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A STUDY OF SINGULARITIES IN NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS IN THE COMPLEX DOMAIN

Research Project

Project/Area Number 19540197
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

TAHARA Hidetoshi  Sophia University, 理工学部, 教授 (60101028)

Co-Investigator(Kenkyū-buntansha) OUCHI Sunao  上智大学, 理工学部, 教授 (00087082)
UCHIYAMA Koichi  上智大学, 理工学部, 名誉教授 (20053689)
KATO Masahide  上智大学, 理工学部, 教授 (90062679)
HIRATA Hitoshi  上智大学, 理工学部, 講師 (20266076)
YOSHINO Kunio  東京都市大学, 知識工学部, 教授 (60138378)
Project Period (FY) 2007 – 2009
Project Status Completed (Fiscal Year 2009)
Budget Amount *help
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2009: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2008: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2007: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywords関数方程式 / 偏微分方程式 / 複素領域 / 特異点 / 形式解 / 正則性 / 合成積方程式 / 正則解 / 変換の方程式
Research Abstract

We studied singularities of solutions of nonlinear partial differential equations in the complex domain, and obtained the following results. (1) In the case of Briot-Bouquet type partial differential equations, we determined all solutions which have singularities on a hypersurface. (2) We proved a sharp form of the uniqueness of the solution of nonlinear totally characteristic type partial differential equations, which yields the non-existence of singularities. (3) We constructed solutions with logarithmic singularities to certain nonlinear partial differential equations. (4) We determined asymptotic expansions of solutions of nonlinear partial differential equations of non-Kowalewskian type by using Mellin transformations.

Report

(4 results)
  • 2009 Annual Research Report   Final Research Report ( PDF )
  • 2008 Annual Research Report
  • 2007 Annual Research Report
  • Research Products

    (43 results)

All 2010 2009 2008 2007

All Journal Article (13 results) (of which Peer Reviewed: 13 results) Presentation (29 results) Book (1 results)

  • [Journal Article] Coupling of two partial differential equations and its application, II-the case of Briot-Bouquet type PDEs-.2009

    • Author(s)
      Hidetoshi Tahara
    • Journal Title

      Publ. Res. Inst. Math. Sci. 45

      Pages: 393-449

    • NAID

      110007121184

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Coupling of two partial differential equations and its application, II-the case of Briot-Bouquet type PDEs-2009

    • Author(s)
      H.Tahara
    • Journal Title

      Publ.Res.Inst.Math.Sci. 45

      Pages: 393-449

    • NAID

      110007121184

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A sharp form of the uniqueness of the solution of nonlinear totally characteristic partial differential equations2008

    • Author(s)
      Hidetoshi Tahara
    • Journal Title

      Tokyo J. Math. 31

      Pages: 195-202

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Logarithmic singularities of solutions to nonlinear partial differential equations.2008

    • Author(s)
      Hidetoshi Tahara-Hideshi Yamane
    • Journal Title

      J. Math. Soc. Japan 60

      Pages: 603-630

    • NAID

      10024331825

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Coupling of two partial differential equations and its application. Algebraic Analysis of Differential Equations (edited by T. Aoki, H. Majima, Y. Takei and N. Tose)2008

    • Author(s)
      Hidetoshi Tahara
    • Journal Title

      Springer

      Pages: 295-306

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] The asymptotic behavior of singular solutions of some nonlinear partial differential equations in the complex domain.2008

    • Author(s)
      Sunao Ouchi
    • Journal Title

      Publ. Res. Inst. Math. Sci. 44

      Pages: 973-1026

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Asymptotic expansions of the solutions to the heat equations with hyperfunctions initial value.2008

    • Author(s)
      Kunio Yoshino-Yasuyuki Oka
    • Journal Title

      Commun. Korean Math. Soc. 23

      Pages: 555-565

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] A proof of Paley-Wiener theorem for Fourier hyperfunctions with support in a proper convex cone by the heat kernel method.2008

    • Author(s)
      Masanori Suwa-Kunio Yoshino
    • Journal Title

      Complex Var. Elliptic Equ. 53

      Pages: 833-841

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Logarithmic singularities of solutions to nonlinear partial differential equations2008

    • Author(s)
      H. Tahara-H. Yamane
    • Journal Title

      J. Math. Soc. Japan 60

      Pages: 603-630

    • NAID

      10024331825

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A sharp form of the uniqueness of the solution of nonlinear totally characteristic partial differential equations2008

    • Author(s)
      H. Tahara
    • Journal Title

      Tokyo J. Math. 31

      Pages: 195-202

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Coupling of two partial differential equations and its application.2007

    • Author(s)
      Hidetoshi Tahara
    • Journal Title

      Publ. Res. Inst. Math. Sci. 43

      Pages: 535-583

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Coupling of two partial differential equations and its application2007

    • Author(s)
      H. Tahara
    • Journal Title

      Publ.Res.Inst.Math.Sci., Kyoto Univ 43

      Pages: 535-583

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On Blanchard manifolds2007

    • Author(s)
      M. Kato-K, Komada
    • Journal Title

      Tokyo J.Math 30

      Pages: 397-401

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Presentation] An analogue of Maillet type theorem in convolution partial differential equations.2010

