Homotopy Perturbation Method

Homotopy Perturbation Method

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

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,

978-1-5225-2713-8.ch002.m01
(1)

With boundary conditions of:

978-1-5225-2713-8.ch002.m02
(2) where A(u) is defined as follows:
978-1-5225-2713-8.ch002.m03
(3) where 978-1-5225-2713-8.ch002.m04 stands for the linear and 978-1-5225-2713-8.ch002.m05 for the nonlinear part. Homotopy perturbation structure is shown as the following equation:
978-1-5225-2713-8.ch002.m06
(4) where,

978-1-5225-2713-8.ch002.m07
(5)

Obviously, using Eq. (4) we have:

978-1-5225-2713-8.ch002.m08
978-1-5225-2713-8.ch002.m09
(6) where 978-1-5225-2713-8.ch002.m10 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 of978-1-5225-2713-8.ch002.m11 changing from 978-1-5225-2713-8.ch002.m12 to 978-1-5225-2713-8.ch002.m13 We consider978-1-5225-2713-8.ch002.m14, as the following:
978-1-5225-2713-8.ch002.m15
(7) and the best approximation for the solution is:

978-1-5225-2713-8.ch002.m16
(8)

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

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