# The Laplace-Adomian-Pade Technique for the ENSO Model

Authors: Yi Zeng
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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]
(1)dTdt=CT+Dh-εT3,dhdt=-ET-Rhh,
where C, D, E, and Rh 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.
WhenD=0 and 0<ε≪1, then (1) can be reduced to
(2)dTdt=CT-εT3.
In order to solve (2) with the LAPM, apply the Laplace transform L to both sided of (2) first and we can derive
(3)sT¯(s)=T(0)+CT(s)-εL[T3],
where T¯(s)=∫0∞e-stT(t)dt. As a result, (3) leads to
(4)T¯(s)=T(0)(s-C)-εL[T3](s-C).
Apply the inverse of the Laplace transformL-1 and expand the nonlinear term as an Adomian series [1, 12]; then (4) can be written as
(5)∑n=0∞vn(t)=L-1[T(0)(s-C)]-L-1[εL[∑n=0∞An](s-C)],
where T(t)=∑n=0∞vn(t) and ∑n=0∞An is the Adomian series of T3; namely,
(6)A0=v03,A1=3v02v1,A2=3v0v12+3v02v2,⋮
Now the iteration formula can be determined for (2) as
(7)vn+1(t)=-L-1[εL[An](s-C)],n≥1,v0=L-1[T(0)(s-C)].AssumingT(0)=C=1, the successive approximate solutions Tn=v0+⋯+vn can be presented as
(8)T0=L-1[1(s-1)]=et,T1=et-εe2tsinh(t),⋮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(9)T7=et-εe2tsinh(t)+38ε2(-2e3t+e5t+et)-516ε3(3e3t-3e5t+e7t-et)+35128ε4(-4e3t+6e5t-4e7t+e9t+et)-63256ε5(5e3t-10e5t+e11t+10e7t-5e9t-et)+2311024ε6(-6e3t+15e5t-6e11t-20e7t+e13t+15e9t+et)-4292048ε7(7e3t-21e5t+21e11t+35e7t-7e13t+e15t-35e9t-et),T12=et-εe2tsinh(t)+38ε2(-2e3t+e5t+et)+⋯+6760394194304ε12(-12e3t-220e19t+66e5t-792e11t+e25t-220e7t-12e23t+924e13t+495e17t-792e15t+495e9t+66e21t+et).Recall that (2) has an exact solution [13]
(10)T=[εC+(1T(0)2-εC)exp(-2Ct)]-(1/2).
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 solutionsT[LAPM]12th and T[ADOM]12th in Table 1, respectively.Table 1
Comparisons betweenT[ADOM], T[LAPM], and the exact solution.
t
T
[
ADOM
]
12
th
T
[
LAPM
]
12
th
Exact solution
0.0
1.00000000000
1.00000000000
1.00000000000
0.5
1.64870710600
1.64870710600
1.64870710600
1.0
2.71819499800
2.71819499800
2.71819499700
1.5
4.48126178100
4.48126146900
4.48126145500
2.0
7.38709279800
7.38707674000
7.38707669500
2.5
12.1738764100
12.1735250400
12.1735245800
3.0
20.0498629300
20.0452439400
20.0452435100
3.5
32.9785428200
32.9355166300
32.9355172400
4.0
54.1154755000
53.8024166600
53.8023905200
4.5
88.4548189000
86.5781681900
86.5781168700
5.0
143.748571000
134.353636200
134.352939400The exact solution (10), the approximate solutions T[ADOM]12th, T[LAPM]12th, and the solution T[LAM]12th 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 Table1 and Figure 1 illustrate that the LAPM has a higher accuracy, respectively.Case II.
For the coefficientsC=D=E=Rh=1 and ε=0.00001, (1) reduces to
(11)dTdt=T+h-0.00001T3,dhdt=-T-h.
Setting the initial condition value T(0)=h(0)=1, we can derive the following iteration formula:
(12)vn+1(t)=-0.01L-1(1s-1L[An]),v0(t)=T(0)L-1(1s-1)+h(0)L-1[1s2-1]=et+sinh(t),hn+1(t)=-L-1[L[vn(t)]s+1],h0=L-1(h(0)s-1)=et.
where h0+⋯+hn 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 functionsf(t) and g(t) as
(13)f=|dTdt-T-h+0.01T3|,g=|dhdt+T+h|.
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 depthh.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(14)dmudtm+R[u(t)]+N[u(t)]=g(t),u(k)(0)=dku(0)dtk,k=0,…,m-1
into the following equivalent integral equation
(15)u=f(t)+∫0t⋯∫0t(-R[u(s)]-N[u(s)]+g(s))ds⋯ds︸m,u(k)(0)=dku(0)dtk,k=0,…,m-1.Here∫0t⋯∫0t·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.



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