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

我们的研究项目不对称解决方案解决方案解决方案的不对称理论有四个目的。关于高阶普通微分方程的复杂WKB方法，我们将联邦理论应用于名为BNR方程的三阶微分方程。对于积分定义的功能的不对称扩展的研究，我们处理了由拉普拉斯积分表达的BNR方程的溶液。在这些分析中，我们有效地使用了可移动的鞍点方法的概念。毕竟，我们可以获得BNR方程的几乎完整的不对称分析，即在BNR方程二十年后首次出现在1982年。在1982年首次出现后，将在整个复杂平面上构建不对称的扩展并获得连接公式。关于该项目的其他两个目的，汇合WKB方法和部分偏微分方程的不对称理论，我们无法获得任何基本进步……更多。我们的分析有两个突破。一个是发现BNR方程的特征根的riemman曲面之间的映射，该映射由复杂Z平面的6张和一张复合物W平面组成。这样，我们可以在一张纸上表达整个Stokes曲线或Stokes域，然后可以构建可允许的域存在，或者以可见的方式存在基本溶液系统的溶液域或规范域。在分析过程中，我们发现在二阶微分方程中不存在的阴影区域的存在。另一个突破是可移动的鞍点方法的概念。通过将可移动的鞍点法应用于Laplace积分表达的BNR方程的溶液中，我们发现Laplace积分具有不对称的扩展，对可允许的域中的Z有效。此外，库奇的整体理论为我们提供了连接公式，该公式描述了BNR方程的几种解决方案之间的线性关系。较少的