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Mathematical analysis and applications of crystal growth and anomalous diffusion

Research Project

Project/Area Number 16F16319
Research Category

Grant-in-Aid for JSPS Fellows

Allocation TypeSingle-year Grants
Section外国
Research Field Mathematical analysis
Research InstitutionThe University of Tokyo

Principal Investigator

山本 昌宏 (2016-2017)  東京大学, 大学院数理科学研究科, 教授 (50182647)

Co-Investigator(Kenkyū-buntansha) 劉 逸侃  東京大学, 大学院数理科学研究科, 特任助教 (70773084)
LIU YIKAN  東京大学, 数理(科)学研究科(研究院), 外国人特別研究員
Project Period (FY) 2016-11-07 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 2018: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2017: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2016: ¥600,000 (Direct Cost: ¥600,000)
Keywords非整数階拡散方程式 / 双曲型方程式 / 逆問題 / 一意性・安定性 / 数値解法 / 一意接続性 / Carleman評価 / Carleman 評価 / 強最大値原理
Outline of Annual Research Achievements

今年度は、非整数階偏微分方程式の順問題および逆問題に対する数学解析を継続し、結晶成長と異常拡散の交差点について研究した。具体的に、異常拡散を表す非整数階偏微分方程式の初期値・境界値問題に関して、次の研究を行った。
1. 順問題:時間微分階数α∈(0,1)かつ解がスカラー値の場合は多くな先行研究があったが、下記の拡張に対する考察を展開した。(a) α∈(1,2)区間に属す場合に対して、坂本-山本による結果を改善し、解の適切性および解析性を証明した。(b) 非整数階反応拡散系を考えるため、ベクトル値の解が満たすカップリング・システムを考え、解の適切性・解析性・漸近挙動を調べた。
2. α∈(0,1)のときの逆問題:(a) 源泉項F(x,t)=f(x)R(x,t)とし、空間成分f(x)を最終時刻の観測データから決定する問題については、解析Fredholm理論によって一意性を示した。(b) 上記と同じ問題で、部分内部領域の観測データによる再構成については、離散化された最適化問題の解の存在性・安定性・収束性を示した。(c) 源泉項および係数を決定する問題に関しては、近年の成果をまとめてレビュー論文を出版した。
3. α∈(1,2]のときの逆問題:順問題の結果を踏まえ、以下の逆問題を考察した。(a) 源泉項が平行移動する場合、ソースの形状を境界全体の近傍の観測で決定する問題について、一意性を証明した。(b) 源泉項がある軌道に沿って移動する場合、有限個の点における観測で軌道を決定する問題について、条件付き安定性を示した。(c) α∈(1,2)の場合、部分境界における一回の観測によって複数の係数を決定する問題については、特殊な境界条件を課すことによって一意性を証明した。

Research Progress Status

平成30年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

平成30年度が最終年度であるため、記入しない。

Report

(3 results)
  • 2018 Annual Research Report
  • 2017 Annual Research Report
  • 2016 Annual Research Report
  • Research Products

    (23 results)

All 2019 2018 2017 2016 Other

All Int'l Joint Research (4 results) Journal Article (10 results) (of which Int'l Joint Research: 9 results,  Peer Reviewed: 10 results,  Open Access: 3 results) Presentation (9 results) (of which Int'l Joint Research: 6 results,  Invited: 2 results)

  • [Int'l Joint Research] 復旦大学/華中師範大学(中国)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] ボイト理工大学(ドイツ)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] Texas A&M University(米国)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] Central China Normal University(China)

    • Related Report
      2016 Annual Research Report
  • [Journal Article] Inverse problems of determining sources of the fractional partial differential equations2019

    • Author(s)
      Liu Yikan、Li Zhiyuan、Yamamoto Masahiro
    • Journal Title

      Handbook of Fractional Calculus with Applications

      Volume: 2 Pages: 411-430

    • DOI

      10.1515/9783110571660-018

    • ISBN
      9783110571660
    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Inverse problems of determining parameters of the fractional partial differential equations2019

    • Author(s)
      Li Zhiyuan、Liu Yikan、Yamamoto Masahiro
    • Journal Title

      Handbook of Fractional Calculus with Applications

      Volume: 2 Pages: 431-442

    • DOI

      10.1515/9783110571660-019

    • ISBN
      9783110571660
    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Theoretical stability in coefficient inverse problems for general hyperbolic equations with numerical reconstruction2018

