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Very recently, the convenient way to calculate the Adomian series was suggested. This paper combines this technique and the Pade approximation to develop some new iteration schemes. Then, the combined method is applied to nonlinear models and the residual functions illustrate the accuracies and conveniences.

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) [

Very recently, for the ADM, Duan [

Recently, Tsai and Chen [

With Duan and Tsai’s idea, this paper suggests a novel approximation scheme for the oscillating physical mechanism of the nonlinear models [

Generally, consider the following nonlinear equation:

Apply the inverse

Consider the basic idea of the Picard method

The classical ADM [

Duan et al. [

Now, we present our analytical schemes using the convenient Adomian series, Laplace transform, and Pade approximation. We adopt the steps in [

(i) Take Laplace transform

(ii) Through the Picard successive approximation, we can obtain the following iteration formula:

(iii) Let

where

(iv) Employ the Pade technique to accelerate the convergence of

In this study, we consider a reduced case where

In order to solve (

Setting

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 (

Define the residual function as

Consider the same

The comparisons of the approximate solutions using (

As a result, we decide to adopt the iteration formula (

Analytical solution of (

The approximate solution is reliable from the error analysis of the iteration formula (

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.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the Open Fund of State Key Laboratory of Oil and Gas Geology and Exploration, Southwest Petroleum University (PLN1309), National Natural Science Foundation of China “Study on wellbore flow model in liquid-based whole process underbalanced drilling” (Grant No. 51204140), the Scientific Research Fund of Sichuan Provincial Education Department (4ZA0244), and the Program for Liaoning Excellent Talents in University under Grant no. LJQ2011136.