Mathematical analysis for fractional diffusion-wave equations and related inverse problems

2022 Fiscal Year Final Research Report

• Project/Area Number: 20K14355
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12040:Applied mathematics and statistics-related
• Research Institution: Hokkaido University
• Principal Investigator: Liu Yikan 北海道大学, 電子科学研究所, 助教 (70773084)
• Project Period (FY): 2020-04-01 – 2023-03-31
• Keywords: 非整数階拡散-波動方程式 / 非局所性 / 一意接続性 / 逆問題 / 一意性

Outline of Final Research Achievements

In this research, qualitative properties of fractional diffusion-wave equations were obtained on a more advanced level, and several inverse problems were solved in more general settings from literature.
On qualitative properties of solutions, we focused on fractional wave equations which were not well-studied previously, and concluded that they coincide with fractional diffusion equations in the aspects of unique continuation and long-time positivity.
On related inverse problems, we also greatly improved preliminary results based on the development of theory above and introducing new methods. First, we obtained the uniqueness on determining orders only by the asymptotic behavior of solutions. Second, we reduced the necessary observation times for proving the uniqueness of simultaneous determination of multiple unknown coefficients from infinity to one. Third, for inverse source problems, we also succeeded in reducing the amount of data and generalizing the assumptions.

Free Research Field

偏微分方程式の逆問題

Academic Significance and Societal Importance of the Research Achievements

本研究は非整数階拡散方程式に関する結果を補完しつつ、先行研究が少なかった非整数階波動方程式に深く踏み入れ、両者の相違と類似点を「一意接続性」と「定符号性」などの観点から見極めた。これらは既存成果と合わせて、時間微分回数が0から2までの発展方程式の統一した定性理論の完成を意味する。また関連する逆問題の側面において、先行研究より一般かつ現実的な問題設定を考案し、さらに数値結果を裏付ける正当化を行うことで、理論と応用の両面から有意義な成果を得た。
本研究で得られた結果は関連分野の発展に寄与するのみならず、不均質媒質における汚染源の同定においても、事故直後のデータ欠落やデータ不足の場合にも役立つ。