The enclosure method for the detection of variable order in fractional diffusion equations

Masaru Ikehata; Yavar Kian

doi:10.3934/ipi.2022036

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2023, Volume 17, Issue 1: 180-202. Doi: 10.3934/ipi.2022036 open access

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The enclosure method for the detection of variable order in fractional diffusion equations

Masaru Ikehata^1, and
Yavar Kian^2, ,

1.
Laboratory of Mathematics, Graduate School of Advanced Science and Engineering, Hiroshima University, Higashihiroshima 739-8527, Emeritus Professor at Gunma University, Maebashi 371-8510, Japan
2.
Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

^* Corresponding author: Yavar Kian
^* Corresponding author: Yavar Kian

Received: February 2022

Revised: May 2022

Early access: June 2022

Published: February 2023

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Abstract

This paper is concerned with a new type of inverse obstacle problem governed by a variable-order time-fraction diffusion equation in a bounded domain. The unknown obstacle is a region where the space dependent variable-order of fractional time derivative of the governing equation deviates from a known homogeneous background one. The observation data is given by the Neumann data of the solution of the governing equation for a specially designed Dirichlet data. Under a suitable jump condition on the deviation, it is shown that the most recent version of the time domain enclosure method enables one to extract information about the geometry of the obstacle and a qualitative nature of the jump, from the observation data.

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Mathematics Subject Classification: Primary: 35R30, 35R10.

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Figure 1. The sets $ \Omega $, $ D $ and $ \mbox{dist}\,(K_{\star},D) $

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