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

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.

Özis and Yildirim 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: 𝑢=𝑓+𝑁(𝑢), (1) 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. We are looking for a solution 𝑢 of (1) belonging to 𝐻. 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: 𝑢=𝑓+𝑁(𝑢). (2) The standard ADM consists of representing the solution of (1) as a series 𝑢(𝑥)=^∞_𝑖=0𝑢_𝑖(𝑥), (3) and the nonlinear function as the decomposed form: 𝑁(𝑢(𝑥))=^∞_𝑖=0𝐴_𝑖, (4) where 𝐴_𝑛, 𝑛=0,1,2,… are the Adomian polynomials of 𝑢₀,𝑢₁,…,𝑢_𝑛 given by [9, 10] 𝐴_𝑛=1𝑑𝑛!𝑑𝑝^𝑛𝑛𝑁^𝑛_𝑖=0𝑢_𝑖𝑝^𝑖_𝑝=0. (5) Substituting (3) and (4) into (1) yields ^∞_𝑖=0𝑢_𝑖(𝑥)=𝑓+^∞_𝑖=0𝐴_𝑖. (6) The convergence of the series in (6) gives the desired relation 𝑢₀𝑢=𝑓,_𝑛+1=𝐴_𝑛,𝑛=0,1,2,…. (7) It should be pointed out that 𝐴₀ depends only on 𝑢₀, 𝐴₁ depends only on 𝑢₀ and 𝑢₁; 𝐴₂ depends only on 𝑢₀, 𝑢₁, and 𝑢₂, 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 𝐿(𝑣)=𝑣(𝑥)−𝑓(𝑥)−𝑁(𝑣)=0, (8) with solution 𝑢(𝑥). The basic idea of the HPM is to construct a homotopy 𝐻(𝑣;𝑝)∶𝑅×[0,1]→𝑅 which satisfies ℋ𝑣;𝑝=1−𝑝𝐹(𝑣)+𝑝𝐿(𝑣)=0, (9) where 𝐹(𝑣) is a proper function with known solution which can be obtained easily. The embedding parameter 𝑝 monotonically increases from 0 to 1 as the trivial problem 𝐹(𝑣)=0 is continuously transformed to the original problem 𝑣−𝑓−𝑁(𝑣)=0. From ℋ(𝑣;𝑝)=0, we have 𝐻(𝑣;0)=𝐹(𝑣)=0 and 𝐻(𝑣;1)=𝑣−𝑓−𝑁(𝑣)=0.

It is better to take 𝐹(𝑣) as a deformation of 𝐿(𝑣). For example, in (9), 𝐹(𝑣)=𝑣−𝑓(𝑥). By selecting 𝐹(𝑣)=𝑣−𝑓(𝑥) we can define another convex homotopy ℋ(𝑣;𝑝) by ℋ𝑣;𝑝=𝑣(𝑥)−𝑓(𝑥)−𝑝𝑁(𝑣)=0. (10) The embedding parameter 𝑝∈(0,1] can be considered as an expanding parameter [14, 15]. The HPM uses the embedding parameter 𝑝 as a “small parameter,” and writes the solution of (10) as a power series of 𝑝, that is, 𝑣=𝑣₀+𝑣₁𝑝+𝑣₂𝑝²+⋯. (11) Setting 𝑝=1 results in the approximate solution of (10): 𝑢=lim_𝑝→1𝑣=𝑣₀+𝑣₁+𝑣₂+⋯. (12) Substituting (11) into (10) and equating the terms with identical powers of 𝑝, we can obtain a series of equations of the following form: 𝑝⁰:𝑣₀(𝑥)𝑝−𝑓=0,₁:𝑣₁−𝐻𝑣₀𝑝=0,₂:𝑣₂−𝐻𝑣₀,𝑣₁𝑝=0,₃:𝑣₃−𝐻𝑣₀,𝑣₁,𝑣₂=0,⋮, (13) where 𝐻(𝑣₀,𝑣₁,…,𝑣_𝑗) depend upon 𝑣₀,𝑣₁,…,𝑣_𝑗. In view of (10) to determine 𝐻(𝑣₀,𝑣₁,…,𝑣_𝑗), we use [16] 𝐻𝑣₀,𝑣₁,…,𝑣_𝑗=1𝜕𝑗!𝜕𝑝^𝑗𝑗𝑁^𝑗_𝑖=0𝑣_𝑖𝑝^𝑖_𝑝=0. (14) It is obvious that the system of nonlinear equations in (13) is easy to solve, and the components 𝑣_𝑖, 𝑖≥0 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: 𝐻_𝑛𝑣₀,…,𝑣_𝑛=1𝜕𝑛!𝜕𝑝^𝑛𝑛𝑁^𝑛_𝑖=0𝑣_𝑖𝑝^𝑖_𝑝=0,𝑛=0,1,2,…. (15)

