American Journal of Engineering Research (AJER) 2015 American Journal of Engineering Research (AJER) e-ISSN: 2320-0847 p-ISSN : 2320-0936 Volume-4, Issue-10, pp-127-133 www.ajer.org Research Paper Open Access www.ajer.org Page 127 Solution of the Linear and Non-linear Partial Differential Equations Using Homotopy Perturbation Method Abaker. A. Hassaballa. 1 Department of Mathematics, Faculty of Science, Northern Border University, Arar, K. S. A. P.O. Box 1321, 2 Department of Mathematics, College of Applied & Industrial Sciences, Bahri University, Khartoum, Sudan Abstract:- In recent years, many more of the numerical methods were used to solve a wide range of mathematical, physical, and engineering problems linear and nonlinear. This paper applies the homotopy perturbation method (HPM) to find exact solution of partial differential equation with the Dirichlet and Neumann boundary conditions. Keywords:- homotopy perturbation method, wave equation, Burgers equation, homogeneous KdV equation. I. INRODUCTION The notion of homotopy is an important part of topology and thus of differential geometry. The homotopy continuation method or shortly speaking homotopy was known as early as in the 1930s. Thus, in 1892, Lyapunov [1] introduced the so called ―artificial small parameters method‖ considering a linear differential equation with variable coefficient in the form ( )u du M t dt  with () M t a time periodic matrix. He replaced this equation with the equation ( )u du M t dt   . To get the solution of the last equation, Lyapunov developed the power series over  for the variable u and then setting 1   . Later, this method was used by kinematicians in the 1960s in the US for solving mechanism synthesis problems [29]. The latest development was done by Morgan at General Motors [3]. There are also two important literature studies by Garcia and Zangwill [5] and Allgower and Georg [8]. The HPM was introduced by Ji-Huan He of Shanghai University in 1998, [9-13]. The HPM is a special case of the homotopy analysis method (HAM) developed by Liao Shijunin 1992 [25]. HPM has been applied by many authors, to solve many types of the linear and nonlinear equations in science and engineering, boundary value problems [2,11], Cauchy reaction–diffusion problem [4], heat transfer[6],nonlinear wave equations [9], non-linear oscillators with discontinuities [12], Sumudu transform[21], and to other fields [13-28]. The method employs a homotopy transform to generate a convergent series solution of linear and nonlinear partial differential equations. The homotopy perturbation method is combination of perturbation and homotopy method II. HOMOTOPY PERTURBATION METHOD To illustrate the basic idea of this method, we consider the following non-linear differential equation: ( ) (r) 0, r A u f    (1) With the following boundary conditions: , 0, r u B u n           (2) Where A is a general differential operator, B is a boundary operator, (r) f is a known analytical function and  is the boundary of the domain  . The operator A can be decomposed into a linear and a non-linear, designated as L and N respectively. The equation (1)can be written as the following form.