I nternational Journal of Application or I nnovation in Engineering & M anagement (I JAI EM ) Web Site: www.ijaiem.org Email: editor@ijaiem.org Volume 3, Issue 4, April 2014 ISSN 2319 - 4847 Volume 3, Issue 4, April 2014 Page 29 ABSTRACT In this article, Homotopy Perturbation Method is modified to some moving boundary value problem in order to obtain an approximate explicit solution with regard to phase change problem with a non-uniform initial temperature distribution. The initial initial approximations of homotpy perturbation method have chosen so that the initial and boundary conditions of the moving boundary value problem are satisfied. The obtained results are tabulated and graphically compared with thise due to some authors. The numerical simulation using present approach show very good agreements with others, even a very small number of bases have being adapted. Keywords: Homotopy perturbation method, moving boundary value problem of partial differential equation. 1. INTRODUCTION The term moving boundary problems (MBP's) is commonly used when the boundary is associated with time dependent problems and the boundary of the domain is not known in advanced but has to be determined as a function of time and space. Moving boundary problem have received much attention due to their practical importance in engineering and science[19].These problems become nonlinear due present of moving boundary [8] and for this reason their analytical explicit solution are difficult to obtain in general. Stefan problems (phase change problems) is one class of moving boundary value problem and as well as application, See Crank[9] and Hill[12]. The class of Stefan problem (MBP’S) is interesting because of its nonlinearity nature that is associated with the moving interface (see [8]). Due to presence of moving interface their exact solution are limited. Therefore, Many approximate solutions have been used to solve this problem numerical [5],[6],[24-27] , Stefan problems with time-dependent boundary condition require some special techniques one can see [27], [28], [31]. Savovic and Caldwell [30] presented finite difference solution of one-dimensional Stefan problem with periodic boundary conditions. Ahmed [4] discussed a new algorithm for moving boundary problem subject to periodic boundary conditions. In 2009, Rajeev et al. [23] used variational iteration method to solve a phase change problem with time dependent boundary condition and the result is obtained in term of Mittag-Leffler function. In 2012 Rajeev and M.S. Kushwaha [22] used adomaian decomposition method to solve a Stefan problem with periodic boundary condition . In this paper an approximate explicit approach is interested via a Homotopy perturbation method with some modifications .The obtained results are compared with the non-classical variational solution obtained in [7]. Since both methods are based on the selection a suitable bases that approximate the solution. 2. DESCRIPTION OF THE PROBLEM (SOLIDIFICATION OF WATER) The problem concerns heat transfer in an ice- water medium occupying the region At any time t. The water that undergoing phase change, is contained in the region ( and the rest of the region outside it, is occupied by ice. Initially, , and the temperature of ice is linear in each of two region in which it lies. The temperature of the water is assumed to be equal to zero which is also the critical temperature of phase change. The fixed surface x=0 and x=1, are maintained at unit negative temperature throughout [7]. Remarks 1: 1. As mentioned in [21], [29] due to the symmetry about of the problem , the problem is reduced to find its solution is the region on the initial condition. 2. The region of initial condition is then written as non-uniform initial temperature region as following: 3. The initial condition may be obtained as follows set so that the Approximate solution to one –dimensional phase change problem with non – uniform initial temperature via homotopy perturbation approach by Radhi Ali Zaboon 1 , Ahmed Ismail Mohammed 2 1 Department of Mathematics, College of Science, AL-Mustansiriyah University, Baghdad, Iraq 2 Department of Mathematics, College of Basic Education, Misan University, Misan, Iraq.