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Enclosures of solution curve of differential equations by computer-assisted proof

Research Project

Project/Area Number 21540134
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionKyushu University

Principal Investigator

WATANABE Yoshitaka  九州大学, 情報基盤研究開発センター, 准教授 (90243972)

Co-Investigator(Renkei-kenkyūsha) NAGATOU Kaori  九州大学, マス・フォア・インダストリ研究所, 准教授 (40326426)
Project Period (FY) 2009 – 2011
Project Status Completed (Fiscal Year 2011)
Budget Amount *help
¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2010: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywords計算機援用証明 / 精度保証付き数値計算 / 非線形偏微分方程式 / 非線形微分方程式 / 解曲線の追跡 / 線形化作用素の可逆性 / 無限次元Newton法 / 陰関数定理 / 固有値問題 / 解の存在検証 / 非自己共役固有値問題 / 固有値の除外 / 線形化作用素のノルム評価
Research Abstract

We developed a computer-assisted theory to verify the existence of the solutions for nonlinear elliptic boundary value problems with local uniqueness. We also proposed verified computations of eigenvalue exclosures for the linearization problems. Finally combining with developed method and the implicit function theorem, we introduced a procedure to enclose solution curves of nonlinear partial differential equations by computer-assisted proof and confirmed its effectiveness by some verified results.

Report

(4 results)
  • 2011 Annual Research Report   Final Research Report ( PDF )
  • 2010 Annual Research Report
  • 2009 Annual Research Report
  • Research Products

    (59 results)

All 2012 2011 2010 2009 Other

All Journal Article (22 results) (of which Peer Reviewed: 7 results) Presentation (30 results) Book (3 results) Remarks (4 results)

  • [Journal Article] Tests of Inequivalence Among Absolutely Nonsingular Tensors Through Geometric Invariants2012

    • Author(s)
      Toshio Sakata, Kazumitsu Maehara, Takeshi Sasaki, Toshio Sumi, Mitsuhiro Miyazaki, Yoshitaka Watanabe, and Makoto Tagami
    • Journal Title

      Universal Journal of Mathematics and Mathematical Sciences

      Volume: Vol.1, No.1 Pages: 1-28

    • Related Report
      2011 Final Research Report
  • [Journal Article] Tests of inequivalence among absolutely nonsingular tensors through geometric invariants2012

    • Author(s)
      T.Sakata, et al
    • Journal Title

      Universal Journal of Mathematics and Mathematical Sciences

      Volume: 1-1 Pages: 1-28

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] 精度保証付き数値計算(4)-区間演算と有限次元問題の精度保証2011

    • Author(s)
      渡部善隆
    • Journal Title

      応用数理

      Volume: Vol.21, No.4 Pages: 57-63

    • Related Report
      2011 Final Research Report
  • [Journal Article] A Theorem for Numerical Verification on Local Uniqueness of Solutions to Fixed-Point Equations2011

    • Author(s)
      Nobito Yamamoto, Mitsuhiro T. Nakao, and Yoshitaka Watanabe
    • Journal Title

      Numerical Functional Analysis and Optimization

      Volume: Vol.32, No.11 Pages: 1190-1204

    • Related Report
      2011 Final Research Report
  • [Journal Article] 精度保証付き数値計算(3)-中尾理論入門2011

    • Author(s)
      渡部善隆
    • Journal Title

      応用数理

      Volume: Vol.21, No.3 Pages: 47-53

    • NAID

      110009426225

    • Related Report
      2011 Final Research Report
  • [Journal Article] 精度保証付き数値計算(2)-線形逆作用素のノルム評価2011

    • Author(s)
      渡部善隆
    • Journal Title

      応用数理

      Volume: Vol.21, No.2 Pages: 54-60

    • NAID

      110008673554

    • Related Report
      2011 Final Research Report
  • [Journal Article] 精度保証付き数値計算(1)-不動点定式化2011

    • Author(s)
      渡部善隆
    • Journal Title

      応用数理

      Volume: Vol.21, No.1 Pages: 40-45

    • NAID

      110008593884

    • Related Report
      2011 Final Research Report
  • [Journal Article] Numerical Verification Methods for Solutions of Semilinear Elliptic Boundary Value Problems2011