    • Author(s)
      田原秀敏
    • Organizer
      研究集会「第2回名古屋微分方程式研究集会」
    • Place of Presentation
      名古屋大学理1号館
    • Year and Date
      2010-03-17
    • Related Report
      2009 Final Research Report
  • [Presentation] An analogue of Maillet type theorem in convolution partial differential equations2010

    • Author(s)
      田原秀敏
    • Organizer
      第2回名古屋微分方程式研究集会
    • Place of Presentation
      名古屋大学
    • Year and Date
      2010-03-17
    • Related Report
      2009 Annual Research Report
  • [Presentation] Gevrey regularities of solutions of nonlinear singular partial differential equations.2009

    • Author(s)
      田原秀敏
    • Organizer
      RIMS研究集会「超局所解析とその周辺」
    • Place of Presentation
      関西学院大学・大阪梅田キャンパス
    • Year and Date
      2009-10-19
    • Related Report
      2009 Final Research Report
  • [Presentation] Gevrey regularities of solutions of nonlinear singular partial differential equations2009

    • Author(s)
      田原秀敏
    • Organizer
      RIMS研究集会「超局所解析とその周辺」
    • Place of Presentation
      関西学院大学(大阪梅田C)
    • Year and Date
      2009-10-19
    • Related Report
      2009 Annual Research Report
  • [Presentation] Gevrey regularities of solutions of nonlinear singular partial differential equations.2009

    • Author(s)
      田原秀敏
    • Organizer
      2009年度日本数学会秋季総合分科会
    • Place of Presentation
      大阪大
    • Year and Date
      2009-09-24
    • Related Report
      2009 Final Research Report
  • [Presentation] Gevrey regularities of solutions of nonlinear singular partial differential equations2009

    • Author(s)
      田原秀敏
    • Organizer
      2009年度日本数学会秋季総合分科会
    • Place of Presentation
      大阪大学
    • Year and Date
      2009-09-24
    • Related Report
      2009 Annual Research Report
  • [Presentation] Gevrey regularities of solutions of nonlinear singular partial differential equations2009

    • Author(s)
      Hidetoshi Tahara
    • Organizer
      7th ISAAC Congress
    • Place of Presentation
      Imperial College, London.
    • Year and Date
      2009-07-14
    • Related Report
      2009 Final Research Report
  • [Presentation] Gevrey regularities of solutions of nonlinear singular partial differential equations2009

    • Author(s)
      田原秀敏
    • Organizer
      7th ISAAC Congress
    • Place of Presentation
      Imperial College, London
    • Year and Date
      2009-07-14
    • Related Report
      2009 Annual Research Report
  • [Presentation] Coupling of two partial differential equations and its application.2008

    • Author(s)
      田原秀敏
    • Organizer
      研究集会「微分方程式の総合的研究」
    • Place of Presentation
      京都大学理学部6号館
    • Year and Date
      2008-12-21
    • Related Report
      2009 Final Research Report
  • [Presentation] Coupling of two partial differential equations and its application2008

    • Author(s)
      田原秀敏
    • Organizer
      研究集会「微分方程式の総合的研究」
    • Place of Presentation
      京都大学理学部6号館
    • Year and Date
      2008-12-21
    • Related Report
      2008 Annual Research Report
  • [Presentation] On a reduction of Briot-Bouquet type partial differential equations in a resonant case.2008

    • Author(s)
      Hidetoshi Tahara
    • Organizer
      Journees franco-japonaises en l'honneur de Kazuo Okamoto
    • Place of Presentation
      Universite Louis Pasteur, Strasbourg (France).
    • Year and Date
      2008-11-11
    • Related Report
      2009 Final Research Report
  • [Presentation] On a reduction of Briot-Bouauet type partial differential equations in a resonant case2008

    • Author(s)
      H. Tahara
    • Organizer
      Journees franco-japonaises en l'honneurde Kazuo Okaioto
    • Place of Presentation
      Universite Louis Pasteur, Strasbourg (France)
    • Year and Date
      2008-11-11
    • Related Report
      2008 Annual Research Report
  • [Presentation] On a reduction of Briot- Bouquet type partial differential equation in a resonant case2008

    • Author(s)
      田原秀敏
    • Organizer
      orkshop on New approach to analytic equations, transformation theory, singular solutions and Stokes problem.
    • Place of Presentation
      数理解析研究所.
    • Year and Date
      2008-09-19
    • Related Report
      2009 Final Research Report
  • [Presentation] On a reduction of Briot-Bouquet type partial differential eauation in a resonant case2008

    • Author(s)
      H. Tahara
    • Organizer
      Workshop on New approach to analytic equations-transformation theory, singular solutions and stokes problem
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2008-09-19
    • Related Report
      2008 Annual Research Report
  • [Presentation] On the singularities of solutions of nonlinear partial differential equations in the complex domain.2008

