Solution of an Inverse Boundary Value Problem of Heat Transfer for an Inhomogeneous Ball

This paper studies the problem of determining a boundary condition for the heat conduction equation for composite materials. Mathematically, this problem is reduced to the heat conduction equation in spherical coordinates for an inhomogeneous ball. The temperature inside the ball is assumed to be unknown in an infinite time interval. To find it, the temperature of the heat flow at the interface between the media is measured at point r = r(0). An analytical study of the direct problem is carried out, which makes it possible to give a rigorous formulation of the inverse problem and to define the functional spaces in which it is natural to solve it. The main difficulty is to obtain an error estimate of the approximate solution. A projection regularization method is used to estimate the modulus of continuity of conditional well-posedness. This allows obtaining order of magnitude estimates.