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

Abstract

<p style='text-indent:20px;'>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 <i>the time domain enclosure method</i> enables one to extract information about the geometry of the obstacle and a qualitative nature of the jump, from the observation data.</p>

Inverse Problem to Determine Two Time-Dependent Source Factors of Fractional Diffusion-Wave Equations from Final Data and Simultaneous Reconstruction of Location and Time History of a Point Source

“…Often parameters of models are unknown. To determine the parameters, inverse problems that use measurements of states of the processes are solved [6][7][8][9][10][11]. Depending on possibilities, the state can be measured at a boundary of a domain where the process is going on, in a subdomain or at fixed time values, e.g., at a final time moment.…”