A boundary value problem for a parabolic-type equation in a non-cylindrical domain
We present an application of the method of the eigenfunction decomposition of a self-adjoint differential operator to the solution of a non-stationary heat transfer problem with phase transition with the freezing process in a continuous medium as an example. An approximate analytical solution of the problem in a non-automodel formulation under special initial conditions is obtained. The solution of the problem starts with its conversion to a region with fixed boundaries. Then, for the solution of the transformed problem, we construct a finite integral transform with unknown core, finding of which is associated with formulation and solution of corresponding spectral problems using the degenerate hypergeometric functions. We find the eigenvalues and eigenfunctions, as well as an inversion formula for the introduced integral transform, which allows us to obtain an analytical solution to the problem and consider a number of special cases. While solving the problem, we establish the parabolic law of motion of the interface between the two phases. Problems of this type arise in the mathematical modeling of heat transfer processes in construction, especially in the permafrost areas, in oil and gas production during drilling and operation of wells, in metallurgy, etc.
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