Recent Advances on Methods and Applications of Nonlinear Differential Equations
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The LaplaceAdomianPade Technique for the ENSO Model
Abstract
The LaplaceAdomianPade method is used to find approximate solutions of differential equations with initial conditions. The oscillation model of the ENSO is an important nonlinear differential equation which is solved analytically in this study. Compared with the exact solution from other decomposition methods, the approximate solution shows the method’s high accuracy with symbolic computation.
1. Introduction
In recent years, El Niño/La NiñaSouthern Oscillation (ENSO) is a quasiperiodic climate pattern that occurs across the tropical Pacific Ocean every five years which has caught more and more attention of researchers due to its great destructions. It is coupled with two phases, the warm oceanic phase, El Niño, and the cold phase, La Niña. Some methods were applied to consider the numerical simulation, among which is the famous Adomian decomposition method (ADOM) [1].
Generally speaking, two aspects affect the accuracy of the ADOM: the calculation of the Adomian decomposition series and the initial iteration value. In view of these points, various modified versions are proposed to solve the nonlinear initial value problems [2–7].
Recently, Tsai and Chen [8–10] suggested a LaplaceAdomianPade method (LAPM) to approximately solve the initial value problems of differential equations. The method holds the following merits: (a) the Laplace transformation can be used to “fully” determine the initial iteration value; (b) the Adomian series is used to linearize the nonlinear terms; (c) the Pade technique is used to accelerate the convergence and enlarge the valid area of the approximate solution.
In this paper, we use the method to approximately solve the ENSO model. The approximate solution is compared with other nonlinear techniques in the high order iteration and the result shows the method’s higher accuracy.
2. Approximate Solutions of the ENSO Model
The airsea coupled dynamical system was used to describe the oscillating physical mechanism of the ENSO [11] where , , , and are physical constants, describes the temperature of the eastern equatorial Pacific sea surface, and is the thermocline depth anomaly. The model (1) shows the variations of both eastern and western Pacific anomaly patterns.
Case I. When and , then (1) can be reduced to
In order to solve (2) with the LAPM, apply the Laplace transform to both sided of (2) first and we can derive
where . As a result, (3) leads to
Apply the inverse of the Laplace transform and expand the nonlinear term as an Adomian series [1, 12]; then (4) can be written as
where and is the Adomian series of ; namely,
Now the iteration formula can be determined for (2) as
Assuming , the successive approximate solutions can be presented as
We can consider a Maple program for the approximation and set the truncated order as 7 and 12, respectively. The 7th term approximation and the 12th term approximation can be obtained as
Recall that (2) has an exact solution [13] Setting in this paper, we apply the Padetechnique to the approximate solution . In order to avoid the tediousness, the detail expression of the result is omitted here.
The approximate solutions from the ADOM and the LAPM are compared using the high iteration solutions and in Table 1, respectively.

The exact solution (10), the approximate solutions , , and the solution without the treatment using the Padetechnique are compared in Figure 1.
The results in Table 1 and Figure 1 illustrate that the LAPM has a higher accuracy, respectively.
Case II. For the coefficients and , (1) reduces to
Setting the initial condition value , we can derive the following iteration formula:
where is the th approximation of . As a result, for , we can obtain the approximate solution by means of the LAPM.
Define the residual functions and as
The plotted functions and show that the iteration formula is reliable (Figure 2). Now we can analytically investigate the relationship between the temperature and the thermocline depth , which is shown in Figure 3.
Remarks. This study only concentrates on the applications of the Adomian series in the linearization of the nonlinear equations. For various calculations of the Adomian series, readers are referred to the recent development of the method in [3, 4, 14–16] and the applications in fractional different equations in [17–19]. It is interesting to point out that the results are the same as those of the one using the variational iteration method [20].
In the classical ADOM, the inverse operator should be used. For example, one can need to transform the differential equation into the following equivalent integral equation
Here is called the inverse operator in the ADOM.
In Tsai and Chen’s method, the solution procedure shows that the LAPM without using the inverse operator still keeps approximate solutions of higher accuracies. Furthermore, the initial iteration function can be readily determined. The method also can be extended to fractional differential equations [21] and difference equations.
3. Conclusions
With symbolic computation, the LAPM is used to approximately solve the ENSO model. We compared the approximate solutions with those from the ADOM and the LAPM, respectively. The results show that the LAPM has higher efficiency which can accelerate the convergence and enlarge the valid area of the approximate solution.
Acknowledgment
This work is supported by the Scientific Research Fund of Sichuan Provincial Education Department (12ZA085).
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Copyright © 2013 Yi Zeng. 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.