﻿ 帕德逼近及在求解非线性系统中的应用 Padé Approximation and Its Application in Solving Nonlinear Systems

Dynamical Systems and Control
Vol.05 No.04(2016), Article ID:18850,18 pages
10.12677/DSC.2016.54018

Padé Approximation and Its Application in Solving Nonlinear Systems

Youhua Qian*, Haixia Fu, Liang Shen

College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua Zhejiang

Received: Oct. 8th, 2016; accepted: Oct. 27th, 2016; published: Oct. 31st, 2016

ABSTRACT

Many problems in engineer science and natural science are often summed up in solving nonlinear systems. For many years, the rational function approximation has attracted more and more attention, which is one of the typical nonlinear approximation approaches. We mainly study one of the classical rational function approximations—Padé approximation in this paper. We take the orthogonal function system as the base function, and study some Padé approximation problems under the base function of orthogonal trigonometric function and orthogonal polynomial function respectively. Then the approximation effect is demonstrated by the concrete examples. Finally, we combine the Padé approximation with the homotopy analysis method to solve the nonlinear system, and verify its effectiveness by a three-degree-of-freedom system.

Keywords:Padé Approximation, Orthogonal Function System, Homotopy Analysis Method, Nonlinear System

1. 引言

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(1) 当时，。求出的值，自然就知道的值；

(2) 当时，，即

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Figure 1. (11,3) order Padé approximation compared with f(x)

Figure 2. (11,4) order Padé approximation compared with f(x)

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Figure 3. (11,2) order Padé approximation compared with f(x)

Figure 4. (10,2) order Padé approximation compared with f(x)

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Figure 5. (10,3) order Padé approximation compared with f(x)

Figure 6. (10,4) order Padé approximation compared with f(x)

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4. 帕德逼近在非线性系统求解中的应用

Figure 7. (5,2) order Padé approximation compared with f(x)

Figure 8. (5,4) order Padé approximation compared with f(x)

Figure 9. (5,1) order Padé approximation compared with f(x)

Figure 10. (5,3) order Padé approximation compared with f(x)

4.1. 同伦分析方法

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Figure 11. (5,5) order Padé approximation compared with f(x)

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,

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，得到零阶形变方程：

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，形变向量导数都存在；

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4.2. 同伦帕德逼近方法

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4.3. 具体例子

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Figure 12. The curves of x1(t), x2(t), x3(t) are obtained by the homotopy analysis method for nonlinear systems

Figure 13. The curves of x1(t), x2(t), x3(t) are obtained by homotopy padé approximation method

Padé Approximation and Its Application in Solving Nonlinear Systems[J]. 动力系统与控制, 2016, 05(04): 161-178. http://dx.doi.org/10.12677/DSC.2016.54018

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