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2 2022

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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.

On the numerical solution of one inverse problem for a linearized two-dimensional system of Navier-Stokes equations

Jenaliyev Muvasharkhan, Ramazanov Murat, Yergaliyev Madi

Opuscula Mathematica

2022

Vol. 42, no. 5

709--725

The paper studies the numerical solution of the inverse problem for a linearized two-dimensional system of Navier-Stokes equations in a circular cylinder with a final overdetermination condition. For a biharmonic operator in a circle, a generalized spectral problem has been posed. For the latter, a system of eigenfunctions and eigenvalues is constructed, which is used in the work for the numerical solution of the inverse problem in a circular cylinder with specific numerical data. Graphs illustrating the results of calculations are presented.