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Study on asymptotic analysis of differential equations

Research Project

Project/Area Number 10640187
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionSophia University

Principal Investigator

UCHIYAMA Koichi  Sophia Univ. Math. Professor, 理工学部, 教授 (20053689)

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)
Project Period (FY) 1998 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥2,400,000 (Direct Cost: ¥2,400,000)
Fiscal Year 1999: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 1998: ¥1,200,000 (Direct Cost: ¥1,200,000)
Keywordscomplex domain / asymptotic analysis / singular point / Stokes phenomenon / saddle point / nonlinear / differential equation
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.

Report

(3 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • Research Products

    (23 results)

All Other

All Publications (23 results)

  • [Publications] K. Uchiyama: "Asymptotic analysis of the modified Bessel function with respect to the parameter"RIMS Kokyuroku. 1088. 55-67 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Uchiyama: "Graphical illustration of Stokes phenomenon of integrals with saddles"Toward the Exact WKB analysis of Differential Equations, Linear or Non-linear, Kyoto UP. 87-95 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Ouchi: "Singular Solution with Asymptotic expansion of Linear Partial Differential Equitations in the Complex Domain"Publ. RIMS, Kyoto Univ.. 34. 291-311 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] R. Gerard(共): "Formal Power Series Solutions of Nonlinear First Order partial Differential Equaitons"Funkcialaj Ekvacioj. 41. 133-166 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H.Tahara: "Uniqueness of the solutions of nonlinear singular partial differential equations"Partial differential equatioins and their applications, world scientific. 297-304 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Sato(共): "Faithful plotting on a two dimensional pixel space"Josai Mathematical Monographs. 2. 77-86 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Uchiyama: "Asymptotic analysis of the modified Bessel function with respect to the parameter"RIMS Kokyuroku. 1088. 55-67 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Uchiyama: "Graphical illustration of Stokes phenomenon of intergrals with saddles, "Toward the Exact WKB Analysis of Differential Equations, Linear or Non-linear""Kyoto Univ. Press. 87-95 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Ouchi: "Singular Solution With Asymptotic Expansion of Linear Partial Differential Equations in the Complex Domain"Publ. RIMS, Kyoto Univ.. 34. 291-311 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] R. Gerard and H. Tahara: "Formal Power Series Solutions of Nonlinear First Order Partial Differential Equations"Funkeialaj Ekvacioj. 41. 133-166 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H. Tahara: "Uniqueness of the solution of nonlinear singular partial differential equations, "Partial differential equations and their applications""World Scientific. 297-304 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Saito, Y. Kondo, Y. Miyoshi and T, Takeshima: "Faithful plotting on a two dimensional pixel space"Josai Mathematical Monographs. 2. 77-86 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N. Hayashi and H. Hirata: "Global existence of small solutions to nonlinear Schrodinger equations"Nonlinear Analysis, Theory & Application. 31. 671-685 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Goto: "Quantum double construction for subfactors arising from periodic commuting squares"J. Math. Soc. Japan. 52. 187-198 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K.Uchiyama: "Graphical illustration of Stokes phenomenon of integral with saddles"Toward the Eact WRB Analysis of Differential Equations,Linear or Non-Linear,Kyoto U.P.. 87-95 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] S.Ouchi: "Growth property and slowly increasing behavior of singular solutions of linear partial differential equations."J.Math.Soc.Japan. (発売予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Tahara: "Uniqueness of the solution of monlinear singular patial differential equations"Partial differential equations and their applications. 297-304 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Saito et al: "Faithful plotting on a two dimensional pixel space"Josai Mathematical Monographs. 2. 77-86 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] N.Hayashi: "Global exstence of small solutions to nonlinear Schrodinger equations"Nonlinear Analysis,Theory,Methods & Applications. 31. 671-685 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] S.Goto: "Quantum double construction for subfactors arising from periodic commuting squares"J.Math.Soc.Japan. 52・1. 187-198 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Tahara: "On the uniqueness theorem for nolinear singular partial differential eqs" J.Math.Sci.Univ.Tokyo. 5. 477-506 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] H.Tahara,R.Gerard: "Formal power series solutions of nonlinear first order partial differential equations" Funkcialaj Ekvacioj. 41. 133-166 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] S.Ouchi: "Singular solutions with Asymptotic Expansion of Linear Partial Differential Equations in the complex domain" Publ. RlMS Kyoto Univ.34. 291-311 (1998)

    • Related Report
      1998 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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