Murat Ramazanov

Karagandy State University · Faculty of Mathematics and Information Technologies

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 constr...

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 prese...

In this paper we study the solvability of a singular integral equation arising in the theory of boundary value problems for the heat equation in an infinite angular domain. The particular case of the corresponding homogeneous integral equation was investigated earlier in [1, 2] and it was shown that in a weight class of essentially bounded function...

We consider a coefficient inverse problem for a parabolic equation in a degenerate angular domain when the moving part of the boundary changes linearly. We show that the inverse problem for the homogeneous heat equation with homogeneous boundary conditions has a nontrivial solution up to a constant factor consistent with an additional condition. Th...

In a degenerate domain, namely the inverted cone, we consider a boundary value problem of heat conduction. For this problem the solvability theorems are established in weighted spaces of essentially bounded functions. The proofs of the theorems are based on the results of the solvability for a nonhomogeneous integral equation of the third kind. The...

In this paper, we study a homogeneous singular integral Volterra equation of the second kind (pseudoVolterra integral equation). The singularity of the integral equation is shown. Properties of its kernel are proved. The characteristic equation is constructed. It is shown that it really is a characteristic equation for the studied integral equation...

In the paper, we consider a coefficient inverse problem for the heat equation in a degenerating angular domain. It has been shown that the inverse problem for the homogeneous heat equation with homogeneous boundary conditions has a nontrivial solution up to a constant factor consistent with the integral condition. Moreover, the solution of the cons...

In addition to the trivial solution in the class of essentially bounded functions with a given weight for the Solonnikov-Fasano homogeneous parabolic problem in an infinite angular domain we establish the existence of the nontrivial solution, up to a constant factor.

In thispaper we consider the questions of solvability of the nonhomogeneous boundary value problem for the Burgers equation in infinite angular domain. It is reduced to the study of the solvability of a system consisting of two homogeneous integral equations. We prove some lemmas which establish properties of integral operators in weighted space of...

Earlier we studied the homogeneous boundary value problem for the heat equation in degenerating domains. For this problem in the weight class of essentially bounded functions it was established the existence of a nontrivial solution up to a constant multiplier. In this paper, on the basis of the above result, we study the issues of the existence of...

In this paper, we study the solvability of a singular integral equation arising in the theory of boundary value problems for the heat equation in an infinite angular domain. With the help of thermal potentials, the boundary value problem of heat conduction is reduced to a singular integral Volterra equation of the second kind. The corresponding hom...

Research of the Burgers equation has a long history. In work of Y. Benia, B.-K. Sadallah in the Sobolev classes there are results on the existence, uniqueness and regularity for the solution to the Burgers equation in non-cylindrical (non-degenerating) domain, that can be converted into a rectangular domain by a regular replacement of the independe...

In this work, the domain has the peculiarity of degeneration at initial time according to the low x(t) = tω, ω > 1/2. These problems have the great importance for applications. We show that the homogeneous boundary value problem for the heat equation in the degenerating domain has a non-trivial solution in addition to the trivial solution.

In this paper, in the class of essentially bounded functions with a given weight the existence of a nontrivial solution for a constant factor and a constant term is set.

The investigated spectral problems arise in the study of the stabilization problem for a loaded heat equation. The dimension of the space variable is equal to two. Stabilization of the solution of the equation is carried out by means of boundary control actions. The solution of this problem can be solved by separation of variables.

The article addresses the singular Volterra integral equation of the second kind, which has the ’incompressible’ kernel. It is shown that the corresponding homogeneous equation on |λ| ≥ exp{|arg λ|}, arg λ ∈ [−π,π] has a continuous spectrum, and the multiplicity of the characteristic numbers grows with increasing |λ|. We use the Carleman-Vekua regu...

Solving the boundary value problems of the heat equation in noncylindrical domains degenerating at the initial moment leads to the necessity of research of the singular Volterra integral equations of the second kind, when the norm of the integral operator is equal to 1. The paper deals with the singular Volterra integral equation of the second kind...

We prove that the operator of a boundary value problem of heat conduction in an infinite angular domain is Noetherian with index −1 in the class of growing functions.

The article addresses the singular Volterra integral equation of the second kind which has the “incompressible” kernel. It is shown that the corresponding homogeneous equation on |λ| > exp{|arg λ|}, arg λ ∈ [−π, π] has a continuous spectrum, and the multiplicity of the characteristic numbers grows with increasing |λ|. The equation is reduced to Abe...

In this paper it is established that in an infinite angular domain for Dirichlet problem of the heat conduction equation the unique (up to a constant factor) non-trivial solution exists, which does not belong to the class of summable functions with the found weight. It is shown that for the adjoint boundary value problem the unique (up to a constan...

The article addresses the singular Volterra integral equation of the second kind which has the "incompressible" kernel. It is shown that the corresponding homogeneous equation for vertical bar lambda vertical bar > 1 has a continuous spectrum, and the multiplicity of the characteristic numbers grows with increasing vertical bar lambda vertical bar....

In this paper, an approval of a stabilizing inverse problem with a boundary condition for the loaded heat equation is given. Theorem on solvability of the stated inverse problem is proved and an algorithm for approximate construction of boundary controls is developed. Numeral calculations show the efficiency of the offered algorithm.

In the paper questions of the spectrum and the solvability of singular Volterra integral equation of the third kind are investigated.

We continue the study of boundary value problems for spectrally loaded heat equations in unbounded domains for the case in which the order of the derivative in the loaded term coincides with that of the differential part of the equation and the motion of the load point with respect to the space variable is given by the law -x(t) = t ω , −∞ < ω < 1...

We study the solvability questions for a particular second kind Volterra integral equation with a spectral parameter λ arising in the theory of boundary value problems for the spectrally loaded parabolic equations in unbounded domains when the order of a derivative in a loaded summand agrees with that in the differential part of the equation. Keyw...

In the present paper, we consider boundary value problems for a loaded spatially one-dimensional heat operator in a quarter plane. The equation in question has the following specific features. First, the spectral parameter serves as the coefficient of the loaded term; second, the order of the derivative in the loaded term is equal to the order of t...

We consider the boundary value problems in a quarter-plane for a loaded heat conduction operator (one-dimensional in the space variable). A peculiarity of the operator in question is as follows: first, the spectral parameter is the coefficient of the loaded summand; second, the order of the derivative in the loaded summand is equal to that of the d...