the cooling process of a body, with a variable specific heat with temperature, immerged in a fluid with a given temperature. The first equation is as follows: ∂u ∂t - ∂ 2 u ∂x 2 = (cos t - sin t) e -2x , (1) ∗ Corresponding author. E-mail address: elqarnia@ucam.ac.ma. Published by World Academic Press, World Academic Union