The enclosure method for inverse obstacle scattering over a finite time interval: VI. Using shell-type initial data

  • 1 Laboratory of Mathematics, Hiroshima University, Graduate School of Engineering, Higashihiroshima, Japan
Masaru IkehataORCID iD: https://orcid.org/0000-0002-1710-5146

Abstract

A simple idea of finding a domain that encloses an unknown discontinuity embedded in a body is introduced by considering an inverse boundary value problem for the heat equation. The idea gives a design of a special heat flux on the surface of the body such that from the corresponding temperature field on the surface one can extract the smallest radius of the sphere centered at an arbitrary given point in the whole space and enclosing unknown inclusions. Unlike before, the designed flux is free from a large parameter. An application of the idea to a coupled system of the elastic wave and heat equations are also given.

  • [1]

    D. E. Carlson, Linear Thermoelasticity, Mechanics of Solids. Volume II: Linear Theories of Elasticity and Thermoelasticity, Linear and Nonlinear Theories of Rods, Plates, and Shells, Springer, Berlin (1984), 297–345.

  • [2]

    W. Dan, On the Neumann problem of linear hyperbolic parabolic coupled systems, Tsukuba J. Math. 18 (1994), no. 2, 371–410.

    • Crossref
    • Export Citation
  • [3]

    M. Ikehata, Extracting discontinuity in a heat conductive body. One-space dimensional case, Appl. Anal. 86 (2007), no. 8, 963–1005.

    • Crossref
    • Export Citation
  • [4]

    M. Ikehata, The framework of the enclosure method with dynamical data and its applications, Inverse Problems 27 (2011), no. 6, Article ID 065005.

  • [5]

    M. Ikehata, On finding an obstacle embedded in the rough background medium via the enclosure method in the time domain, Inverse Problems 31 (2015), no. 8, Article ID 085011.

  • [6]

    M. Ikehata, The enclosure method for inverse obstacle scattering over a finite time interval: IV. Extraction from a single point on the graph of the response operator, J. Inverse Ill-Posed Probl. 25 (2017), no. 6, 747–761.

  • [7]

    M. Ikehata, On finding a cavity in a thermoelastic body using a single displacement measurement over a finite time interval on the surface of the body, J. Inverse Ill-Posed Probl. 26 (2018), no. 3, 369–394.

    • Crossref
    • Export Citation
  • [8]

    M. Ikehata, The enclosure method for the heat equation using time-reversal invariance for a wave equation, preprint (2018), https://arxiv.org/abs/1806.10774.

  • [9]

    M. Ikehata, Prescribing a heat flux coming from a wave equation, J. Inverse Ill-Posed Probl. 27 (2019), no. 5, 731–744.

    • Crossref
    • Export Citation
  • [10]

    M. Ikehata, The enclosure method for inverse obstacle scattering over a finite time interval: V. Using time-reversal invariance, J. Inverse Ill-Posed Probl. 27 (2019), no. 1, 133–149.

    • Crossref
    • Export Citation
  • [11]

    M. Ikehata and H. Itou, On reconstruction of a cavity in a linearized viscoelastic body from infinitely many transient boundary data, Inverse Problems 28 (2012), no. 12, Article ID 125003.

  • [12]

    M. Ikehata and M. Kawashita, On the reconstruction of inclusions in a heat conductive body from dynamical boundary data over a finite time interval, Inverse Problems 26 (2010), no. 9, Article ID 095004.

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This journal presents original articles on the theory, numerics and applications of inverse and ill-posed problems. These inverse and ill-posed problems arise in mathematical physics and mathematical analysis, geophysics, acoustics, electrodynamics, tomography, medicine, ecology, financial mathematics etc. Articles on the construction and justification of new numerical algorithms of inverse problem solutions are also published.

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