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 𝑁(𝑢) and the parameterized representation of 𝑣 are 𝑣(𝑝)=∑^∞_𝑖=0𝑣_𝑖𝑝^𝑖, where 𝑝 is a parameter, then we have 𝜕^𝑛𝑁𝑣𝑝𝜕𝑝^𝑛_𝑝=0=𝜕^𝑛𝑁∑^∞_𝑖=0𝑣_𝑖𝑝^𝑖𝜕𝑝^𝑛_𝑝=0=𝜕^𝑛𝑁∑^𝑛_𝑖=0𝑣_𝑖𝑝^𝑖𝜕𝑝^𝑛_𝑝=0. (16)

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 𝑁(𝑣), we have 𝑁(𝑣)=𝑁𝑣₀+𝑁^′𝑣₀𝑣−𝑣₀+1𝑁2!^′′𝑣₀𝑣−𝑣₀²+⋯ (17) substituting (11) in (17) and expanding it in terms of 𝑝 leads to 𝑁^∞_𝑖=0𝑣_𝑖𝑝^𝑖=𝑁𝑣₀+𝑁^′𝑣₀𝑣₁𝑝+𝑣₂𝑝²+1+⋯𝑁2!^′′𝑣₀𝑣₁𝑝+𝑣₂𝑝²+⋯²𝑣+⋯=𝑁₀+𝑁^′𝑣₀𝑣₁𝑝+𝑁^′𝑣₀𝑣₂+1𝑁2!^′′𝑣₀𝑣²₁𝑝²+⋯=𝐻₀+𝐻₁𝑝+𝐻₂𝑝²+⋯, (18) where 𝐻_𝑖, 𝑖=0,1,2,…, depends only on 𝑣₀,𝑣₁,…,𝑣_𝑖.
In order to obtain 𝐻_𝑛, we give 𝑛-order derivative of both sides of (18) with respect to 𝑝 and let 𝑝=0, that is, 𝜕^𝑛𝑁𝑣𝑝𝜕𝑝^𝑛_𝑝=0=𝜕^𝑛∑^∞_𝑖=0𝐻_𝑖𝑝^𝑖𝜕𝑝^𝑛_𝑝=0. (19) According to Theorem 2𝜕^𝑛𝑁∑^∞_𝑖=0𝑣_𝑖𝑝^𝑖𝜕𝑝^𝑛_𝑝=0=𝜕^𝑛𝑁∑^𝑛_𝑖=0𝑣_𝑖𝑝^𝑖𝜕𝑝^𝑛_𝑝=0,𝜕^𝑛∑^∞_𝑖=0𝐻_𝑖𝑝^𝑖𝜕𝑝^𝑛_𝑝=0=𝜕^𝑛∑^𝑛_𝑖=0𝐻_𝑖𝑝^𝑖𝜕𝑝^𝑛_𝑝=0=𝑛!𝐻_𝑛. (20) We know that 𝐻_𝑖 just depends on 𝑣₀,𝑣₁,…,𝑣_𝑖 so (𝜕^𝑛∑^𝑛_𝑖=0𝐻_𝑖𝑝^𝑖)/𝜕𝑝^𝑛|_𝑝=0=𝑛!𝐻_𝑛. Substituting (20) in (19) leads us to find 𝐻_𝑖 in the following form: 𝐻_𝑛=1𝜕𝑛!^𝑛𝑁∑^𝑛_𝑖=0𝑣_𝑖𝑝^𝑖𝜕𝑝^𝑛_𝑝=0 (21)

