The Asymptotic Theory of Solutions of Differential Equations
Project/Area Number 
11640193

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Basic analysis

Research Institution  Kochi University of Technology 
Principal Investigator 
NISHIMOTO Toshihiko Kochi University of Technology Engineering, Professor, 工学部, 教授 (60016061)

CoInvestigator(Kenkyūbuntansha) 
KASAHARA Yasushi Kochi University of Technology, Engineering, Lecturer, 工学部, 講師 (80299370)
SEKIGUCHI Kouji Kochi University of Technology, Engineering, Asso. Professor, 工学部, 助教授 (80163096)
INOUE Masaaki Kochi University of Technology, Engineering, Asso. Professor, 工学部, 助教授 (50168465)

Project Period (FY) 
1999 – 2001

Project Status 
Completed (Fiscal Year 2001)

Budget Amount *help 
¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 2001: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2000: ¥900,000 (Direct Cost: ¥900,000)

Keywords  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 
Research Abstract 
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 wplane. 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

Report
(4 results)
Research Products
(6 results)