    • Author(s)
      Mitsuhiro T. Nakao, and Yoshitaka Watanabe
    • Journal Title

      Nonlinear Theory and Its Applications, IEICE

      Volume: Vol.2, No.1 Pages: 2-31

    • NAID

      130000903439

    • Related Report
      2011 Final Research Report
  • [Journal Article] A Computer-assisted Stability Proof for the Orr-Sommerfeld Problem with Poiseuille Flow2011

    • Author(s)
      Yoshitaka Watanabe, Kaori Nagatou, Michael Plum, and Mitsuhiro T. Nakao
    • Journal Title

      Nonlinear Theory and Its Applications, IEICE

      Volume: Vol.2, No.1 Pages: 123-127

    • NAID

      130000903447

    • Related Report
      2011 Final Research Report
  • [Journal Article] A theorem for numerical verification on local uniqueness of solutions to fixed-point equations2011

    • Author(s)
      N.Yamamoto, et al
    • Journal Title

      Numerical Functional Analysis and Optimization

      Volume: 32-11 Issue: 11 Pages: 1190-1204

    • DOI

      10.1080/01630563.2011.594348

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] 無限次元固有値問題に対する固有値の非存在証明,研究集会・数値解析と数値計算アルゴリズムの最近の展開2010

    • Author(s)
      渡部善隆, 長藤かおり, Michael Plum, 中尾充宏
    • Journal Title

      京都大学数理解析研究所講究録

      Volume: Vol.1719 Pages: 118-129

    • Related Report
      2011 Final Research Report
  • [Journal Article] Computer Assisted Proofs of Bifurcating Solutions for Nonlinear Heat Convection Problems2010

    • Author(s)
      Mitsuhiro T. Nakao, Yoshitaka Watanabe, Nobito Yamamoto, Takaaki Nishida and Myoung-Nyoun Kim
    • Journal Title

      Journal of Scientific Computing

      Volume: Vol.43, No.3 Pages: 388-401

    • Related Report
      2011 Final Research Report 2010 Annual Research Report
  • [Journal Article] Numerical Verification Methods for Solutions of Semilinear Elliptic Boundary Value Problems2010

    • Author(s)
      M.T.Nakao, Y.Watanabe
    • Journal Title

      Nonlinear Theory and Its Applications, IEICE

      Volume: Vol.2, No.1 Pages: 2-31

    • NAID

      130000903439

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A Computer-assisted Stability Proof for the Orr-Sommerfeld Problem with Poiseuille Flow2010

    • Author(s)
      Y.Watanabe, et al.
    • Journal Title

      Nonlinear Theory and Its Applications, IEICE

      Volume: Vol.2, No.1 Pages: 123-127

    • NAID

      130000903447

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Numerical Verification Method of Solutions for Elliptic Equations and Its Application to the Rayleigh-Benard Problem2009

    • Author(s)
      Yoshitaka Watanabe and Mitsuhiro T. Nakao
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics

      Volume: Vol.26, No.2-3 Pages: 443-463

    • NAID

      10028169295

    • Related Report
      2011 Final Research Report
  • [Journal Article] A Numerical Verification Method for Two-Coupled Elliptic Partial Differential Equations2009

    • Author(s)
      Yoshitaka Watanabe
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics

      Volume: Vol.26, No.2-3 Pages: 233-247

    • NAID

      10028168961

    • Related Report
      2011 Final Research Report
  • [Journal Article] Numerical Verification Method of Solutions for Elliptic Equations and Its Application to the Rayleigh-Benard Problem2009

    • Author(s)
      Y.Watanabe, et al.
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics Vol.26, No.2-3

      Pages: 443-463

    • NAID

      10028169295

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A Numerical Verification Method for Two-Coupled Elliptic Partial Differential Equations2009

    • Author(s)
      Y.Watanabe
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics Vol.26, No.2-3

      Pages: 233-247

    • NAID

      10028168961

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Computer Assisted Proofs of Bifurcating Solutions for Nonlinear Heat Convection Problems2009

    • Author(s)
      M.T.Nakao, et al.
    • Journal Title

      Journal of Scientific Computing online

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A Numerical Verification Method for a System of FitzHugh-Nagumo Type

    • Author(s)
      Chuting Cai, Kaori Nagatou, and Yoshitaka Watanabe
    • Journal Title

