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The Asymptotic Theory of Solutions of Differential Equations

微分方程解的渐近理论

基本信息

批准号：
11640193
负责人：
NISHIMOTO Toshihiko
金额：
$ 1.34万
依托单位：
Kochi University of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640193/
关键词：
The complex WKB methods Asymptotic expansion The Stokes curves The canonical domain Turning point Movable saddle point method Connection formulas Fedoryuk理論鞍部点法 shadow region リーマン面新ストークス曲線

项目摘要

There are four purposes of studies of our Research Project Asymptotic Theory of Solutions of Differential Equations. Regarding the complex WKB method for higher order ordinary differential equations, We applied Fedoryuk theory to the third order differential equation named BNR equation. And as for studies of the asymptotic expansion of the functions defined by the integrals, we treated solutions of BNR equation expressed by the Laplace integral.In these analyses we effectively use the notion of movable saddle point method. After all, we could obtain almost complete asymptotic analyses for the BNR equation, that is to construct asymptotic expansions on the whole complex plane and to get the connectIon formulas, after twenty years from the BNR equation firstly appeared in 1982.Our results will be published in the near future. Regarding other two purposes of the project, confluent WKB method, and asymptotic theory for partial differential equation, we could not obtain any essential progre … More ss.There are two break through in our analyses. The one is the discovery of the mapping between a Riemman surfase of characteristic roots of BNR equation, which composed of 6 sheets of the complex z plane, and one sheet of the complex w-plane. By this, we can express whole Stokes curves or Stokes domains on one sheet of paper, and then it becomes possible to construct admissible domains where asymptotic expansions of solutions exist, or canonical domains where fundamental systems of solutions exist, in a visible manner. In the course of the analyses, we find the existence of shadow zone which does not exist in the case of second order differential equations.The another break through is the notion of the movable saddle point method. By applying the movable saddle point method to the solution of the BNR equation expressed by the Laplace integral, we find that the Laplace integral has asymptotic expansion uniformly valid for z in the admissible domain. Moreover, the Cauchy s integral theorem gives us connection formulas which describe linear relation between several solutions of the BNR equation. Less

我们的研究计画“微分方程解的渐近理论”有四个研究目的。关于高阶常微分方程的复WKB方法，我们将Fedoryuk理论应用于三阶常微分方程BNR方程。在研究由积分定义的函数的渐近展开式时，我们处理了由拉普拉斯积分表示的BNR方程的解，在这些分析中，我们有效地使用了活动鞍点方法的概念。毕竟，从1982年BNR方程首次出现至今，经过20年的时间，我们已经可以对BNR方程进行几乎完全的渐近分析，即在整个复平面上构造渐近展开式，得到连通公式，我们的结果将在不久的将来发表。关于项目的另外两个目的，合流WKB方法和偏微分方程的渐近理论，我们没有得到任何实质性的进展。 ...更多信息本文的分析有两个突破。一是发现了BNR方程的特征根的Riemman曲面（由复z平面的6片组成）与复w平面的1片之间的映射;这样，我们可以在一张纸上表示整个Stokes曲线或Stokes域，然后可以以可见的方式构造存在解的渐近展开式的容许域或存在基本解系的正则域。在分析过程中，我们发现了二阶微分方程不存在的阴影区的存在性，另一个突破是提出了可移动鞍点方法的概念。将可移动鞍点方法应用于由拉普拉斯积分表示的BNR方程的求解，发现在容许区域内，拉普拉斯积分具有对z一致有效的渐近展开式.此外，柯西积分定理给出了描述BNR方程多个解之间线性关系的联系公式。少