    • Author(s)
      田原秀敏
    • Organizer
      nternational Workshop on Global Behaviors of Differential Equations- Singularity and Transformation Theory-
    • Place of Presentation
      広島大学理学研究科.
    • Year and Date
      2008-09-10
    • Related Report
      2009 Final Research Report
  • [Presentation] On the singularities of solutions of nonlinear partial differential eauations in the complex domain2008

    • Author(s)
      H. Tahara
    • Organizer
      International Workshop on Global Behaviors of Differential Equations-Singularity and Transforiation Theory-
    • Place of Presentation
      広島大学理学研究科
    • Year and Date
      2008-09-10
    • Related Report
      2008 Annual Research Report
  • [Presentation] Coupling of two singular partial differential equations and its application.2008

    • Author(s)
      Hidetoshi Tahara
    • Organizer
      Banach Center Conference "Formal and Analytic Solutions of Differential Equations"
    • Place of Presentation
      Mathematical Research and Conference Center in Bedlewo (Poland).
    • Year and Date
      2008-08-12
    • Related Report
      2009 Final Research Report
  • [Presentation] Coupling of two singular partial differential equations and its application2008

    • Author(s)
      H. Tahara
    • Organizer
      Banach Center Conference 「Forial and Analytic Solutions of Differential Eauations」
    • Place of Presentation
      Matheiatical Research and Conference Center in Bedlewo (Poland)
    • Year and Date
      2008-08-12
    • Related Report
      2008 Annual Research Report
  • [Presentation] Coupling of two singular partial differential equations and its application.2008

    • Author(s)
      田原秀敏
    • Organizer
      RIMS研究集会「WKB解析と超局所解析」
    • Place of Presentation
      数理解析研究所.
    • Year and Date
      2008-05-29
    • Related Report
      2009 Final Research Report
  • [Presentation] Coupling of two singular partial differential equations and its application2008

    • Author(s)
      田原秀敏
    • Organizer
      研究集会「WKB解析と超局所解析」
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2008-05-29
    • Related Report
      2008 Annual Research Report
  • [Presentation] 非線型偏微分方程式と対数的特異性.2008

    • Author(s)
      山根英司-田原秀敏
    • Organizer
      RIMS研究集会「WKB解析と超局所解析」
    • Place of Presentation
      数理解析研究所.
    • Year and Date
      2008-05-27
    • Related Report
      2009 Final Research Report
  • [Presentation] 非線型偏微分方程式と対数的特異性2008

    • Author(s)
      山根英司-田原秀敏
    • Organizer
      研究集会「WKB解析と超局所解析」
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2008-05-27
    • Related Report
      2008 Annual Research Report
  • [Presentation] Coupling of two singular partial differential equations and its application.2008

    • Author(s)
      田原秀敏
    • Organizer
      2008年度日本数学会年会
    • Place of Presentation
      近畿大.
    • Year and Date
      2008-03-23
    • Related Report
      2009 Final Research Report
  • [Presentation] 非線型偏微分方程式の解が持つ対数的特異性について.2008

    • Author(s)
      山根英司, 田原秀敏
    • Organizer
      2008年度日本数学会年会
    • Place of Presentation
      近畿大.
    • Year and Date
      2008-03-23
    • Related Report
      2009 Final Research Report
  • [Presentation] 非線型偏微分方程式の解が持つ対数的特異性について2008

    • Author(s)
      山根 英司・田原 秀敏
    • Organizer
      2008年度日本数学会年会
    • Place of Presentation
      近畿大
    • Year and Date
      2008-03-23
    • Related Report
      2007 Annual Research Report
  • [Presentation] Coupling of two singular partial differential equations and its application2008

    • Author(s)
      田原 秀敏
    • Organizer
      2008年度日本数学会年会
    • Place of Presentation
      近畿大
    • Year and Date
      2008-03-23
    • Related Report
      2007 Annual Research Report
  • [Presentation] On local solutions to a radial p-Laplace equation2008

    • Author(s)
      内山 康一
    • Organizer
      偏微分方程式姫路研究集会
    • Place of Presentation
      姫路市イーグレ姫路
    • Year and Date
      2008-02-21
    • Related Report
      2007 Annual Research Report
  • [Presentation] Coupling of two singular partial differential equations and its application- a dream of R. Gerard-2008

    • Author(s)
      Hidetoshi Tahara
    • Organizer
      Conference on holomorphic PDEs, small divisors and summability
    • Place of Presentation
      CIRM at Luminy (Marseille, France).
    • Year and Date
      2008-01-31
    • Related Report
      2009 Final Research Report
  • [Presentation] Coupling of two singular partial differential equations and its application-adream of R.Gerard2008

    • Author(s)
      H. Tahara
    • Organizer
      Conference on holomorphic PDEs, small divisors and summability
    • Place of Presentation
      CIRM at Luminy(Marselle,France)
    • Year and Date
      2008-01-31
    • Related Report
      2007 Annual Research Report
  • [Book] Algebraic Analysis of Differential Equations(edited by T. Aoki, H. Majima, Y. Takei and N. Tose)2008

    • Author(s)
      H. Tahara
    • Publisher
      Springer
    • Related Report
      2008 Annual Research Report

URL: 

Published: 2007-04-01   Modified: 2016-04-21  

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