On the error in determining the protective layer boundary in the inverse heat problem

The paper studies the problem of determining the error introduced by inaccuracy in determining the thickness of a protective heat-resistant coating of composite materials. The mathematical problem is the heat equation on an inhomogeneous half-line. The temperature on the outer side of the half-line ($x = 0$) is considered unknown over an infinite time interval. To find it, the temperature is measured at the interface of the media at the point $x = x_0$. An analytical study of the direct problem is carried out and enables a rigorous statement of the inverse problem and determining the functional spaces in which it is natural to solve the inverse problem. The main difficulty that the present paper aims at solving is obtaining an estimate for the error of the approximate solution. To estimate the conditional correctness modulus, the projection regularization method is used; this allows obtaining order-accurate estimates.