AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 601643 10.1155/2014/601643 601643 Research Article A Convenient Adomian-Pade Technique for the Nonlinear Oscillator Equation http://orcid.org/0000-0001-7045-5519 Wei Na 1 Liu Cheng-Xi 2 Hu Teng 2 You Fu-Cai 3 Baleanu Dumitru 1 State Key Laboratory of Oil and Gas Reservoirs Geology and Exploration Southwest Petroleum University Chengdu 610500 China swpi.edu.cn 2 Key Laboratory of Numerical Simulation of Sichuan Province and College of Mathematics and Information Science Neijiang Normal University Neijiang 641112 China njtc.edu.cn 3 Department of Basic Sciences Shenyang Institute of Engineering Shenyang 110136 China sie.edu.cn 2014 18 3 2014 2014 26 02 2014 12 03 2014 12 03 2014 9 4 2014 2014 Copyright © 2014 Na Wei et al. 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.

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) . 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 .

Very recently, for the ADM, Duan  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 . The technique has been successfully extended to fractional differential equations and boundary value problems.

Recently, Tsai and Chen  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  (1)dTdt=CT+Dh-εT3,T(0)=1,dhdt=-ET-Rhh,h(0)=1, where C,D,E, and Rh are physical constants, T describes the temperature of the eastern equatorial Pacific sea surface, and h is the thermocline depth anomaly.

2. Preliminaries of the Adomian Series

Generally, consider the following nonlinear equation: (2)L[u]+R[u]+N[u]=g(t), where L is the highest derivative of u, for example, m order, R is the remaining linear part containing the lower order derivatives, and N is the nonlinear operator.

Apply the inverse L-1 of the linear operator L in (2), and we can obtain (3)u=u(0)+u(0)t++u(m-1)(0)tm-1(m-1)!+L-1(g(t))-L-1(R[u])-L-1(N[u]).

Consider the basic idea of the Picard method (4)un+1=u(0)+u(0)t++u(m-1)(0)tm-1(m-1)!+L-1(g(t))-L-1(R[un])-L-1(N[un]), and assume that (5)u=i=0vi,un=i=0nvi.

The classical ADM  supposes that the nonlinear term N[u] can be expanded approximately as (6)N[u]=n=0An, where An is calculated by (7)An=1n!nλn[N(k=0vkλk)]λ=0. For example, n=0An is the Adomian series of T3; namely, (8)A0=v03,A1=3v02v1,A2=3v0v12+3v02v2,

Duan et al.  very recently suggested a convenient way to calculate the Adomian series as (9)An=1nk=0n-1(k+1)vk+1dAn-1-kdv0, as well as the case of the m-variable (10)An=1ni=1mk=0n-1(k+1)vi,k+1An-1-kvi,0. For the single variable case, N[u]=f(u), the first three components are listed as (11)A1=v1df(v0)dv0,A2=12v12d2f(v0)dv02+v2df(v0)dv0,A3=16v13d3f(v0)dv03+v2v1d2f(v0)dv02+v3df(v0)dv0,A4=124v14d4f(v0)dv04+12v2v12d3f(v0)dv03+v3v1d2f(v0)dv02+12v22d2f(v0)dv02+v4df(v0)dv0. And for the two-variable case, N[u]=f(u1,u2), the first three components are listed as (12)A1=v1,1f(v1,0,v2,0)v1,0+v2,1f(v1,0,v2,0)v2,0,A2=12v1,122f(v1,0,v2,0)v1,02+v1,1v2,12f(v1,0,v2,0)v2,0v1,0+v1,2f(v1,0,v2,0)v1,0+12v2,122f(v1,0,v2,0)v2,02+v2,2f(v1,0,v2,0)v2,0,A3=16v1,133f(v1,0,v2,0)v1,03+12v1,12v2,13f(v1,0,v2,0)v2,0v1,02+v1,1v1,22f(v1,0,v2,0)v1,02+12v1,1v2,123f(v1,0,v2,0)v2,02v1,0+v1,1v2,22f(v1,0,v2,0)v2,0v1,0+v1,2v2,12f(v1,0,v2,0)v2,0v1,0+v1,3f(v1,0,v2,0)v1,0+16v2,133f(v1,0,v2,0)v2,03+v2,1v2,22f(v1,0,v2,0)v2,02+v2,3f(v1,0,v2,0)v2,0. The above formulae (9) spend less time deriving the An. 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 . Considering (2), we show the following iteration schemes.

(i) Take Laplace transform L~ to both sides: (13)L~[L[u]+R[u]]+L~[N[u]]=L~[g(t)].

We can have iteration formula (4) through inverse of Laplace transform L~-1: (14)u(t)=f(t)+L~-1[λ~(s)L~[N[u]]], where f(t) and λ~(s) can be determined by calculation of Laplace transform to L[u], R[u], and g(t). The calculation of λ~(s) is similar to the determination of the Lagrange multiplier of the variational iteration method in .

(ii) Through the Picard successive approximation, we can obtain the following iteration formula: (15)un+1=f(t)+L~-1[λ~(s)L~[N[un]]].

(iii) Let un=i=0nvi and apply the Adomian series to expand the term N[u] as i=0Ai. Then, the iteration formula reads (16)vn+1=L~-1[λ~(s)L~[An]],v0=f(t),

where Ai are calculated by (17)An=1nk=0n-1(k+1)vk+1dAn-1-kdv0.

(iv) Employ the Pade technique to accelerate the convergence of un=i=0nvi.

4. Applications of the Iteration Formulae

In this study, we consider a reduced case where D=0 and 0<ε1 in (1) as follows: (18)dTdt=CT-εT3,C=1,T(0)=1.

In order to solve (18) with the Maple software, apply Laplace transform L to both sides firstly. This step can fully and optimally determine the initial iteration. We can derive (19)vn+1(t)=-L~-1[εL~[An](s-C)],n1,v0=L~-1[T(0)(s-C)].

Setting ε=0.00001 in the model (18), now we can obtain the first few as (20)T0=v0=et,T1=v0+v1=et-0.0001  e2tsinh(t), Apply the Pade technique to Tn and denote the result as PT1,n[p/q].

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 (21)vn+1(t)=C0tvndτ-ε0tAndτ,n1,v0=T(0). Also apply the Pade technique to Tn and denote the result as PT2,n[p/q].

Define the residual function as (22)gn=log|d(PTi,n[p/q])dt-C(PTi,n[pq])=log=+ε(PTi,n[pq])3|,i=1,2.

Consider the same n=20 and p=q=20; from the comparison illustrated through Figure 1, we can see that the iteration formula (19) has a higher accuracy almost in the interval [0,5].

The comparisons of the approximate solutions using (19) and (21).

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.

Analytical solution of (18) via (19).

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.