Geodesic
Exact and approximate solutions to the Stefan problem in ellipsoidal coordinates
Eurasian mathematical journal, Tome 13 (2022) no. 3, pp. 51-66.

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In this paper, we present exact and approximate solutions of the Stefan problems in ellipsoidal coordinates. We consider two models of electrical contact heating for melting process. The first problem describes the contact heating for liquid and solid zones based on the two-phase Stefan problem, where time $t$ is present as a parameter. Contact heating including softening processes are described by a mathematical model based on the three-phase Stefan problem for the ellipsoidal heat equation. Numerical results are presented and discussed. 
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S. A. Kassabek; S. N. Kharin; D. Suragan. Exact and approximate solutions to the Stefan problem in ellipsoidal coordinates. Eurasian mathematical journal, Tome 13 (2022) no. 3, pp. 51-66. http://geodesic.mathdoc.fr/item/EMJ_2022_13_3_a4/

[1] T. R. Goodman, “The heat-balance integral and its application to problems involving a change of phase”, Trans. ASME, J. Heat Transfer, 80 (1958), 335–342 | MR

[2] T. R. Goodman, “Application of integral methods in transient non-linear heat transfer”, Advances in Heat Transfer, 1, eds. T. F. Irvine, J.P. Hartnett, Academic Press, New York, 1964, 51–122 | DOI

[3] S. C. Gupta, The classical Stefan problem. Basic Concepts, Modeling and Analysis, Elsevier, Amsterdam, 2003 | MR

[4] R. Holm, Electrical contacts, 4-th Edition, Springer-Verlag, Berlin-Heidelberg, 1967

[5] S. N. Kharin, H. Nouri, T. Davies, “The mathematical models of welding dynamics in closed and switching electrical contact”, Proc. 49th IEEE Holm Conference on Electrical Contacts (Washington, USA, 2003), 128–146

[6] S. N. Kharin, Mathematical models of phenomena in electrical contacts, A.P. Ershov Institute of Informatics System, Russian Academy of Sciences, Siberian Branch, Novosibirsk, 2017

[7] S. N. Kharin, M. M. Sarsengeldin, S. A. Kassabek, T. Nauryz, “The model of melting and welding of closed electrical contacts with softening contact zone”, 29th International conference on electrical contacts and the 64th IEEE Holm conference on electrical contacts (Albuquerque, New Mexico, USA), 2018, 38–45

[8] S. L. Mitchell, T. G. Myers, “Application of standard and refined heat balance integral methods to one-dimensional Stefan problems”, SIAM Rev., 52:1 (2010), 57–86 | DOI | MR | Zbl

[9] A. D. Polyanin, Handbook of linear partial differential equations for engineers and scientists, A CRC Press LLC, Boca Raton, 2002 | MR | Zbl

[10] H. S. Ren, “Application of the heat-balance integral to an inverse Stefan problem”, Int. J. Thermal Sci., 46:2 (2007), 118–127 | DOI

[11] L. I. Rubinstain, The Stefan problem, American Mathematical Society, Providence, RI, 1971 | MR

[12] N. Sadoun, E. K. Si-Akhmed, P. Colinet, “On the refined integral method for the one-phase Stefan problem with timedependent boundary conditions”, Appl. Math. Modelling, 25 (2006), 531–544 | DOI

[13] S. A. Kassabek, S. N. Kharin, D. Suragan, “A heat polynomials method for inverse cylindrical one-phase Stefan problems”, Inverse Problems in Science and Engineering, 29 (2021), 3423–3450 | DOI | MR | Zbl

[14] S. A. Kassabek, D. Suragan, “Numerical approximation of the one-dimensional inverse Cauchy-Stefan problem using heat polynomials methods”, Computational and Applied Mathematics, 41:4 (2022), 1–19 | DOI | MR

[15] S. A. Kassabek, D. Suragan, “Two-phase inverse Stefan problems solved by heat polynomials method”, Journal of Computational and Applied Mathematics, 2022, 114854 | MR

[16] J. W. Sitison, D. A. Edwards, “The heat balance integral method for cylindrical extruders”, Journal of Engineering Mathematics, 122 (2020), 1–16 | DOI | MR | Zbl

[17] A. S. Wood, “A new look at the heat balance integral method”, Appl. Math. Modelling, 25 (2001), 815–824 | DOI | Zbl