A Convenient Adomian-Pade Technique for the Nonlinear Oscillator Equation

and Applied Analysis 3 3. Iteration Schemes Based on the Convenient Adomian Series Now, we present our analytical schemes using the convenient Adomian series, Laplace transform, and Pade approximation. We adopt the steps in [16]. Considering (2), we show the following iteration schemes. (i) Take Laplace transform ?̃? to both sides: ?̃? [L [u] + R [u]] + ?̃? [N [u]] = ?̃? [g (t)] . (13) We can have iteration formula (4) through inverse of Laplace transform ?̃?: u (t) = f (t) + ?̃? −1 [?̃? (s) ?̃? [N [u]]] , (14) where f(t) and ?̃?(s) can be determined by calculation of Laplace transform to L[u], R[u], and g(t). The calculation of ?̃?(s) is similar to the determination of the Lagrangemultiplier of the variational iteration method in [17]. (ii) Through the Picard successive approximation, we can obtain the following iteration formula: u n+1 = f (t) + ?̃? −1 [?̃? (s) ?̃? [N [un]]] . (15) (iii) Let u n = ∑ n i=0 V i and apply the Adomian series to expand the termN[u] as ∑∞ i=0 A i . Then, the iteration formula reads V n+1 = ?̃? −1 [?̃? (s) ?̃? [An]] , V 0 = f (t) , (16) where A i are calculated by


Introduction
Analytical methods for nonlinear systems have caught much attention due to their convenience for obtaining solutions in real engineering problems.One of the most often used methods is the Adomian decomposition method (ADM) [1].Also due to the rapid development of the computer science, various modifications of these nonlinear analytical methods have been proposed and have been extensively applied to various nonlinear systems [2][3][4][5][6][7][8][9][10][11][12][13][14].
Very recently, for the ADM, Duan [4][5][6] suggested a convenient Adomian calculation scheme.The method can help us get a higher accuracy and can hand higher order approximation problem due to its easier calculation of the Adomian series than the classical one [1].The technique has been successfully extended to fractional differential equations and boundary value problems.
Recently, Tsai and Chen [11] proposed a Laplace-Adomian-Pade method (LAPM).The method holds the following merits: (a) Laplace transform can be used to determine the initial iteration value; (b) the Pade technique is adopted to accelerate the convergence.
With Duan and Tsai's idea, this paper suggests a novel approximation scheme for the oscillating physical mechanism of the nonlinear models [15] where , , , and  ℎ are physical constants,  describes the temperature of the eastern equatorial Pacific sea surface, and ℎ is the thermocline depth anomaly.

Preliminaries of the Adomian Series
Generally, consider the following nonlinear equation: where  is the highest derivative of , for example,  order,  is the remaining linear part containing the lower order derivatives, and  is the nonlinear operator.

Abstract and Applied Analysis
Apply the inverse  −1 of the linear operator  in (2), and we can obtain Consider the basic idea of the Picard method and assume that The classical ADM [1] supposes that the nonlinear term [] can be expanded approximately as where   is calculated by For example, ∑ ∞ =0   is the Adomian series of  3 ; namely, Duan et al. [4][5][6] very recently suggested a convenient way to calculate the Adomian series as as well as the case of the -variable For the single variable case, [] = (), the first three components are listed as And for the two-variable case, [] = ( 1 ,  2 ), the first three components are listed as The above formulae (9) spend less time deriving the   .On the other hand, this provides a possible tool to investigate the higher order approximation solution.

Iteration Schemes Based on the Convenient Adomian Series
Now, we present our analytical schemes using the convenient Adomian series, Laplace transform, and Pade approximation.We adopt the steps in [16].Considering (2), we show the following iteration schemes.
(i) Take Laplace transform L to both sides: We can have iteration formula (4) through inverse of Laplace transform L−1 : where () and λ() can be determined by calculation of Laplace transform to [], [], and ().The calculation of λ() is similar to the determination of the Lagrange multiplier of the variational iteration method in [17].
(ii) Through the Picard successive approximation, we can obtain the following iteration formula: (iii) Let   = ∑  =0 V  and apply the Adomian series to expand the term [] as ∑ ∞ =0   .Then, the iteration formula reads where   are calculated by (iv) Employ the Pade technique to accelerate the convergence of   = ∑  =0 V  .

Applications of the Iteration Formulae
In this study, we consider a reduced case where  = 0 and 0 <  ≪ 1 in (1) as follows: In order to solve (18) with the Maple software, apply Laplace transform  to both sides firstly.This step can fully and optimally determine the initial iteration.We can derive Formula ( 19) Formula ( 21) Setting  = 0.00001 in the model ( 18), now we can obtain the first few as We now can compare the accuracies of the different versions of the Adomian decomposition methods.
For example, we can write out the classical Adomian formula for (18) as Consider the same  = 20 and  =  = 20; from the comparison illustrated through Figure 1, we can see that the iteration formula (19) has a higher accuracy almost in the interval [0, 5].
As a result, we decide to adopt the iteration formula (19) and give the numerical simulation of (18) in the case of the higher order approximation.The analytical solution is illustrated in Figure 2.
The approximate solution is reliable from the error analysis of the iteration formula (19) in Figure 1.

Conclusions
The approximate solution is compared with the nonlinear techniques in higher order iteration and the result shows the new way's higher accuracy to calculate the Adomian series.In view of this point, the comparison of different versions of the Adomian method is possible.The results show that the iteration formula fully using all the linear parts has a higher accuracy.It provides an efficient tool to select a suitable algorithm when solving engineering problems.
21) Also apply the Pade technique to   and denote the result as  2, [/].