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Application of the method of fundamental solutions for abnormal diffusion equations in the layered medium

Research Project

Project/Area Number 18K03438
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 12040:Applied mathematics and statistics-related
Research InstitutionOkayama University of Science

Principal Investigator

Ohe Takashi  岡山理科大学, 理学部, 教授 (90258210)

Co-Investigator(Kenkyū-buntansha) 町田 学  近畿大学, 工学部, 准教授 (40396916)
Project Period (FY) 2018-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords異常拡散方程式 / 基本解近似解法 / 代用電荷法 / Convolution Quadrature / 後退差分公式 / 陰的Runge-Kutta法 / 時間依存基本解 / 拡散方程式 / CQM / 異常拡散現象 / 数値解法 / 基本解 / 基本解解法 / 数値計算法 / 層状領域 / 逆問題
Outline of Final Research Achievements

In this project, we develop an application of the method of fundamental solutions (charge simulation method) for the initial-boundary value problem for the abnormal diffusion equation. At first, we apply a naive implementation for the problem, but we find some numerical instability under the small time-step condition. To avoid this numerical instability, we apply the Convolution Quadrature Method (CQM) to discretize the integration in time. Numerical experiments show that our method is stable even if the time-step is small, and we can obtain a high-precision numerical solution if we apply the implicit Runge-Kutta method in CQM.

Academic Significance and Societal Importance of the Research Achievements

非定常問題に対する代用電荷法(基本解近似解法)の適用に関する従来の研究は、時間離散化について差分法等を用いることで、時間依存の基本解を利用しないものがほとんどであった。これに対し、本研究では時間依存の基本解を用いた直接的な離散化手法について検討した。また、その際に生じる数値的不安定性を除去する手法についても併せて開発した。異常拡散方程式の数値解法は近赤外線を用いたCT法において必要とされており、時に短時間の挙動の解析が需要となっている。この問題に対し、一つの解決手法を与えたことは大きな意義があるものと考えられる。

Report

(7 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (21 results)

All 2024 2023 2022 2021 2019 2018 Other

All Journal Article (2 results) (of which Peer Reviewed: 2 results) Presentation (16 results) (of which Int'l Joint Research: 7 results,  Invited: 10 results) Remarks (3 results)

  • [Journal Article] Algebraic Reconstruction of a Dipolar Wave Source from Observations on Several Points2023

    • Author(s)
      Takashi Ohe, Misa Yokoyama
    • Journal Title

      Practical Inverse Problems and Their Prospects, Mathematics for Industry

      Volume: 37 Pages: 247-261

    • DOI

      10.1007/978-981-99-2408-0_15

    • ISBN
      9789819924073, 9789819924080
    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Real-time reconstruction of moving point/dipole wave sources from boundary measurements2019

    • Author(s)
      Takashi Ohe
    • Journal Title

      Inverse Problems in Science and Engineering

      Volume: - Issue: 8 Pages: 1057-1102

    • DOI

      10.1080/17415977.2019.1696787

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Presentation] 陰的Runge-Kutta法を用いたCQMの拡散方程式に対する基本解解法への適用2024

    • Author(s)
      大江貴司
    • Organizer
      第29回計算工学講演会
    • Related Report
      2023 Annual Research Report
  • [Presentation] 拡散および異常拡散方程式に対する基本解解法におけるCQMの適用2023

    • Author(s)
      大江貴司
    • Organizer
      第28回計算工学講演会
    • Related Report
      2023 Annual Research Report
  • [Presentation] 少数の点における波動場の情報に基づく源泉項の再構成2023

    • Author(s)
      大江貴司
    • Organizer
      愛媛大学解析セミナー
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] 拡散および異常拡散方程式に対する基本解近似解法におけるConvolution Quadrature Methodの適用2023

    • Author(s)
      大江貴司
    • Organizer
      金沢大学解析セミナー
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Direct reconstruction methods for moving sources in the wave equation2023

    • Author(s)
      Takashi Ohe
    • Organizer
      ICIAM2023
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 少数の点における観測に基づく双極子波源の推定2023

    • Author(s)
      大江貴司
    • Organizer
      第12回福島応用数学研究集会
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Algebraic reconstruction of a dipolar wave source from observations on several points2022

    • Author(s)
      Takashi Ohe, Misa Yokoyama
    • Organizer
      IMI workshop "Practical inverse problems and their prospects”,
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 非整数階拡散方程式に対する代用電荷法の適用について2021

    • Author(s)
      大江 貴司
    • Organizer
      日本応用数理学会2021年度年会
    • Related Report
      2021 Research-status Report
  • [Presentation] 非整数階拡散方程式に対する時間依存基本解を用いた代用電荷法の適用2021

    • Author(s)
      大江 貴司
    • Organizer
      日本応用数理学会環瀬戸内応用数理研究部会第25 回シンポジウム
    • Related Report
      2021 Research-status Report
  • [Presentation] Reconstruction of a dipole wave source from point observations2019

    • Author(s)
      Takashi Ohe, Misa Yokoyama
    • Organizer
      The 10th Applied Inverse Problems Conference
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Direct reconstruction of time dependent moving point/dipole wave sources from boundary measurements2019

    • Author(s)
      Takashi Ohe
    • Organizer
      Summer School on Applied Inverse Problems and Related Topics
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 数点における観測値に基づく双極子波源の代数的推定法2019

    • Author(s)
      大江貴司, 横山美沙
    • Organizer
      日本応用数理学会 2019年度年会
    • Related Report
      2019 Research-status Report
  • [Presentation] 数点における観測値に基づく双極子波源の代数的推定法とその数値実験2019

    • Author(s)
      大江貴司,横山美沙
    • Organizer
      第23回 環瀬戸内応用数理研究部会シンポジウム
    • Related Report
      2019 Research-status Report
  • [Presentation] Real-time reconstruction of moving directional wave sources from boundary measurements2019

    • Author(s)
      Takashi Ohe
    • Organizer
      A3 Workshop in Applied Inverse Problems
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Comparison of two types of reconstruction formula in the enclosure method2018

    • Author(s)
      Takashi Ohe, Masaru Ikehata
    • Organizer
      The 9th International Conference ”Inverse Problems: Modeling and Simulation”
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Numerical comparison of various reconstruction formulae based on the enclosure method2018

    • Author(s)
      Takashi Ohe
    • Organizer
      Inverse Problems for Partial Differential Equations at TUS
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Remarks] 日本応用数理学会環瀬戸内応用数理研究部会 第25回シンポジウム

    • URL

      https://sites.google.com/view/kanseto-jsiam-2021/

    • Related Report
      2021 Research-status Report
  • [Remarks] (1) RIMS共同研究(公開型)「偏微分方程式における逆問題とその応用のさらなる展開」

    • URL

      https://www.xmath.ous.ac.jp/~ohe/RIMS_Jan2021/index_jp.html

    • Related Report
      2020 Research-status Report
  • [Remarks] (2) 日本応用数理学会環瀬戸内応用数理研究部会 第24回シンポジウム

    • URL

      https://sites.google.com/view/kanseto-jsiam-2020/

    • Related Report
      2020 Research-status Report

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Published: 2018-04-23   Modified: 2025-01-30  

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