On Solving Problems of Thermal Conductivity on an Anisotropic Plane with a Weakly Permeable Film

The problem of thermal conductivity on an anisotropic plane (x; y) divided into two halfplanes D1(1 < x < 0; y 2 R) and D2(0 < x < 1; y 2 R) by a weakly permeable film x = 0 is considered at given heat sources and a given initial temperature. The anisotropy ellipses are arbitrary (in magnitude and direction) and are the same at all points of the plane. Using the method of convolution of Fourier expansions, the solution of the problem is expressed in single quadratures through the well-known solution of the classical Cauchy problem on an isotropic plane without a film. The results obtained are of practical interest in the problems of heat propagation and conservation in materials with anisotropic properties (crystalline, fibrous materials), in the presence of a thermal insulation film.