Difference method for calculating the heat flux at an inaccessible boundary in the problem of heat conduction

The continuation problem for the heat equation is considered. Determining the heat flux at an inaccessible boundary reduces to an inverse problem. For the numerical solution of the inverse problem, an implicit difference scheme is used. At each time step, for the difference analogue of the elliptic equation, the heat flux at the inaccessible boundary is calculated by an economical direct method. The proposed algorithm significantly expands the range of tasks to be solved and can be used for creation of devices capable of real-time determination of the heat flux on inaccessible measurements of parts of inhomogeneous structures. For example, to determine the heat flux on the internal radius of a pipe made of various materials.