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

摘要: 工程和自然科学中的许多问题常常可以归结为非线性系统的求解。多年来，作为非线性逼近的典型之一的有理函数逼近愈来愈引起人们的关注。本文主要研究了帕德逼近这一经典的有理函数逼近。我们以正交函数系作为基函数，分别研究了正交三角函数系和正交多项式函数系下的帕德逼近问题，并通过具体的例子展示了其逼近效果。最后，我们将帕德逼近与同伦分析方法结合来求解非线性系统，并通过一个三自由度系统验证了它的有效性。

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.