The Laplace-Adomian-Pade Technique for the ENSO Model

Abstract

The Laplace-Adomian-Pade 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 Niña-Southern 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 Laplace-Adomian-Pade 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 air-sea coupled dynamical system was used to describe the oscillating physical mechanism of the ENSO [11] 𝑑𝑇𝑑𝑡=𝐶𝑇+𝐷ℎ−𝜀𝑇³,𝑑ℎ𝑑𝑡=−𝐸𝑇−𝑅_ℎℎ, (1) where 𝐶, 𝐷, 𝐸, and 𝑅_ℎ are physical constants, 𝑇 describes the temperature of the eastern equatorial Pacific sea surface, and ℎ is the thermo-cline depth anomaly. The model (1) shows the variations of both eastern and western Pacific anomaly patterns.

Case I. When 𝐷=0 and 0<𝜀≪1, then (1) can be reduced to 𝑑𝑇𝑑𝑡=𝐶𝑇−𝜀𝑇³. (2)
In order to solve (2) with the LAPM, apply the Laplace transform 𝐿 to both sided of (2) first and we can derive 𝑠𝑇(𝑠)=𝑇(0)+𝐶𝑇(𝑠)−𝜀𝐿𝑇³, (3) where 𝑇(𝑠)=∫^∞₀𝑒^−𝑠𝑡𝑇(𝑡)𝑑𝑡. As a result, (3) leads to 𝑇(𝑠)=𝑇(0)()−𝑇𝑠−𝐶𝜀𝐿³().𝑠−𝐶 (4)
Apply the inverse of the Laplace transform 𝐿⁻¹ and expand the nonlinear term as an Adomian series [1, 12]; then (4) can be written as ^∞_𝑛=0𝑣_𝑛(𝑡)=𝐿⁻¹𝑇(0)(𝑠−𝐶)−𝐿⁻¹𝜀𝐿∑^∞_𝑛=0𝐴_𝑛(),𝑠−𝐶 (5) where 𝑇(𝑡)=∑^∞_𝑛=0𝑣_𝑛(𝑡) and ∑^∞_𝑛=0𝐴_𝑛 is the Adomian series of 𝑇³; namely, 𝐴₀=𝑣³₀,𝐴₁=3𝑣²₀𝑣₁,𝐴₂=3𝑣₀𝑣²₁+3𝑣²₀𝑣₂,⋮ (6) Now the iteration formula can be determined for (2) as 𝑣_𝑛+1(𝑡)=−𝐿⁻¹𝜀𝐿𝐴_𝑛()𝑣𝑠−𝐶,𝑛≥1,₀=𝐿⁻¹𝑇(0)().𝑠−𝐶 (7)

Assuming 𝑇(0)=𝐶=1, the successive approximate solutions 𝑇_𝑛=𝑣₀+⋯+𝑣_𝑛 can be presented as 𝑇₀=𝐿⁻¹1(𝑠−1)=𝑒^𝑡,𝑇₁=𝑒^𝑡−𝜀𝑒^2𝑡(𝑡),⋮sinh (8)

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 𝑇₇=𝑒^𝑡−𝜀𝑒^2𝑡(𝑡)+3sinh8𝜀²−2𝑒^3𝑡+𝑒^5𝑡+𝑒^𝑡−5𝜀16³3𝑒^3𝑡−3𝑒^5𝑡+𝑒^7𝑡−𝑒^𝑡+35𝜀128⁴−4𝑒^3𝑡+6𝑒^5𝑡−4𝑒^7𝑡+𝑒^9𝑡+𝑒^𝑡−63𝜀256⁵5𝑒^3𝑡−10𝑒^5𝑡+𝑒^11𝑡+10𝑒^7𝑡−5𝑒^9𝑡−𝑒^𝑡+231𝜀1024⁶−6𝑒^3𝑡+15𝑒^5𝑡−6𝑒^11𝑡−20𝑒^7𝑡+𝑒^13𝑡+15𝑒^9𝑡+𝑒^𝑡−429𝜀2048⁷7𝑒^3𝑡−21𝑒^5𝑡+21𝑒^11𝑡+35𝑒^7𝑡−7𝑒^13𝑡+𝑒^15𝑡−35𝑒^9𝑡−𝑒^𝑡,𝑇₁₂=𝑒^𝑡−𝜀𝑒^2𝑡(𝑡)+3sinh8𝜀²−2𝑒^3𝑡+𝑒^5𝑡+𝑒^𝑡++⋯676039𝜀4194304¹²−12𝑒^3𝑡−220𝑒^19𝑡+66𝑒^5𝑡−792𝑒^11𝑡+𝑒^25𝑡−220𝑒^7𝑡−12𝑒^23𝑡+924𝑒^13𝑡+495𝑒^17𝑡−792𝑒^15𝑡+495𝑒^9𝑡+66𝑒^21𝑡+𝑒^𝑡. (9)

