College of Science, Shanghai Institute of Technology, Shanghai 201418, P.R. China
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.
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The power series method (PSM) is classical in resolution of differential equations. For a nonlinear differential equation, such as
where An, depending on a0, a1, . . . , an, are called the Adomian polynomials, which were defined as 
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 . For practical calculation and programming, the Adomian polynomials can be expressed as
The first five Adomian polynomials are
A2=f′(a0)a2 + f′′(a0)
A3=f′(a0)a3 + f′′(a0)a1a2 + f′′′(a0)
A4=f′(a0)a4 + f′′(a0)
We observe that (5)
where are the sums of all possible products of k components from a1, a2, • • • , an−k+1, whose subscripts sum to n, divided by the factorial of the number of repeated subscripts , which is called Rach’s Rule [7,8].
Algorithm 1 .
for n ≥ 2 and
where p1 → p1 + 1 stands for replacing by
Algorithm 2 .
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
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, ti+1] we apply the m-term approximation (t; ti, Ci), where i=0, 1, . . . , and C0 is the initial value while Ci, i>0, is the value at t=ti of the last approximation (t; ti−1,Ci−1).
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