Total development of the time domain enclosure method

2022 Fiscal Year Final Research Report

• Project/Area Number: 17K05331
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Hiroshima University
• Principal Investigator: Ikehata Masaru 広島大学, 先進理工系科学研究科(工), 教授 (90202910)
• Project Period (FY): 2017-04-01 – 2023-03-31
• Keywords: inverse problems / enclosure method / inverse obstacle problem / wave equation / heat equation / fractional diffusion / Maxwell system / thermoelasticity

Outline of Final Research Achievements

1. The enclosure method using a wave observed over a finite time interval at the same place where the wave originated is developed.
The governing equations of the wave are scalar wave equations or the Maxwell system in an exterior or the whole space. The method enables us to extract various information about an unknown obstacle, such as location, shape and the state of the surface.
2. The enclosure method in a bounded domain is developed. The domain contains an unknown discontinuity, and the method employs a signal propagating inside the domain which is raised by prescribing a suitable input on the outer surface and observed on the same surface. A general and natural idea of the construction of the needed input is introduced. It is shown that the idea covers various inverse obstacle problems, in which the signals governed by the heat equation, wave equation, a thermoelastic system of equations and a diffusion equation with a space dependent fractional power time derivative.

Free Research Field

偏微分方程式に対する逆問題

Academic Significance and Societal Importance of the Research Achievements

非破壊検査等さまざまな応用を持つ障害物逆問題において有限時間領域におけるデータをどう使うかという問いに対する接近方法として, 囲い込み法が様々な問題を通してさらに展開された. 熱方程式, 波動方程式, 熱弾性体の方程式, Maxwell方程式系等によって支配された逆問題に取り組む過程で, 囲い込み法は新たなアイデアを加えてさらに柔軟な方法になった. 今後ますます応用範囲を見出すことは疑いの余地がない.