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Analysis of a global-in-time solution for reaction-diffusion system using verified numerical computation

Research Project

Project/Area Number 18K13462
Research Category

Grant-in-Aid for Early-Career Scientists

Allocation TypeMulti-year Fund
Review Section Basic Section 12040:Applied mathematics and statistics-related
Research InstitutionChuo University (2021-2022)
Waseda University (2018-2020)

Principal Investigator

Mizuguchi Makoto  中央大学, 理工学部, 助教 (90801241)

Project Period (FY) 2018-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2021: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2020: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywords計算機援用証明 / 解の精度保証付き数値計算法 / 放物型偏微分方程式 / 偏微分方程式 / 誤差定数 / 爆発解 / 藤田型方程式 / 半離散近似解 / 爆発時間 / 反応拡散系 / 精度保証付き数値計算 / ソボレフ定数の評価 / 大域解
Outline of Final Research Achievements

In this research, we mainly aim to improve numerical verification method for solutions of parabolic partial differential equations including reaction-diffusion systems, and to establish a method for verifying the existence of special solutions such as global-in-time solutions and blow-up solutions. First, for improving the verification method, we obtained the best value of the error constant of the semi-discrete approximation of the parabolic equation.The improved method were finally able to clarify the range of the explosion time of the blow-up solution, which could not be clarified by previous mathematical methods, of a parabolic equation.

Academic Significance and Societal Importance of the Research Achievements

一般的な非線形偏微分方程式の解を解析的に解くことは難しい. しかし解の精度保証付き数値計算法を用いれば偏微分方程式の解の厳密な存在範囲を明確に示すことができる. そのため方程式の解の存在だけでなく, 数値シミュレーション結果の妥当性を保証するといった工学面に対する応用も可能である. この解の精度保証付き数値計算法の改良かつその手法の適応範囲の拡大が本研究の主な目的である. 本研究の最も重要な成果はその計算手法を用いてある放物型方程式の解の爆発時間の範囲を得たことである. それは既存の数学的手法では得られなかった現象(爆発現象)の一端が解明できたことを意味する.

Report

(6 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (13 results)

All 2022 2021 2020 2019 2018

All Journal Article (3 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 3 results,  Open Access: 2 results) Presentation (10 results) (of which Int'l Joint Research: 2 results,  Invited: 2 results)

  • [Journal Article] Rigorous numerical inclusion of the blow-up time for the Fujita-type equation2022

    • Author(s)
      Mizuguchi Makoto, Sekine Kouta, Hashimoto Kouji, Nakao Mitsuhiro T., Oishi Shin’ichi
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics

      Volume: 40 Issue: 1 Pages: 665-689

    • DOI

      10.1007/s13160-022-00545-8

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Error Constants for the Semi-Discrete Galerkin Approximation of the Linear Heat Equation2021

    • Author(s)
      M. Mizuguchi, M. T. Nakao, K. Sekine, S. Oishi
    • Journal Title

      Journal of Scientific Computing

      Volume: 89 Issue: 2

    • DOI

      10.1007/s10915-021-01636-3

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Estimation of Sobolev embedding constant on a bounded convex domain2018

    • Author(s)
      Makoto Mizuguchi, Kazuaki Tanaka, Kouta Sekine, and Shin'ichi Oishi
    • Journal Title

      book of abstracts scan 2018

      Volume: - Pages: 164-165

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Presentation] 放物型方程式の全離散近似に対する誤差評価について2022

    • Author(s)
      水口 信 , 中尾 充宏 , 橋本 弘治 , 関根 晃太 , 大石 進一
    • Organizer
      日本応用数理学会2022年度年会
    • Related Report
      2022 Annual Research Report
  • [Presentation] 3つのHilbert空間の組における最良な射影誤差定数について2022

    • Author(s)
      高橋 宗久 , 関根 晃太 , 水口 信
    • Organizer
      日本応用数理学会2022年度年会
    • Related Report
      2022 Annual Research Report
  • [Presentation] Numerical verification method for a blow-up solution of Fujita-type equation2022

    • Author(s)
      Makoto Mizuguchi
    • Organizer
      International Workshop on Reliable Computing and Computer-Assisted Proofs (ReCAP 2022)
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research
  • [Presentation] 楕円型方程式と放物型方程式に対する半離散ガレルキン近似の誤差定数について2021

    • Author(s)
      水口 信, 中尾 充宏, 関根 晃太, 大石 進一
    • Organizer
      日本応用数理学会2021年度年会
    • Related Report
      2021 Research-status Report
  • [Presentation] 藤田型方程式の解の爆発時間に対する数値的検証法2021

    • Author(s)
      水口 信
    • Organizer
      有限時間特異性の包括的記述に向けた 数学解析・計算機援用解析の展開「有限時間特異性」勉強会 第3回
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] 放物型方程式の半離散近似に対する誤差定数値の評価について2021

    • Author(s)
      水口 信
    • Organizer
      第5回 精度保証付き数値計算の実問題への応用研究集会 (NVR 2021) ・ JST/CREST「モデリングのための精度保証付き数値計算論の展開」成果報告会
    • Related Report
      2021 Research-status Report
  • [Presentation] 藤田型方程式の解の爆発時間に対する計算機を用いた数値的包含方法について2020

    • Author(s)
      水口 信
    • Organizer
      2020年度 応用数学合同研究集会
    • Related Report
      2020 Research-status Report
  • [Presentation] 線形熱方程式の解と半離散近似解との誤差評価の改善2019

    • Author(s)
      水口 信
    • Organizer
      日本応用数理学会2019年度年会
    • Related Report
      2019 Research-status Report
  • [Presentation] Estimation of Sobolev embedding constant on a bounded convex domain2018

    • Author(s)
      Makoto Mizuguchi, Kazuaki Tanaka, Kouta Sekine, and Shin'ichi Oishi
    • Organizer
      The 18th International Symposium on Scientific Computing, Computer Arithmetic, and Verified Numerical Computations(SCAN 2018)
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] 半線形熱方程式の解の精度保証付き数値計算法について2018

    • Author(s)
      水口 信, 関根 晃太, 中尾充宏, 大石進一
    • Organizer
      第2回 精度保証付き数値計算の実問題への応用研究集会
    • Related Report
      2018 Research-status Report
    • Invited

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Published: 2018-04-23   Modified: 2024-01-30  

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