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 ℋ(𝑣;𝑝) given by ℋ𝑣;𝑝=𝑣−𝑓(𝑥)−𝑝𝑁(𝑣). (22)

Proof. Substituting (11) and (18) into (10) and equating the terms with the identical powers of 𝑝, we have ℋ𝑣;𝑝=^∞_𝑖=0𝑣_𝑖𝑝^𝑖−𝑓(𝑥)−𝑝^∞_𝑖=0𝐻_𝑖𝑝^𝑖ℋ=0,𝑣;𝑝=𝑣₀−𝑓(𝑥)+^∞_𝑖=0𝑣_𝑖+1−𝐻_𝑖𝑝^𝑖+1𝑝=0,₀:𝑣₀(𝑥)𝑝−𝑓=0,_𝑛+1:𝑣_𝑛+1−𝐻_𝑛=0,𝑛=0,1,2,… (23)(24) From (24) we have 𝑣₀(𝑥),𝑣=𝑓_𝑛+1=𝐻_𝑛,𝑛=0,1,2,… (25) According to Theorem 3 we have 𝐻_𝑛=𝐴_𝑛. Substituting (25) in (11) leads us to 𝑣=𝑣₀+𝑣₁𝑝+𝑣₂𝑝²(𝑥)+⋯=𝑓+𝐴₀𝑝+𝐴₁𝑝²+⋯, (26) so lim_𝑝→1𝑣=𝑓(𝑥)+𝐴₁+𝐴₂(𝑥)++⋯=𝑓^∞_𝑖=0𝐴_𝑖=^∞_𝑖=0𝑢_𝑖=𝑢. (27) Therefore, by letting ℋ𝑣;𝑝=𝑣−𝑓(𝑥)−𝑝𝑁(𝑣), (28) we observe that the power series 𝑣₀+𝑣₁𝑝+𝑣₂𝑝²+⋯ corresponds to the solution of the equation ℋ(𝑣;𝑝)=𝑣−𝑓(𝑥)−𝑝𝑁(𝑣)=0 and becomes the approximate solution of (1) if 𝑝→1. This shows that the homotopy perturbation method is the Adomian decomposition method with the homotopy ℋ(𝑣;𝑝) 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: 𝑦(𝑥)=𝑥+^𝑥₀𝑦²(𝑡)𝑑𝑡, (29) with the exact solution 𝑦(𝑥)=tan𝑥.

We apply standard ADM and HPM. For applying standard HPM, we construct following homotopy: ℋ𝑢;𝑝=𝑢(𝑥)−𝑥−𝑝^𝑥₀[𝑢(𝑡)]²𝑑𝑡=0. (30) In view of (13), we have 𝑝⁰:𝑣₀(𝑥)𝑝−𝑥=0,_𝑛:𝑣_𝑛+1(𝑥)−^𝑥₀𝐻𝑣₀,𝑣₁,…,𝑣_𝑛𝑑𝑡=0,𝑛≥0. (31) Now if we apply ADM for solving (29), substituting (3) and (4) in (29) leads to ^∞_𝑖=0𝑢_𝑖(𝑥)=𝑥+^𝑥₀^∞_𝑖=0𝐴_𝑖𝑑𝑡. (32) In view of (7), we have following recursive formula: 𝑢₀(𝑥)𝑢=𝑥,_𝑛+1(𝑥)=^𝑥₀𝐴_𝑛𝑑𝑡𝑛≥0. (33) According to Theorem 3, we have 𝐴_𝑛=𝐻(𝑣₀,𝑣₁,…,𝑣_𝑛). By solving (31) and (33), we have 𝑢(𝑥)=^∞_𝑖=0𝑢_𝑖(𝑥)=lim_𝑝→1𝑣₀+𝑣₁𝑝+𝑣₂𝑝²𝑥+⋯=𝑥+³3+2𝑥⁵+1517𝑥⁷+31562𝑥⁹2835+⋯=tan𝑥. (34)

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.