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A Convenient Adomian-Pade Technique for the Nonlinear Oscillator Equation
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.
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 [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 . 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  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  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 . 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 . (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
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.
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.
- G. Adomian, Solving Frontier Problems of Physics: the Decomposition Method, Kluwer Academic Publishers, Boston, Mass, USA, 1994.
- S. S. Ray and R. K. Bera, “An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 167, no. 1, pp. 561–571, 2005.
- H. Jafari and V. Daftardar-Gejji, “Revised Adomian decomposition method for solving a system of nonlinear equations,” Applied Mathematics and Computation, vol. 175, no. 1, pp. 1–7, 2006.
- J.-S. Duan, “Recurrence triangle for Adomian polynomials,” Applied Mathematics and Computation, vol. 216, no. 4, pp. 1235–1241, 2010.
- J.-S. Duan, “An efficient algorithm for the multivariable Adomian polynomials,” Applied Mathematics and Computation, vol. 217, no. 6, pp. 2456–2467, 2010.
- J. S. Duan, R. Rach, D. Baleanu, and A. M. Wazwaz, “A review of the Adomian decomposition method and its applications to fractional differential equations,” Communications in Fractional Calculus, vol. 3, no. 2, pp. 73–99, 2012.
- G.-C. Wu, “Adomian decomposition method for non-smooth initial value problems,” Mathematical and Computer Modelling, vol. 54, no. 9-10, pp. 2104–2108, 2011.
- V. Daftardar-Gejji and H. Jafari, “Adomian decomposition: a tool for solving a system of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 301, no. 2, pp. 508–518, 2005.
- H. Jafari and V. Daftardar-Gejji, “Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition,” Applied Mathematics and Computation, vol. 180, no. 2, pp. 488–497, 2006.
- S. Momani and Z. Odibat, “Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 488–494, 2006.
- P.-Y. Tsai and C.-K. Chen, “An approximate analytic solution of the nonlinear Riccati differential equation,” Journal of the Franklin Institute, vol. 347, no. 10, pp. 1850–1862, 2010.
- D. Q. Zeng and Y. M. Qin, “The Laplace-Adomian-Pade technique for the seepage flows with the Riemann-Liouville derivatives,” Communications in Fractional Calculus, vol. 3, no. 1, pp. 26–29, 2012.
- H. Jafari, C. M. Khalique, and M. Nazari, “Application of the Laplace decomposition method for solving linear and nonlinear fractional diffusionwave equations,” Applied Mathematics Letters, vol. 24, no. 11, pp. 1799–1805, 2011.
- H. Jafari, M. Nazari, D. Baleanu, and C. M. Khalique, “A new approach for solving a system of fractional partial differential equations,” Computers & Mathematics with Applications, vol. 66, no. 5, pp. 838–843, 2013.
- J. Q. Mo and W.-T. Lin, “Generalized variation iteration solution of an atmosphere-ocean oscillator model for global climate,” Journal of Systems Science and Complexity, vol. 24, no. 2, pp. 271–276, 2011.
- Y. Zeng, “The Laplace-Adomian-Pade technique for the ENSO model,” Mathematical Problems in Engineering, vol. 2013, Article ID 954857, 4 pages, 2013.
- G. C. Wu, “Challenge in the variational iteration method—a new approach to identification of the Lagrange multipliers,” Journal of King Saud University—Science, vol. 25, no. 2, pp. 175–178, 2013.
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