WKB analysis for highorder ordinary differencial equations with a large parameter
Project/Area Number 
12640195

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Basic analysis

Research Institution  KINKI UNIVERSITY 
Principal Investigator 
AOKI Takashi KINKI UNIVERSITY, SCHOOL OF SCIENCE AND ENGINEERING, PROFESSOR, 理工学部, 教授 (80159285)

CoInvestigator(Kenkyūbuntansha) 
TAZAWA Shinsei KINKI UNIVERSITY, SCHOOL OF SCIENCE AND ENGINEERING, PROFESSOR, 理工学部, 教授 (80098657)
IZUMI Shuzo KINKI UNIVERSITY, SCHOOL OF SCIENCE AND ENGINEERING, PROFESSOR, 理工学部, 教授 (80025410)

Project Period (FY) 
2000 – 2002

Project Status 
Completed (Fiscal Year 2002)

Budget Amount *help 
¥2,500,000 (Direct Cost: ¥2,500,000)
Fiscal Year 2002: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2001: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2000: ¥1,200,000 (Direct Cost: ¥1,200,000)

Keywords  differential equations of higherorder / differntial equations of infinite order / exact WKB analysis / local theory / turning points / Stokes curves / connection problems / WKB solutions / 無段階微分方程式 / 高階微分方程式 / 最急降下法 / ストークス現象 / 積分表示 / レベル交叉 / WKB解析 
Research Abstract 
The purpose of this research was to establish the exact WKB analysis for ordinary differential equations of higherorder with a large parameter. Regarding local theory, we have achieved this purpose. The results we have obtained are as follows: (1) We established a new method of describing Stokes geometry for differential equations of higherorder with a large parameter. This method is called the exact steepest descent method. By using this method, we can find complete Stokes geometry for linear ordinary differential equations of arbitrary order with quadratic coefficients. This is a natural generalization of the steepest descent method. (2) We consider linear ordinary differential equation of infinite order with a large parameter satisfying the following condition: Regarding the large parameter as a differential operator with respect to the variable of the Borel plane, they are microdifferential operators of order 0 and they do not contain that variable. The category of such operators c
… More
ontains linear ordinary differential operators of arbitrary order with the large parameter. For such an operator, we have defined the notions of WKB solutions, turning points and Stokes curves. We have also introduced the notion of simplicity of turning points. These notions are natural extension of that for finiteorder case, (3) Such an equation in the class mentioned above may admit infinitely many phases. After fixing one of it, we have constructed the WKB solution of the difierential equation. This construction is applicable to linear ordinary differential equations of arbitrary order. (4) We have established local decomposition theorem near turning points. If we consider a simple turning point of a differential equation of infinite order, we can decompose the relevant differential operator into the product of two operators; one is invertible and another is of second order whose turning point is exactly the same as the simple turning point. Moreover, the phase of the second order operator is the coincident with the original phase. This implies that an infiniteorder differential equation is reduced to a secondorder equation near a simple turning point. Thus we have obtained local connection formulas of infiniteorder equations near simple turning points. Less

Report
(4 results)
Research Products
(19 results)