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Application of thermal potentials to the solution of the problem of heat conduction in a region degenerates at the initial moment. (English) Zbl 1499.45008

Summary: In this paper, the boundary value problem for the heat equation in the region which degenerates at the initial time is considered. Such problems arise in mathematical models of the processes occurring by opening of electric contacts, in particular, at the description of the heat transfer in a liquid metal bridge and electric arcing. The boundary value problem is reduced to a Volterra integral equation of the second kind which has a singular feature. The class of solutions for the integral equation is defined and the constructive method of its solution is developed.

MSC:

45D05 Volterra integral equations
45P05 Integral operators
35K05 Heat equation
80A05 Foundations of thermodynamics and heat transfer
80A19 Diffusive and convective heat and mass transfer, heat flow
80A21 Radiative heat transfer
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[1] M.M. Amangaliyeva, M.T. Jenaliyev, M.T. Kosmakova, M. I. Ramazanov, About Dirichlet boundary value problem for the heat equation in angular domain, Boundary Value Problems (2014), 2014:213. · Zbl 1304.35322
[2] R. Holm, Electrical Contacts, IL, Moscow, 1961 (in Russian).
[3] S.N. Kharin, The thermal processes in the electrical contacts and the related singular integral equations, Ph.D. thesis, Alma-Ata, 1970 (in Russian).
[4] M.L. Krasnov, Integral Equations, Nauka, Moscow, 1975 (in Russian).
[5] T.E. Omarov, M.O. Otelbaev, On a class of singular integral equations of Volterra type II, Math. Invest. V3, Karaganda, (1976) 12-19 (in Russian).
[6] M.I. Ramazanov, Investigation of eigenvalues and eigen-functions of the singular integral equation of the Volterra second kind, Differential Eq. Appl. Alma-Ata (1979) 83-90 (in Russian).
[7] Yu.R. Shpadi, A Heat equation problems in domains with varying cross-section, Ph.D. thesis, Almaty, 1998 (in Russian).
[8] A.N. Tikhonov, A.A. Samarskii, Equations of Mathematical Physics, (5th edition), Nauka, Moscow, 1977 (in Russian). · Zbl 0044.09302
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