    • Author(s)
      Yu Jie、Liu Yikan、Yamamoto Masahiro
    • Journal Title

      Inverse Problems

      Volume: 34 Issue: 4 Pages: 045001-045001

    • DOI

      10.1088/1361-6420/aaa4a0

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Weak unique continuation property and a related inverse source problem for time-fractional diffusion-advection equations2017

    • Author(s)
      Jiang Daijun、Li Zhiyuan、Liu Yikan、Yamamoto Masahiro
    • Journal Title

      Inverse Problems

      Volume: 33 Issue: 5 Pages: 055013-055013

    • DOI

      10.1088/1361-6420/aa58d1

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A new unique continuation property for the Lamé system in two dimensions (in Chinese)2017

    • Author(s)
      Yanbo Wang、Yikan Liu、Jin Cheng
    • Journal Title

      Scientia Sinica Mathematica

      Volume: 47 Issue: 10 Pages: 1327-1334

    • DOI

      10.1360/n012016-00195

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Reconstruction of the temporal component in the source term of a (time-fractional) diffusion equation2017

    • Author(s)
      Yikan Liu、Zhidong Zhang
    • Journal Title

      Journal of Physics A: Mathematical and Theoretical

      Volume: 50 Issue: 30 Pages: 305203-305203

    • DOI

      10.1088/1751-8121/aa763a

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Inverse source problem for the hyperbolic equation with a time-dependent principal part2017

    • Author(s)
      Daijun Jiang, Yikan Liu, Masahiro Yamamoto
    • Journal Title

      Journal of Differential Equations

      Volume: 262 Issue: 1 Pages: 653-681

    • DOI

      10.1016/j.jde.2016.09.036

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Inverse source problem for a wave equation with final observation data2017

    • Author(s)
      Daijun Jiang, Yikan Liu, Masahiro Yamamoto
    • Journal Title

      Mathematics for Industry

      Volume: 26 Pages: 153-164

    • DOI

      10.1007/978-981-10-2633-1_11

    • ISBN
      9789811026324, 9789811026331
    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Strong maximum principle for multi-term time-fractional diffusion equations and its application to an inverse source problem2017

    • Author(s)
      Yikan Liu
    • Journal Title

      Computers & Mathematics with Applications

      Volume: 73 Issue: 1 Pages: 96-108

    • DOI

      10.1016/j.camwa.2016.10.021

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Strong maximum principle for fractional diffusion equations and an application to an inverse source problem2016

    • Author(s)
      Y. Liu, W. Rundell and M. Yamamoto
    • Journal Title

      Fract. Calc. Appl. Anal.

      Volume: 19 Issue: 4 Pages: 888-906

    • DOI

      10.1515/fca-2016-0048

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Maximum principle for time-fractional diffusion equations and a related inverse problem2019

    • Author(s)
      Liu Yikan
    • Organizer
      偏微分方程式の最大値原理とその周辺 3
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] A new unique continuation property for two-dimensional anisotropic elasticity systems2018

    • Author(s)
      Liu Yikan
    • Organizer
      9th International Conference "Inverse Problems: Modeling and Simulation"
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Inverse problems for hyperbolic-type equations with time-dependent principal parts2018

    • Author(s)
      Liu Yikan
    • Organizer
      The 10th Annual Meeting on Inverse Problems
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Time-fractional diffusion equations: Maximum principle and inverse problem2018

    • Author(s)
      Yikan Liu
    • Organizer
      草津セミナー2018
    • Related Report
      2017 Annual Research Report
  • [Presentation] Reconstruction of the temporal component in the source term of a fractional diffusion equation2017

    • Author(s)
      Yikan Liu
    • Organizer
      Applied Inverse Problem Conference 2017
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Inverse problems for the acoustic equation with a time-dependent principal part2017

    • Author(s)
      Yikan Liu
    • Organizer
      CSRC Summer School on Applied Inverse Problems
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research
  • [Presentation] A new unique continuation property for anisotropic elasticity systems in two dimensions2017

    • Author(s)
      Yikan Liu
    • Organizer
      A3 Workshop on Modeling and Computation of Applied Inverse Problems, CSIAM 2017
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Coefficient inverse problem for hyperbolic equations with time-dependent principal parts2017

    • Author(s)
      Yikan Liu
    • Organizer
      Taiwan-Japan Joint Workshop on Inverse Problems
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research
  • [Presentation] 非整数階拡散方程式のソース項決定逆問題2016

    • Author(s)
      Yikan Liu
    • Organizer
      日本数学会異分野・異業種研究交流会
    • Place of Presentation
      明治大学中野キャンパス (東京都中野区)
    • Related Report
      2016 Annual Research Report

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Published: 2016-11-08   Modified: 2024-03-26  

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