# Vol 25, No 4 (2019)

**Year:**2019**Articles:**6**URL:**https://journals.ssau.ru/est/issue/view/429

## Full Issue

## Статьи

### CORRECTNESS OF A MIXED PROBLEM FOR DEGENERATE THREE-DIMENSIONAL HYPERBOLIC-PARABOLIC EQUATIONS

#### Abstract

In mathematical modeling of electromagnetic fields in space, the nature of electromagnetic process is determined by the properties of the medium. If the medium is non-conducting, we obtain degenerate three-dimensional hyperbolic equations. If the medium has a high conductivity, then we come to degenerate

three-dimensional parabolic equations. Consequently, the analysis of electromagnetic fields in complex media (for example, if the medium’s conductivity changes) is reduced to degenerate three-dimensional hyperbolic-parabolic equations. The mixed problem for multidimensional hyperbolic equations is well studied and has been previously considered in the works of various authors. In the articles of Professor S.A. Aldashev, the unique solvability of the mixed problem for degenerate multidimensional hyperbolic equations is proved. It is known that mixed problems for multidimensional hyperbolic-parabolic equations have not been studied much. The paper finds a new class of degenerate three-dimensional hyperbolic-parabolic equations for which the mixed problem has a unique solution and gives an explicit representation of its classical solution.

**Vestnik of Samara University. Natural Science Series**. 2019;25(4):7-13

### GENERALIZATIONS TO SOME INTEGRO-DIFFERENTIAL EQUATIONS EMBODYING POWERS OF A DIFFERENTIAL OPERATOR

#### Abstract

The abstract equations containing the operators of the second, third and fourth degree are investigated in this work.

The necessary conditions for the solvability of the abstract equations, containing the operators of the second and fourth degree, are proved without using linear independence of the vectors included in these equations. Previous authors have essentially used the linear independence of the vectors to prove the necessary

solvability condition.

The present paper also gives the correctness criterion for the abstract equation, containing the operators of the third degree with arbitrary vectors, and its exact solution in terms of these vectors in a Banach space.

The theory presented here, can be useful for investigation of Fredholm integro-differential equations embodying powers of an ordinary differential operator or a partial differential operator.

**Vestnik of Samara University. Natural Science Series**. 2019;25(4):14-21

### SOLVABILITY OF A NONLOCAL PROBLEM WITH INTEGRAL CONDITIONS OF THE II KIND FOR ONE-DIMENSIONAL HYPERBOLIC EQUATION

#### Abstract

In this paper we consider a nonlocal problem with integral conditions of the II kind for one-dimensional hyperbolic equation. Nonlocal conditions of the second kind differ in type of non-integral terms, that may contain traces of required solution and traces of derivatives. This difference turns out to be significant for choosing a method for investigating the solvability of the problem. In this work we consider the case when

nonintegral terms are traces of required solution on boundary of the domain. To investigate the solvability of the problem we use method of reduction to the boundary problem for loaded equation. This method allowed us to define a generalized solution, to obtaim apriori estimates and to prove existence of unique generalized solution of the given problem.

**Vestnik of Samara University. Natural Science Series**. 2019;25(4):22-28

### ABOUT THE SYSTEMS WITH FULL SPARK

#### Abstract

Frames of a finite-dimensional Euclidean and unitary spaces composed of discrete Fourier transform matrices are considered. The relationship of phaseless reconstruction systems with the alternative completeness property is presented. In the complex case, alternative completeness is only a necessary condition for phaseless reconstruction. A system of vectors is constructed such that each of its subsystems with a volume equal to the dimension of space is linearly independent. These systems are called systems with full spark. In particular, such systems are optimal for phase retrieval.

**Vestnik of Samara University. Natural Science Series**. 2019;25(4):29-35

### PROBLEMS OF DIFFERENTIAL AND TOPOLOGICAL DIAGNOSTICS. PART 3. THE CHECKING PROBLEM

#### Abstract

Proposed work is the third in the cycle, therefore, we explain such notions as checking sphere, checking ellipsoid and checking tubes. The checking problem is stated and the algorithms for solving it are formulated. The criterion for a malfunction in a controlled system whose motion is described by ordinary differential equations is taken to be the attainment of a checking surface by the checking vector. We first propose the methods for solving the checking problems in which the checking surfaces are chosen in the form of a checking sphere, checking ellipsoid or checking tube. Then we consider the general techniques for constructing the checking surface by using the statistical testing method. We also give the extended statement of the checking problem. And we also prepare the material for the consideration of the problem of diagnostics.

**Vestnik of Samara University. Natural Science Series**. 2019;25(4):36-47

### EXPERIMENTAL TECHNIQUE FOR DETERMINING THE EVOLUTION OF THE BENDING SHAPE OF THIN SUBSTRATE BY THE COPPER ELECTROCRYSTALLIZATION IN AREAS OF COMPLEX SHAPES

#### Abstract

The present paper is aimed of experimental technique of local incompatible deformations’ identification in thin layers obtained as a result of electrocrystallization. The process of electrocrystallization is carried on thin substrates. Changes in time of form of these thin substrates are registered during the experiment. Identification of local incompatible deformations’ parameters is carried out from the condition of the minimum deflection of experimentally detected displacements and displacements that were determined by theoretical relations. As such a relationship the solution of a boundary value problem for a layer by layer growing plate is used in the paper. Significant difference of suggested technique from known methods is that testing electrocrystallization is carried out in areas of various forms. It allows to provide analysis of the influence that corner points of deposition area’s boundary have on incompatible deformations caused by electrochemical process.

**Vestnik of Samara University. Natural Science Series**. 2019;25(4):48-73