Homotopy Perturbation Method

Homotopy Perturbation Method

DOI: 10.4018/978-1-5225-2713-8.ch002
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Abstract

The homotopy perturbation method (HPM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing on the problem. It has been attempted to show the capabilities and wide-range applications of the homotopy perturbation method in comparison with the previous ones in solving heat transfer problems. The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations.
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Introduction

Basic Idea of Homotopy Perturbation Method

To illustrate the basic ideas of the new method, we consider the following nonlinear differential equation,

(1)

With boundary conditions of:

(2) where A(u) is defined as follows:
(3) where stands for the linear and for the nonlinear part. Homotopy perturbation structure is shown as the following equation:
(4) where,

(5)

Obviously, using Eq. (4) we have:

(6) where is an embedding parameter and u0 is the first approximation that satisfies the boundary condition. The process of changes in p from zero to unity is that of changing from to We consider, as the following:
(7) and the best approximation for the solution is:

(8)

The above convergence is discussed in Cai et. Al (2006) and Cveticanin (2005).

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