Solving the Inverse Heat Conduction Boundary Problem for Composite Materials

Abstract

The paper deals with the problem of determining the boundary condition in the heat equation consisting of homogeneous parts with different thermal properties. As boundary conditions, the Dirichlet condition at the left end of the rod (at \(x=0\)) corresponding to the heating of this end and the linear condition of the third kind at the right end (at \(x=1\)) corresponding to the cooling when interacting with the environment are considered. In a point of discontinuity of heat transfer properties (at \(x=x_0\)) conditions of continuity for temperature and heat flow are set. In the inverse problem, the boundary condition at the left is considered unknown over the entire infinite time interval. To find it, the value of the direct problem solution at the point of \(x_0\), that is, the point of the rod division into two homogeneous sections, is specified. In this paper, an analytical study of the direct problem was carried out, which allowed us to apply the time Fourier transform to the inverse boundary value problem. The inverse heat conduction boundary problem was solved using the projection-regularization method and order-accurate error estimates of this solution were obtained.