©Freund Publishing House Ltd. International Journal of Nonlinear Sciences and Numerical Simulation, 8(2), 229-232, 2007
He’s Homotopy Perturbation Method for Calculating Adomian
Polynomials
Asghar Ghorbani
*
, Jafar Saberi-Nadjafi
Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
*
E-mail address: as_gh56@yahoo.com (A. Ghorbani)
Abstract
The Adomian method is widely used in approximate calculation, its main demerit is that it is very
difficult and complex to calculate Adomian’s polynomials. This paper introduces the homotopy
perturbation method for overcoming completely the disadvantage. The solution procedure is very effective
and straightforward.
Keywords: Homotopy perturbation method; Adomian polynomials
1. Introduction
Homotopy perturbation method (HPM) [1-10]
proposed by Ji-Huan He has been recently
intensively studied by scientists and engineers
and used for solving various nonlinear problems
[11-18]. The method has been shown to solve
effectively, easily, and accurately a large class of
nonlinear problems, generally one or two
iterations lead to high accurate solutions. This
method is, in fact, a coupling of the traditional
perturbation method and homotopy in topology.
The Adomian method [19,20] is also a technique
for solving functional equations of various kinds
in the form
f u N u = - ) ( ,
where N is a nonlinear operator from Hilbert
space H to H , u is an unknown function, and f is
a known function in H. The decomposition
method decomposes u as a series with
components u
n
, and N(u) as a series with
components A
n
.
The A
n
's are the so called Adomian’s
polynomials which can be calculated using the
formula
0
0
) (
!
1
=
∞
=
=
λ
λ
λ
i
i
i
n
n
n
u N
d
d
n
A .
Generally speaking the solution procedure of
the Adomian’s polynomials is remarkably
complex and difficult, though there are some
alternative approaches to calculating the
Adomian’s polynomials (for example, see [21]
and [22]), the problem keeps unsolved. In order
to overcome the problem completely, here we
apply the homotopy perturbation method for
calculating Adomian’s polynomials, examples
will be given to show the effectiveness and
convenience of the present method.
2. Homotopy perturbation method
We consider the following general equation
) (u N R = , (1)
where N is a nonlinear operator. To illustrate
the homotopy perturbation method (HPM), we
consider (1) as
0 ) ( ) ( = - = u N G G L , (2)
with solution R . As a possible remedy, we can
define homotopy ) , ( p G H as follows
) ( ) 1 , ( ), ( ) 0 , ( G L G H G F G H = = ,