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**Jun-Sheng Duan ^{*}**

College of Science, Shanghai Institute of Technology, Shanghai 201418, P.R. China

- *Corresponding Author:
- Jun-Sheng Duan

College of Science

Shanghai Institute of Technology

Shanghai 201418, P.R. China

**E-mail**: [email protected]

**Received Date:** June 28, 2012; **Accepted Date:** June 30, 2012; **Published Date:** July 04, 2012

**Citation:**Duan JS (2012) On the Power Series Expansion of a Nonlinear Function of a Power Series. J Applied Computat Mathemat 1:e109. doi: 10.4172/2168-9679.1000e109

**Copyright:** © 2012 Duan JS. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** Journal of Applied & Computational Mathematics

**Introduction**

The power series method (PSM) is classical in resolution of differential equations. For a nonlinear differential equation, such as

du/dt=f(u), u(t_{0})=C,(1)

where f(u) is an analytical nonlinearity, the PSM requires to expand the nonlinear function of a power series into a power series. Adomian and Rach [1,2] gave the formula we require

(2)

where An, depending on a_{0}, a_{1}, . . . , a_{n}, are called the Adomian polynomials, which were defined as [3]

(3)

We note that the Adomian polynomials were initially used in the Adomian decomposition method [3,4], and they are expressed in the components uj of the Adomian decomposition series. The PSM combined with the Adomian polynomials is called the modified decomposition method [5]. For practical calculation and programming, the Adomian polynomials can be expressed as

(4)

The first five Adomian polynomials are

A_{0}=f(a_{0}),

A_{1}=f′(a_{0})a_{1}

A_{2}=f′(a_{0})a_{2} + f′′(a_{0})

A_{3}=f′(a_{0})a_{3} + f′′(a_{0})a_{1}a_{2} + f′′′(a_{0})

A_{4}=f′(a_{0})a_{4} + f′′(a_{0})

We observe that (5)

where are the sums of all possible products of k components from a_{1}, a_{2}, • • • , a_{n−k+1}, whose subscripts sum to n, divided by the factorial of the number of repeated subscripts [6], which is called Rach’s Rule [7,8].

Other different algorithms for the Adomian polynomials have been developed by Rach [9], Wazwaz [10], Abdelwahid [11], and several others [12-17].

We review the new fast algorithms for the Adomian polynomials. In [15-17] recursion relations for in (5) have been presented.

**Algorithm 1 [15].**

For

for n ≥ 2 and

for

where p_{1} → p_{1} + 1 stands for replacing by

**Algorithm 2 [17].**

For

for

In the two algorithms the recursion operation does not involve the differentiation, but only requires the operations of addition and multiplication, which greatly facilitates calculation and programming. In most practical cases, the exact solution of a nonlinear differential equation is unknown. We obtain the m-term approximation for the solution

(6)

With the fast algorithms for the Adomian polynomials we can efficiently calculate the for large *m*. Further we can use the acceleration convergence techniques, such as the Pade´ approximants and the iterated Shanks transforms, to extend the effective region of convergence and increase the accuracy for the approximate solution.

Another important application is to derive the high-order numeric scheme for nonlinear differential equations more efficiently. For each subinterval [ti, t_{i+1}] we apply the m-term approximation (t; t_{i}, C_{i}), where i=0, 1, . . . , and C_{0} is the initial value while C_{i}, i>0, is the value at t=t_{i} of the last approximation (t; t_{i−1},C_{i−1}).

For the MATHEMATICA subroutine for generating the Adomian polynomials and further readings we suggest readers to refer to [16-18].

- Adomian G, Rach R (1991) Transformation of series. Appl Math Lett 4: 69-71.
- Adomian G, Rach R (1992) Nonlinear transformation of series-part II. Comput Math Appl23: 79-83.
- Adomian G, Rach R (1983) Inversion of nonlinear stochastic operators. J Math Anal Appl 91: 39-46.
- Adomian G (1983) Stochastic Systems. Academic New York
- Rach R, Adomian G, Meyers RE (1992) A modified decomposition. Comput Math Appl 23: 17-23.
- Rach R (1984) A convenient computational form for the Adomian polynomials. J Math Anal Appl 102: 415-419.
- Adomian G (1989) Nonlinear Stochastic Systems Theory and Applications to Physics. Kluwer Academic Dordrecht.
- Adomian G (1994) Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic Dordrecht.
- Rach R (2008) A new definition of the Adomian polynomials. Kybernetes 37: 910-955
- Wazwaz AW (2000) A new algorithm for calculating adomian polynomials for nonlinear operators. Appl Math Comput 111: 33-51.
- Abdelwahid F (2003) A mathematical model of Adomian polynomials. Appl Math Comput 141: 447-453.
- Abbaoui K, Cherruault Y, Seng V (1995) Practical formulae for the calculus of multivariable Adomian polynomials. Math Comput Modelling 22: 89-93.
- Zhu Y, Chang Q, Wu S (2005) A new algorithm for calculating Adomian polynomials. Appl Math Comput 169: 402-416.
- Azreg-A¨inou M (2009) A developed new algorithm for evaluating Adomian polynomials. CMES-Comput Model Eng Sci 42: 1-18.
- Duan JS (2010) Recurrence triangle for Adomian polynomials. Appl Math Comput 216: 1235-1241.
- Duan JS (2010) An efficient algorithm for the multivariable Adomian polynomials. Appl Math Comput 217: 2456-2467.
- Duan JS (2011) Convenient analytic recurrence algorithms for the Adomian polynomials. Appl Math Comput 217: 6337-6348
- Duan JS, Rach R (2011) New higher-order numerical one-step methods based on the Adomian and the modified decomposition methods. Appl Math Comput 218: 2810-2828.

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