      A Numerical Verification Method for a System of FitzHugh-Nagumo Type

    • Related Report
      2011 Final Research Report
  • [Journal Article] 微分方程式の精度保証付き数値計算-逐次反復に基づく計算機援用証明

    • Author(s)
      渡部善隆
    • Journal Title

      京都大学数理解析研究所別冊,掲載決定

    • Related Report
      2011 Final Research Report
  • [Journal Article] A Posteriori Estimates of Inverse Operators for Boundary Value Problems in Linear Elliptic Partial Differential Equations

    • Author(s)
      Yoshitaka Watanabe, Takehiko Kinoshita, and Mitsuhiro T. Nakao
    • Journal Title

      Mathematics of Computation

      Volume: (accepted)

    • Related Report
      2011 Final Research Report
  • [Presentation] 無限次元作用素に対する固有値の除外理論と精度保証付き数値計算,九州大学数値解析学セミナー2012

    • Author(s)
      渡部善隆,長藤かおり, Michael Plum,中尾充宏
    • Organizer
      九州大学大学院数理学研究院
    • Year and Date
      2012-04-21
    • Related Report
      2011 Final Research Report
  • [Presentation] exflibにおける多倍長区間演算Fortranインターフェースとその応用2012

    • Author(s)
      渡部善隆,藤原宏志
    • Organizer
      日本応用数理学会2012年研究部会連合発表会
    • Place of Presentation
      九州大学
    • Related Report
      2011 Final Research Report
  • [Presentation] Computer-assisted stability and instability proofs for the Orr-Sommerfeld problem with Poiseuille flow, Workshop on" Avoided? Crossing of Eigenvalue Curves2012

    • Author(s)
      Yoshitaka Watanabe
    • Organizer
      Non-commutative Harmonic Oscillator, Special Functions and Number Theory-, Kyushu University
    • Related Report
      2011 Final Research Report
  • [Presentation] 無限次元作用素に対する固有値の計算機援用非存在証明2011

    • Author(s)
      渡部善隆,長藤かおり, Michael Plum,中尾充宏
    • Organizer
      応用数学合同研究集会報告集
    • Place of Presentation
      龍谷大学
    • Year and Date
      2011-12-16
    • Related Report
      2011 Final Research Report
  • [Presentation] Computer-assisted proof for functional equations based on infinite dimensional sequential iteration2011

    • Author(s)
      Y.Watanabe
    • Organizer
      Japanese-German Workshop on Computer-Assisted Proofs and Verification Methods
    • Place of Presentation
      Karlsruhe Institute of Technology, Germany
    • Year and Date
      2011-09-22
    • Related Report
      2011 Annual Research Report
  • [Presentation] rr-Sommerfeld方程式に対する局所一意性付き計算機援用証明2011

    • Author(s)
      渡部善隆,木下武彦,中尾充宏
    • Organizer
      日本数学会2011年度秋季総合分科会応用数学分科会講演アブストラクト
    • Place of Presentation
      信州大学
    • Related Report
      2011 Final Research Report
  • [Presentation] 線形楕円型偏微分作用素の逆作用素に対する事後誤差評価について2011

    • Author(s)
      木下武彦,渡部善隆,中尾充宏
    • Organizer
      日本数学会2011年度秋季総合分科会応用数学分科会講演アブストラクト
    • Place of Presentation
      信州大学
    • Related Report
      2011 Final Research Report
  • [Presentation] Computer-assisted proof for functional equations based on infinite dimensional sequential iteration2011

    • Author(s)
      Yoshitaka Watanabe
    • Organizer
      Japanese-German Workshop on Computer-Assisted Proofs and Verification Methods, Karlsruhe Institute of Technology, Germany
    • Related Report
      2011 Final Research Report
  • [Presentation] 線形楕円型偏微分作用素の逆作用素に対する高精度な事後評価について2011

    • Author(s)
      木下武彦,渡部善隆,中尾充宏
    • Organizer
      日本応用数理学会2011年度年会講演予稿集
    • Place of Presentation
      同志社大学
    • Related Report
      2011 Final Research Report
  • [Presentation] 逐次反復に基づく関数方程式の計算機援用証明2011

    • Author(s)
      渡部善隆
    • Organizer
      日本応用数理学会2011年度年会講演予稿集
    • Place of Presentation
      同志社大学
    • Related Report
      2011 Final Research Report
  • [Presentation] 微分方程式の精度保証付き数値計算,研究集会2011

