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Solution of the boundary value problem of heat conduction in a cone

Ramazanov Murat, Jenaliyev Muvasharkhan, Gulmanov Nurtay

Opuscula Mathematica

2022

Vol. 42, no. 1

75--91

In the paper we consider the boundary value problem of heat conduction in a non-cylindrical domain, which is an inverted cone, i.e. in the domain degenerating into a point at the initial moment of time. In this case, the boundary conditions contain a derivative with respect to the time variable; in practice, problems of this kind arise in the presence of the condition of the concentrated heat capacity. We prove a theorem on the solvability of a boundary value problem in weighted spaces of essentially bounded functions. The issues of solvability of the singular Volterra integral equation of the second kind, to which the original problem is reduced, are studied. We use the Carleman–Vekua method of equivalent regularization to solve the obtained singular Volterra integral equation.

Stefan problems in non-cylindrical domains arising in Czochralski process of crystal growth

Fukao T., Kenmochi N., Pawłow I.

Control and Cybernetics

2003

Vol. 32, no 2

201-221

In this paper we discuss a two-phase Stefan problem with convection in a non-cylindrical (time-dependent) domain. This work is motivated by phase change phenomenon arising in the Czochralski process of crystal growth. The time-dependence of domain is a mathematical description of the situation in which the material domain changes its shape with time by crystal growth. We consider the so-called enthalpy formulation for it and give its solvability, assuming that the time-dependence of the material domain is prescribed and smooth enough in time, and the convective vector is prescribed, too. Our main idea is to apply the theory of quasi-linear equations of parabolic type.

The problem without initial data for linear Sobolev-Halpern system

Lavrenjuk S., Kolinko M.

Demonstratio Mathematica

1998

Vol. 31, nr 1

25-32

Some sufficient conditions of the existence and uniqueness of a weak solution of