Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 943232, 4 pages

http://dx.doi.org/10.1155/2013/943232

Research Article

A Comparison between Adomian’s Polynomials and He’s Polynomials for Nonlinear Functional Equations

Hossein Jafari,1,2 Saber Ghasempoor,¹ and Chaudry Masood Khalique²

¹Department of Mathematics, University of Mazandaran, P.O. Box 47416-95447, Babolsar, Iran

²International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Mmabatho 2735, South Africa

Received 20 March 2013; Revised 11 May 2013; Accepted 2 June 2013

Academic Editor: Mufid Abudiab

Copyright © 2013 Hossein Jafari et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We will compare the standard Adomian decomposition method and the homotopy perturbation method applied to obtain the solution of nonlinear functional equations. We prove analytically that the two methods are equivalent for solving nonlinear functional equations. In Ghorbani (2009), Ghorbani presented a new definition which he called as He’s polynomials. In this paper, we also show that He’s polynomials are only the Adomian polynomials.

1. Introduction

The Adomian decomposition method (ADM) and the homotopy perturbation method (HPM) are two powerful methods which consider the approximate solution of a nonlinear equation as an infinite series usually converging to the accurate solution. These methods have been used in obtaining analytic and approximate solutions to a wide class of linear and nonlinear, differential, and integral equations.

Öziş and Yıldırım compared Adomian’s method and He’s homotopy perturbation method [1] for solving certain nonlinear problems. Li also has shown that the ADM and HPM for solving nonlinear equations are equivalent [2]. In [3], Ghorbani has presented a definition which he called it as He’s polynomials.

Consider the following nonlinear functional equation:

where is a nonlinear operator from Hilbert space to , is an unknown function, and is a known function in . We are looking for a solution of (1) belonging to . We will suppose that (1) admits a unique solution. If (1) does not possess a unique solution, the ADM and HPM will give a solution among many (possible) other solutions. However, relatively few papers deal with the comparison of these methods with other existing techniques. In [4], a useful comparison between the decomposition method and the perturbation method showed the efficiency of the decomposition method compared to the tedious work required by the perturbation techniques. In [5], the advantage of the decomposition method over the Picard’s method has been emphasized. Sadat has shown that the Adomian decomposition method and perturbation method are closely related and lead to the same solution in many heat conduction problems [6]. In [7, 8] the HPM has compared with Liao’s homotopy analysis method and showed the HPM is special case of HAM, and the advantage of the HAM over the HPM has been emphasized.

In this paper, we want to prove that He’s polynomials are only Adomian’s polynomials. We will also show that the standard Adomian decomposition method and the standard HPM are equivalent when applied for solving nonlinear functional equations.

2. Adomian’s Decomposition Method (ADM)

Let us consider the nonlinear equation (1) which can be written in the following canonical form:

The standard ADM consists of representing the solution of (1) as a series

and the nonlinear function as the decomposed form:

where , are the Adomian polynomials of given by [9, 10]

Substituting (3) and (4) into (1) yields

The convergence of the series in (6) gives the desired relation

It should be pointed out that depends only on , depends only on and ; depends only on , , and , and so on. The Adomian technique is very simple in its principles. The difficulties consist in proving the convergence of the introduced series.

3. Homotopy Perturbation Method (HPM)

This is a basic idea of homotopy method which is to continuously deform a simple problem easy to solve into the difficult problem under study.

In this section, we apply the homotopy perturbation method [11–13] to the discussed problem. To illustrate the homotopy perturbation method (HPM), we consider (1) as

with solution . The basic idea of the HPM is to construct a homotopy which satisfies

where is a proper function with known solution which can be obtained easily. The embedding parameter monotonically increases from to as the trivial problem is continuously transformed to the original problem . From , we have and .

It is better to take as a deformation of . For example, in (9), . By selecting we can define another convex homotopy by

The embedding parameter can be considered as an expanding parameter [14, 15]. The HPM uses the embedding parameter as a “small parameter,” and writes the solution of (10) as a power series of , that is,

Setting results in the approximate solution of (10):

Substituting (11) into (10) and equating the terms with identical powers of , we can obtain a series of equations of the following form:

where depend upon . In view of (10) to determine , we use [16]

It is obvious that the system of nonlinear equations in (13) is easy to solve, and the components , of the homotopy perturbation method can be completely determined, and the series solutions are thus entirely determined. For the convergence of the previous method we refer the reader to the work of He [12, 17, 18].

4. Equivalence between ADM and HPM

In this section, we prove that the HPM and the ADM give same solution for solving nonlinear functional equations. We also show that the He polynomials are like the Adomian polynomials. In [3], Ghorbani has presented the following definition.

Definition 1 (see [3]). The He polynomials are defined as follows:

Note 1. Comparison between (5) and (15) has shown that the He polynomials are only Adomian’s polynomials, and it is calculated like Adomian’s polynomials.

Theorem 2. Suppose that nonlinear function and the parameterized representation of are , where is a parameter, then we have

Proof (see [3, 19]). In Theorem 3, we prove that the He polynomials are the Adomian polynomials.

Theorem 3. The He polynomials which are given by (15) are the Adomian polynomials.

Proof. From Taylor’s expansion of , we have

substituting (11) in (17) and expanding it in terms of leads to

where , , depends only on .

In order to obtain , we give -order derivative of both sides of (18) with respect to and let , that is,

According to Theorem 2

We know that just depends on so . Substituting (20) in (19) leads us to find in the following form:

which is called for the first time by Ghorbani as the He polynomials [3]!

Theorem 4. The homotopy perturbation method for solving nonlinear functional equations is the Adomian decomposition method with the homotopy given by

Proof. Substituting (11) and (18) into (10) and equating the terms with the identical powers of , we have

From (24) we have

According to Theorem 3 we have . Substituting (25) in (11) leads us to

so

Therefore, by letting

we observe that the power series corresponds to the solution of the equation and becomes the approximate solution of (1) if . This shows that the homotopy perturbation method is the Adomian decomposition method with the homotopy given by (28). The proof of Theorem 4 is completed.

These two approaches give the same equations for high-order approximations. This is mainly because the Taylor series of a given function is unique, which is a basic theory in calculus. Thus, nothing is new in Ghorbani’s definition, except the new name “He’s polynomial.” He just employed the early ideas of ADM.

Example 5 (see [20]). Consider the following nonlinear Volterra integral equation:

with the exact solution .

We apply standard ADM and HPM. For applying standard HPM, we construct following homotopy:

In view of (13), we have

Now if we apply ADM for solving (29), substituting (3) and (4) in (29) leads to

In view of (7), we have following recursive formula:

According to Theorem 3, we have . By solving (31) and (33), we have

5. Conclusion

It has been shown that the standard HPM provides exactly the same solutions as the standard Adomian decomposition method for solving functional equations. It has been proved that He’s polynomials are only Adomian’s polynomials with different name.

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