Solution of the unsteady-state heat conduction problem for a two-dimensional region with a moving boundary

With the use of the convolution-type functional a variational description is given for the process of unsteady-state heat conduction with the first-kind boundary conditions for a two-dimensional region whose boundary moves in time according to the familiar arbitrary law. Based on the Galerkin-Kantorovich method, a corresponding system of Euler equations is written the solution of which (numerical or analytical) is required to determine the temperature field in each specific case. As an example, the first and second analytic approximations to the solution of the above problem are obtained for the case of the deformation of a prism having initially a circular cross-section.