Solution of Partial Differential Equations with Nonlocal Conditions by Combine Homotopy Perturbation Method and Laplace Transform

Abstract

In this thesis, the homotopy perturbation method (HPM) was presented, and applied for solving some differential and integral equations with non-local conditions (linear and nonlinear). This method provides an analytical approximate solution of the differential equations. A combined form of the Laplace transform method and homotopy perturbation method, called the homotopy perturbation transform method (HPTM) was introduced and used to solve nonlinear equations. The nonlinear terms of the nonlinear equations was easily handled and treated by the use of He’s polynomials. One of the significant advantages of this method its ability to find solutions without any discretization or restrictive assumptions avoiding the round-off errors. The fact that the proposed technique solves nonlinear problems without using Adomian’s polynomials can also be considered as an additional advantage of this algorithm over the decomposition method. Furthermore, a new approach to solving non-local initial-boundary value problems for linear and nonlinear parabolic and hyperbolic partial differential equations subject to initial and nonlocal boundary conditions of integral type was introduced. The technique of transforming the given non-local initial-boundary value problems of an integral type, into local Dirichlet initial-boundary value problems was implemented for both linear and nonlinear parabolic and hyperbolic partial differential equations, and then the homotopy perturbation method (HPM) was applied to their problems.