An algorithm for inhomogeneous medium reconstruction in case of unsteady particle transport

We consider the problem of X-ray tomography that is the inverse problem for the non-stationary differential transport equation. We study an equation in which the coefficients and the unknown function depend on time, while the coefficients can undergo a discontinuity of the first kind in the spatial variable. The desired object is the set on which the coefficients of the transport equation undergo a discontinuity, that corresponds to the search of boundaries between various substances contained in the probed medium. To this end, we consider a special function-an indicator of medium heterogeneity. Using the explicit solutions of the direct and inverse problems, we can indicate the main property of that function: it takes unlimited values on the desired sets. Our main result is a numerical demonstration of the properties of that function. Several examples are given.