    • Author(s)
      渡部善隆
    • Organizer
      可積分系数理の進化,京都大学数理解析研究所
    • Related Report
      2011 Final Research Report
  • [Presentation] An enclosure method for solutions of FitzHugh-Nagumo equation2010

    • Author(s)
      Shuting Cai,長藤かおり,渡部善隆
    • Organizer
      応用数学合同研究集会報告集
    • Place of Presentation
      龍谷大学
    • Year and Date
      2010-12-16
    • Related Report
      2011 Final Research Report
  • [Presentation] A self-validating norm computation of inverse for infinite dimensional linear operators and its applications2010

    • Author(s)
      Y.Watanabe
    • Organizer
      SCAN 2010
    • Place of Presentation
      ENS de Lyon, France
    • Year and Date
      2010-09-30
    • Related Report
      2010 Annual Research Report
  • [Presentation] 非自己共役作用素に対する固有値の除外法2010

    • Author(s)
      渡部善隆,長藤かおり,中尾充宏
    • Organizer
      日本応用数理学会2010年度年会
    • Place of Presentation
      明治大学
    • Year and Date
      2010-09-08
    • Related Report
      2011 Final Research Report
  • [Presentation] MATLABで偏微分方程式の精度保証プログラムを作ってみました2010

    • Author(s)
      渡部善隆
    • Organizer
      九州大学数値解析学セミナー,九州大学大学院数理学研究院
    • Year and Date
      2010-07-27
    • Related Report
      2011 Final Research Report
  • [Presentation] Computer assisted proofs for partial differential equations2010

    • Author(s)
      Kaori Nagatou and Yoshitaka Watanabe
    • Organizer
      Seminar at University of Kent, UK
    • Year and Date
      2010-03-31
    • Related Report
      2011 Final Research Report
  • [Presentation] Computer-assisted existence proofs with local uniqueness for the Orr-Sommerfeld problem2010

    • Author(s)
      Y, Watanabe
    • Organizer
      International Workshop on Numerical Verification and its Applications 2010
    • Place of Presentation
      Hotel Lido Azzurro, Tokyo, Japan
    • Year and Date
      2010-03-11
    • Related Report
      2009 Annual Research Report
  • [Presentation] Legendre多項式を用いた高精度品質保証-およびその課題2010

    • Author(s)
      渡部善隆,木下武彦
    • Organizer
      日本応用数理学会2010年研究部会連合発表会
    • Place of Presentation
      筑波大学
    • Year and Date
      2010-03-08
    • Related Report
      2011 Final Research Report
  • [Presentation] 逐次反復に基づく関数方程式の精度保証付き数値計算2010

    • Author(s)
      渡部善隆
    • Organizer
      研究集会:「科学計算の信頼性とその周辺に関するワークショップ」,西海国立公園九十九島ビジターセンター
    • Place of Presentation
      佐世保市
    • Related Report
      2011 Final Research Report
  • [Presentation] 無限次元固有値問題に対する固有値の除外法2010

    • Author(s)
      渡部善隆
    • Organizer
      研究集会:「数値解析と計算の信頼性評価」,ハウステンボス,ユトレヒト
    • Place of Presentation
      佐世保市
    • Related Report
      2011 Final Research Report
  • [Presentation] A self-validating norm computation of inverse for infinite dimensional linear operators and its applications2010

    • Author(s)
      Yoshitaka Watanabe
    • Organizer
      14th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics(SCAN 2010)
    • Place of Presentation
      ENS de Lyon, France
    • Related Report
      2011 Final Research Report
  • [Presentation] Computer-assisted existence proofs with local uniqueness for the Orr-Sommerfeld problem2010

    • Author(s)
      Yoshitaka Watanabe
    • Organizer
      International Workshop on Numerical Verification and its Applications 2010(INVA2010)
    • Place of Presentation
      Hotel Lido Azzurro, Hachijyo Island, Tokyo, Japan
    • Related Report
      2011 Final Research Report
  • [Presentation] ある非自己共役作用素の複素固有値の除外に対する数値的検証法II2009

