An inverse heat conduction boundary problem for a two-part rod with different thermal conductivity

Abstract

The paper studies the problem of determining the boundary condition in the heat conduction equation for a rod consisting of homogeneous parts with different thermophys- ical properties. We consider the Dirichlet condition at the left end of the rod (at x = 0) corresponding to the heating of this end and the homogeneous condition of the first kind at the right end (at x = 1) corresponding to cooling during interaction with the environment as boundary conditions. At the point of discontinuity of the thermophysical properties (at x = x0), the conditions for the continuity of temperature and heat flux are set. In the inverse problem, the boundary condition at the left end is assumed to be unknown. To find it, the value of the direct problem solution at the point x0, i.e., the point of separation of the rod into two homogeneous sections, is set. In this work, we carried out an analytical study of the direct problem, which allowed us to apply the time Fourier transform to the inverse boundary value problem. The projection-regularization method is used to solve the inverse boundary value problem for the heat equation and obtain error estimates of this solution correct to the order.