1999 Fiscal Year Final Research Report Summary
Study on asymptotic analysis of differential equations
| Project/Area Number |
10640187
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Sophia University |
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Principal Investigator |
UCHIYAMA Koichi Sophia Univ. Math. Professor, 理工学部, 教授 (20053689)
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| Co-Investigator(Kenkyū-buntansha) |
YOSHINO Kunio Sophia Univ. Math. Lecturer, 理工学部, 講師 (60138378)
TAHARA Hidetoshi Sophia Univ. Math. Professor, 理工学部, 教授 (60101028)
OUCHI Sunao Sophia Univ. Math. Professor, 理工学部, 教授 (00087082)
SAITO Tomokatsu Sophia Univ. Math. Assistant, 理工学部, 助手 (00119132)
HIRATA Hitoshi Sophia Univ. Math. Assistant, 理工学部, 助手 (20266076)
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| Project Period (FY) |
1998 – 1999
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| Keywords | complex domain / asymptotic analysis / singular point / Stokes phenomenon / saddle point / nonlinear / differential equation |
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| Research Abstract |
A. Asymptotic analysis of differential equations in complex domain.
Uchiyama analyzed an integral representation with infinite saddles of the modified Bessel functions to give all illustration of new phenomena of Stokes geometry by hyperasymptotic analysis, WKB method and computer aided graphics. He also gave an overview of Stokes phenomenon of integrals with saddles. Ouchi studied linear partial differential equations in the complex domain. Supposing that the inhomogenious term and a solution are holomorphic outside of a hypersurface K, he proved that if the inhomogenious term has a certain asymptotic property near K, the solution has the asymptotic property of the same type. Tahara extended the classical Maillet type theorem to obtain that if a formal solution exists, it belongs to a certain Gevrey class. Furthermore, he gave uniqueness theorems, existence of a holomrophic solution, existence of singular points of the solution. Yoshino computed concretely the Jost function of a harmonic oscillator proposed by A. Voros using the Binet formula for Euler's Gamma function
B. Applied analysis in real domain.
Hirata constructed a time global solution for small initial data using smoothing property. Saito constructed an algorithm of drawing of curves defined as real zeros of real algebraic polynomials with real coefficients and implemented it .
Goto generalized a new construction of subfactors by Erlijman in operator algebra.
Saito and Kobayashi helped to keep and enhance the computer environments for research.
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