Unsteady heat conduction problem for a plane with a crack at the interface between two inhomogeneous materials

An unsteady heat conduction problem is studied in a plane consisting of two half-planes filled with different inhomogeneous materials with exponential internal thermal conductivities. It is assumed that there is a crack at the boundary of the half-planes, i.e., inhomogeneous transmission conditions are set. In the upper and lower half-planes, the heat equations are supplemented with conditions on the differences between the temperatures and heat fluxes in the upper and lower crack faces. Homogeneous initial conditions are specified. Integral representations for the components of the solution to the problem are presented, and the boundary and initial conditions are proved to hold. For the solution to the problem, after making a change of variables, even continuations of the studied functions to the upper half-plane are constructed. The problem is reduced to a generalized one. The Fourier transform with respect to space variables and the Laplace transform with respect to time are applied to the latter problem, so that the properties of these transforms can be used to obtain the solution. Integral representations of the solution to the original problem are found using the inverse transforms. This paper is the first of two addressing this subject. In the second work, we intend to construct singular components of asymptotic expansions of the solution with respect to the distance to the interface line.