Recall that (2) has an exact solution [13] 𝜀𝑇=𝐶+1𝑇(0)²−𝜀𝐶exp(−2𝐶𝑡)^−(1/2). (10) Setting 𝜀=0.00001 in this paper, we apply the Pade-technique to the approximate solution 𝑇₁₂. 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 𝑇^12th_[LAPM] and 𝑇^12th_[ADOM] in Table 1, respectively.

Table 1 

Comparisons between 𝑇_[ADOM], 𝑇_[LAPM], and the exact solution.

The exact solution (10), the approximate solutions 𝑇^12th_[ADOM], 𝑇^12th_[LAPM], and the solution 𝑇^12th_[LAM] without the treatment using the Pade-technique are compared in Figure 1.

Figure 1 

The comparisons of the approximate solutions using different methods.

The results in Table 1 and Figure 1 illustrate that the LAPM has a higher accuracy, respectively.

Case II. For the coefficients 𝐶=𝐷=𝐸=𝑅_ℎ=1 and 𝜀=0.00001, (1) reduces to 𝑑𝑇𝑑𝑡=𝑇+ℎ−0.00001𝑇³,𝑑ℎ𝑑𝑡=−𝑇−ℎ. (11) Setting the initial condition value 𝑇(0)=ℎ(0)=1, we can derive the following iteration formula: 𝑣_𝑛+1(𝑡)=−0.01𝐿⁻¹1𝐿𝐴𝑠−1^𝑛,𝑣₀(𝑡)=𝑇(0)𝐿⁻¹1(0)𝐿𝑠−1+ℎ⁻¹1𝑠²−1=𝑒^𝑡(𝑡),ℎ+sinh_𝑛+1(𝑡)=−𝐿⁻¹𝐿𝑣_𝑛(𝑡)𝑠+1,ℎ₀=𝐿⁻¹ℎ(0)𝑠−1=𝑒^𝑡. (12) where ℎ₀+⋯+ℎ_𝑛 is the 𝑛th approximation of ℎ(𝑡). As a result, for 𝑛=12, we can obtain the approximate solution by means of the LAPM.
Define the residual functions 𝑓(𝑡) and 𝑔(𝑡) as 𝑓=𝑑𝑇𝑑𝑡−𝑇−ℎ+0.01𝑇³,𝑔=𝑑ℎ.𝑑𝑡+𝑇+ℎ (13) The plotted functions 𝑓(𝑡) and 𝑔(𝑡) show that the iteration formula is reliable (Figure 2). Now we can analytically investigate the relationship between the temperature 𝑇 and the thermo-cline depth ℎ, which is shown in Figure 3.

Figure 2 

The residual functions: 𝑓(𝑡) and 𝑔(𝑡).

Figure 3 

The temperature versus the thermo-cline depth ℎ.

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 𝑑^𝑚𝑢𝑑𝑡^𝑚[𝑢(𝑡[𝑢(𝑡(𝑡),𝑢+𝑅)]+𝑁)]=𝑔_(𝑘)(0)=𝑑^𝑘𝑢(0)𝑑𝑡^𝑘,𝑘=0,…,𝑚−1 (14) into the following equivalent integral equation 𝑢=𝑓(𝑡)+^𝑡₀⋯^𝑡₀−𝑅[𝑢(𝑠)]−𝑁[𝑢(𝑠)]+𝑔(𝑠)𝑑𝑠⋯𝑑𝑠_𝑚,𝑢^(𝑘)(0)=𝑑^𝑘𝑢(0)𝑑𝑡^𝑘,𝑘=0,…,𝑚−1. (15)

Here ∫^𝑡₀⋯∫^𝑡₀⋅𝑑𝑠⋯𝑑𝑠_𝑚 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 𝑞-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).