Abstract
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.
1. 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 [2–14].
Very recently, for the ADM, Duan [4–6] 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 are physical constants, describes the temperature of the eastern equatorial Pacific sea surface, and is the thermocline depth anomaly.
2. 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.
Apply the inverse 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, is the Adomian series of ; namely,
Duan et al. [4–6] 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, , 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.
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: We can have iteration formula (4) through inverse of Laplace transform : where and can be determined by calculation of Laplace transform to , , and . The calculation of 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 and apply the Adomian series to expand the term as . Then, the iteration formula reads where are calculated by (iv) Employ the Pade technique to accelerate the convergence of .
4. Applications of the Iteration Formulae
In this study, we consider a reduced case where and 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
Setting in the model (18), now we can obtain the first few as Apply the Pade technique to and denote the result 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 Also apply the Pade technique to and denote the result as .
Define the residual function as
Consider the same and ; from the comparison illustrated through Figure 1, we can see that the iteration formula (19) has a higher accuracy almost in the interval .
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.
5. 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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
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.