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

TL;DR

This paper introduces a novel inverse obstacle problem for variable-order fractional diffusion equations, using the enclosure method to identify obstacle geometry and jump characteristics from boundary observations.

Contribution

It extends the enclosure method to variable-order fractional diffusion equations, enabling obstacle detection with variable fractional derivatives.

Findings

Successfully detects obstacle geometry from boundary data.

Identifies the nature of the variable-order jump.

Demonstrates effectiveness of the method for variable-order equations.

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.