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On the Power Series Expansion of a Nonlinear Function of a Power Series | OMICS International
ISSN: 2168-9679
Journal of Applied & Computational Mathematics
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On the Power Series Expansion of a Nonlinear Function of a Power Series

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.

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Nonlinear Differential Equation

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(t0)=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

Equation(2)

where An, depending on a0, a1, . . . , an, are called the Adomian polynomials, which were defined as [3]

Equation(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

Equation(4)

The first five Adomian polynomials are

A0=f(a0),

A1=f′(a0)a1

A2=f′(a0)a2 + f′′(a0) Equation

A3=f′(a0)a3 + f′′(a0)a1a2 + f′′′(a0) Equation

A4=f′(a0)a4 + f′′(a0) Equation

We observe that Equation(5)

where Equation 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 [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].

New Fast Algorithms and Applications

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

Algorithm 1 [15].

For Equation

for n ≥ 2 and Equation

for Equation

where p1 → p1 + 1 stands for replacing Equation by

Algorithm 2 [17].

For Equation

for Equation

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

Equation(6)

With the fast algorithms for the Adomian polynomials we can efficiently calculate the Equation 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 Equation (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 Equation (t; ti−1,Ci−1).

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

References

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