On estimating the errors of approximate solution methods for an inverse problem

We study an inverse boundary problem for the heat equation for an inhomogeneous rod composed of two materials. We consider three different situations of temperature measurements inside the rod, from which we must reconstruct one of the boundary values of the problem. Similar problems arise in the test benching of rocket engines, and their solutions are required to be very precise. We solve the problems by the projection regularization method, and for their solutions we obtain estimates that are precise up to the order of magnitude.