# A boundary value problem for a parabolic-type equation in a non-cylindrical domain

### Abstract

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.

### References

[1]

Aksenov B. G. and Karyakin Yu. E., “Numerical simulation of one-dimensional multi-front Stefan problems [in Russian],” Vestn. Tyumen. Gos. Univ., Fiz.-Mat. Modelir., 3, No. 3, 8–16 (2017).

[2]

Vasilyev V. I., Vasilyeva M. V., Sirditov I. K., Stepanov S. P., and Tseeva A. N., “Mathematical modeling of temperature regime of soils of foundation on permafrost,” Vestn. Mosk. Gos. Tekh. Univ. im. N. E. Baumana, Estestv. Nauki No. 1, 142–159 (2017).

[3]

Arutunyan R. V., “Integral equations of the Stefan problem and their application in modeling soil thawing [in Russian],” Nauka i Obrazovanie, Mosk. Gos. Tekh. Univ. im. N. E. Baumana, Electron. J., No. 10, 419–437 (2015).

[4]

Gornov V. F., Stepanov S. P., Vasilyeva M. V., et al., “Mathematical modeling of heat transfer problems in the permafrost,” AIP Conf. Proc., 1629, pp. 424–431, AIP (2014).

[5]

Aksenov B. G. and Karyakina S. V., “Stefan problem as a limiting case of the problem of phase transition in the temperature spectrum [in Russian],” Vestn. Tyumen. Gos. Univ., Fiz.-Mat. Nauki, Inform., No. 7, 133–140 (2013).

[6]

Alghalith M., “A new stopping time model: A solution to a free-boundary problem,” J. Optim. Theory Appl., 152, No. 1, 265–270 (2012).

[7]

Chernov I. A., “Classical solution of one-dimensional parabolic boundary value problem with nonlinear boundary conditions and moving boundary [in Russian],” Differ. Equ., 46, No. 7, 1044–1052 (2010).

[8]

Dancer E. N. and Yihong Du, “A uniqueness theorem for a free-boundary problem,” Proc. Amer. Math. Soc., 134, No. 11, 3223–3230 (2006).

[9]

Borisovich A. and Friedman A., “Symmetry-breaking bifurcations for free boundary problems,” Indiana Univ. Math. J., 54, No. 3, 927–947 (2005).

[10]

Baconneau O. and Lunardi A., “Smooth solutions to a class of free boundary parabolic problems,” Trans. Amer. Math. Soc., 356, No. 3, 987–1007 (2004).

[11]

Kartashov E. M., “Analytical methods for solving boundary value problems of unsteady thermal conductivity in a region with moving boundaries [in Russian],” Izv. Ross. Akad. Nauk, Ser. Energetika, No. 5, 3–34 (1999).

[12]

Sadovnichy V. A. and Dubrovsky V. V., “Remark on a new method of calculation of eigenvalues and eigenfunctions for discrete operators,” J. Math. Sci., New York, 75, No. 3, 1770–1772 (1995).

[13]

Kadchenko S. I., “The method of regularized traces [in Russian],” Vestn. Yuzhno-Ural. Gos. Univ., Ser. Mat. Model. Program., No. 37 4–23 (2009).

[14]

Zaynullin R. G. and Fazullin Z. Y., “Description of temperature fields of heat transfer process with phase transition [in Russian],” Uspekhi Sovremen. Nauki, Fiz.-Mat. Nauki, No. 7, 82–91 (2016).

[15]

Zaynullin R. G., “On an analitycal approach to solving one-dimensional heat transfer problem with free boundaries [in Russian],” Izv. Vuzov, Mat., No. 2, 24–31 (2008).

[16]

Shafeev M. N., “Solution of one plane Stefan problem by WGGP [in Russian],” Inzh.-Fiz. Zhurn., 34, No. 4, 713–722 (1978).

[17]

Shafeev M. N., “The solution of one nonlinear problem by the method of WGHP [in Russian],” Izv. Vuzov, Mat., No. 12, 73–75 (1980).

[18]

Khakimov R. H., Freezing of Soils for Construction Purposes [in Russian], Gosstroiizdat, Moscow (1962).

[19]

Abramovich M. A. and Stigan I., Handbook of Special Functions with Formulas, Graphs and Tables [in Russian], Nauka, Moscow (1979).

[20]

Prudnikov A. P., Brychkov Yu. A., and Marichev O. I., Integrals and Series, Additional Chapters [in Russian], Nauka, Moscow (1986).

[21]

Prudnikov A. P., Brychkov Yu. A., and Marichev O. I., Integrals and Series, Special Functions [in Russian], Nauka, Moscow (1983).

[22]

Slater L. D., Degenerate Hypergeometric Functions [in Russian], Izdat. Vychisl. Tsentra Akad. Nauk SSSR, Moscow (1968).

[23]

Tikhonov A. N. and Samarskii A. A., Equations of Mathematical Physics [in Russian], Nauka, Moscow (1972).

[24]

Coddington E. A. and Levinson N., Theory of Ordinary Differential Equations [in Russian], Izdat. Inostr. Lit., Moscow (1958).

[25]

Koshlyakov N. S., Gleaner E. B., and Smirnov M. M., Partial Differential Equations of Mathematical Physics [in Russian], Vysshaya Shkola, Moscow (1970).

[26]

Nikiforov A. F. and Uvarov V. B., Special Functions of Mathematical Physics [in Russian], Nauka, Moscow (1984).

*Mathematical notes of NEFU*, 27(2), pp. 3-20. doi: https://doi.org/10.25587/SVFU.2020.72.83.001.

This work is licensed under a Creative Commons Attribution 4.0 International License.