Abstract <p>We present the approximate analytic solution for two-dimensional unsteady single-phase heat transfer problem with free boundary under special boundary conditions in the non-automodel formulation using the eigenfunction decomposition method of a self-adjoint differential operator. The solution gives the parabolic law of movement of the free boundary and its rectilinear shape. The case is considered when the equations for the kernels of integral transformations are subjected to the requirement of self-conjugancy. It makes possible to apply the mentioned method. One of the kernels is expressed in the terms of confluent hypergeometric function. Problems of this type arise in the mathematical modeling of the heat exchange processes in construction, especially in permafrost areas, in oil and gas production during drilling and operation of wells, in metallurgy, etc.</p>