    • Author(s)
      渡部善隆,中尾充宏, Michael Plum,長藤かおり
    • Organizer
      応用数学合同研究集会報告集
    • Place of Presentation
      龍谷大学
    • Year and Date
      2009-12-17
    • Related Report
      2011 Final Research Report
  • [Presentation] Orr-Sommerfeld問題の解に対する局所一意性付き数値的検証2009

    • Author(s)
      渡部善隆
    • Organizer
      研究集会:「数値解析の現状と展望」-九州大学数値解析セミナーとその協力者からの発信
    • Place of Presentation
      公立はこだて未来大学
    • Year and Date
      2009-11-22
    • Related Report
      2011 Final Research Report
  • [Presentation] Orr-Sommerfeld方程式の複素固有値の除外に対する計算機援用証明2009

    • Author(s)
      渡部善隆,中尾充宏, Michael Plum,長藤かおり
    • Organizer
      日本応用数理学会2009年度年会
    • Place of Presentation
      大阪大学
    • Year and Date
      2009-09-30
    • Related Report
      2011 Final Research Report
  • [Presentation] 楕円型作用素の逆作用素の作用素ノルムの評価の改良について2009

    • Author(s)
      木下武彦,渡部善隆,中尾充宏
    • Organizer
      日本数学会2009年度秋季総合分科会
    • Place of Presentation
      大阪大学
    • Year and Date
      2009-09-26
    • Related Report
      2011 Final Research Report
  • [Presentation] Spectral problem for 3-D photonic crystals, Proceedings of the international Conference on Numerical Analysis and Applied Mathematics 2009(ICNAAM-2009)2009

    • Author(s)
      Kaori Nagatou, Malcolm Brown, Myoungnyoun Kim, Yoshitaka Watanabe and Ian Wood
    • Organizer
      September 18-22, Rethymno, Crete, Greece. AIP Conference Proceedings
    • Year and Date
      2009-09-21
    • Related Report
      2011 Final Research Report
  • [Presentation] 熱対流問題の精度保証-計算の品質評価2009

    • Author(s)
      渡部善隆
    • Organizer
      九州大学産業数理研究センター第7回ワークショップ
    • Year and Date
      2009-09-03
    • Related Report
      2011 Final Research Report
  • [Presentation] 計算機援用証明による並行Poiseuille流れの安定性・不安性解析,九州大学数値解析学セミナー2009

    • Author(s)
      渡部善隆
    • Organizer
      九州大学大学院数理学研究院
    • Year and Date
      2009-04-28
    • Related Report
      2011 Final Research Report
  • [Presentation] An eigenvalue excluding method for the Orr-Sommerfeld problem2009

    • Author(s)
      Yoshitaka Watanabe
    • Organizer
      International Conference on Engineering and Computational Mathematics(ECM2009)
    • Place of Presentation
      The Hong Kong Polytechnic University, Hong Kong, China
    • Related Report
      2011 Final Research Report
  • [Book] シミュレーション辞典,微分方程式と数値計算(日本シミュレーション学会編)2012

    • Author(s)
      渡部善隆
    • Total Pages
      454
    • Publisher
      コロナ社
    • Related Report
      2011 Final Research Report
  • [Book] 実例で学ぶ精度保証付き数値計算~理論と実装~,臨時別冊・数理科学2011

    • Author(s)
      中尾充宏, 渡部善隆
    • Total Pages
      216
    • Publisher
      サイエンス社
    • Related Report
      2011 Final Research Report
  • [Book] 実例で学ぶ精度保証付き数値計算~理論と実装~2011

    • Author(s)
      中尾充宏, 渡部善隆
    • Total Pages
      216
    • Publisher
      サイエンス社
    • Related Report
      2011 Annual Research Report
  • [Remarks]

    • URL

      http://www.cc.kyushu-u.ac.jp/RD/watanabe/

    • Related Report
      2011 Final Research Report
  • [Remarks]

    • URL

      http://www.cc.kyushu-u.ac.jp/RD/watanabe/

    • Related Report
      2011 Annual Research Report
  • [Remarks]

    • URL

      http://www.cc.kyushu-u.ac.jp/RD/watanabe/

    • Related Report
      2010 Annual Research Report
  • [Remarks]

    • URL

      http://www.cc.kyushu-u.ac.jp/RD/watanabe/

    • Related Report
      2009 Annual Research Report

URL: 

Published: 2009-04-01   Modified: 2016-04-21  

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