# Vol 23, No 3 (2017)

**Year:**2017**Articles:**8**URL:**https://journals.ssau.ru/est/issue/view/268

## Full Issue

## Статьи

### THE CORRECTNESS OF THE DIRICHLET PROBLEM IN A CYLINDRICAL DOMAIN FOR DEGENERATE MULTIDIMENSIONAL ELLIPTIC-PARABOLIC EQUATIONS

#### Abstract

The correctness of boundary value problems on the plane for elliptic equations by the method of the theory of analytic functions of a complex variable has been well studied. When investigating similar questions, when the number of independent variables is greater than two, problems of a fundamental nature arise. A very attractive and convenient method of singular integral equations loses their validity due to the absence of any full theory of multidimensional singular integral equations. Boundary value problems for second-order elliptic equations in domains with edges have been studied in detail. In the author’s papers explicit forms of classical solutions of Dirichlet problems in cylindrical domains for multidimensional elliptic equations are found. In this paper we use the method proposed in the author’s works, we show the unique solvability and obtain an explicit form of classical solution of the Dirichlet problem in a cylindrical domain for degenerate multidimensional elliptico-parabolic equations.

**Vestnik of Samara University. Natural Science Series**. 2017;23(3):7-11

### ON CERTAIN CONTROL PROBLEM OF DISPLACEMENT AT ONE ENDPOINT OF A THIN BAR

#### Abstract

In this paper, we study an inverse problem for hyperbolic equation. This problem arises when we consider vibration of a thin bar if one endpoint is fixed but behavior of the other is unknown and is the subject to find. Overdetermination is given in the form of integral with respect to spacial variable. The problem is reduced to the second kind Volterra integral equation. Special case is considered.

**Vestnik of Samara University. Natural Science Series**. 2017;23(3):12-17

### NONLOCAL PROBLEM WITH DYNAMICAL BOUNDARY CONDITIONS FOR HYPERBOLIC EQUATION

#### Abstract

In this article, we consider a boundary-value problem with nonlocal dynamical conditions for hyperbolic equation. A feature of such conditions is the presence of both first and second order derivatives with respect to time-variable. Furthermore, boundary conditions are nonlocal to the extent that their representation is a relation between values of the derivatives on different parts of the boundary. The problem under consideration arise when we study vibration of a bar with damping and point masses. The existence and uniqueness of a generalized solution are proved. The proof is based on apriori estimates and Galerkin procedure.

**Vestnik of Samara University. Natural Science Series**. 2017;23(3):18-25

### PROBLEM WITH NONLOCAL BOUNDARY CONDITION FOR A HYPERBOLIC EQUATION

#### Abstract

In this paper we consider an initial-boundary problem with nonlocal boundary condition for one-dimensional hyperbolic equation. Nonlocal condition is dynamic so as represents a relation between values of derivatives with respect of spacial variables of a required solution, first-order derivatives with respect to time variable and an integral of a required solution of spacial variable. We prove the existence and uniqueness of a generalized solution, which belongs to the Sobolev space. To prove uniquely solvability of the problem techniques developed specifically for research nonlocal problems are used. The application of these methods allowed us to obtain a priori estimates, through which the uniqueness of the solution is proved. The proof is based on the a priori estimates obtained in this paper and Galyorkin’s procedure.

**Vestnik of Samara University. Natural Science Series**. 2017;23(3):26-33

### ON THE EXTENSION OF NON-ADDITIVE SET FUNCTIONS

#### Abstract

In this paper we prove theorems on the extension of non-additive set functions domain of definition of which, generally speaking, is not a ring, on the sigma-ring of sets. It is shown that continuous from the top at zero, the exhaustive compositional submersion of the first or second kind can be continued from the multiplicative class of sets to the sigma-ring of sets to a complete quasitriangular submerse complete at zero. Conditions are found under which the composition sub-measure of the first (second) kind extends to the composition sub-measure of the same kind. The continuation of the composite submerses obtained in the work is, in general, not unique. Some particular types of submeasures are considered, for which uniqueness of continuation takes place.

**Vestnik of Samara University. Natural Science Series**. 2017;23(3):34-40

### ON A PENDULUM MOTION IN MULTI-DIMENSIONAL SPACE. PART 1. DYNAMICAL SYSTEMS

#### Abstract

In the proposed cycle of work, we study the equations of the motion of dynamically symmetric fixed n-dimensional rigid bodies–pendulums located in a nonconservative force fields. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of a medium. In parallel, we study the problem of the motion of a free n-dimensional rigid body also located in a similar force fields. Herewith, this free rigid body is influenced by a nonconservative tracing force; under action of this force, either the magnitude of the velocity of some characteristic point of the body remains constant, which means that the system possesses a nonintegrable servo constraint. In thit work, we derive the general multi-dimensional dynamic equations of the systems under study.

**Vestnik of Samara University. Natural Science Series**. 2017;23(3):41-64

### COMPARATIVE ASSESSMENT OF VARIABILITY OF THE MORPHOLOGICAL TRAITS OF SOME SPECIES OF THE GENUS NEPETA L. AT INTRODUCTION IN MOUNTAIN CONDITIONS OF DAGESTAN

#### Abstract

Interspecific variability of three species of the genus Nepeta L. (N. pamirica, N. pannonica and N. subsessilis) was studied with their introduction in the mountainous conditions of Dagestan on the basis of a complex of morphological features of generative shoot. The evaluation of the examined features revealed that along the length and mass of the shoot these species superior are widespread and most productive species N. grandiflora. Weight features of all three species are at a high level of variability. Most of the traits studied are in a positive correlation with each other. Reproductive effort positively correlates with the mass of shoot, but the foliage — not always. The results of the studies make it possible to evaluate the studied species as being resistant to the conditions of introduction, and recommend for cultivation in the mountainous zone of Dagestan.

**Vestnik of Samara University. Natural Science Series**. 2017;23(3):65-70

### MATHEMATICAL DESCRIPTION OF NON-LINEAR RELAXATING POLARIZATION IN DIELECTRICS WITH HYDROGEN BONDS

#### Abstract

Analytical investigating of the patterns of relaxation (volume-charge) polarization in dielectric materials class hydrogen bonded crystals (HBC) in the wide range of temperature (1–1500 K) and polarizing field strengths (100 kV/m–100 MV/m) in alternating field at frequencies of about 1 kHz–10 MHz is made. The generalized nonlinear by the polarizing field the semi-classical kinetic equation of proton relaxation, having (in this model) sense the protons current continuity equation solving by method of successive approximation by decomposition in infinite power series in comparison parameter is built. It is established that in the range of low fields (100–1000 kV/m) and high temperatures (100–250 K) the generalized kinetic equation is converted to the linearized Fokker-Planck equation and at low (70–100 K) and sufficiently high (250–450 K) temperatures are showed the nonlinear polarization effects caused respectively by proton tunneling and volume charge relaxation. With ultra - low (1–10 K) and ultra-high (500–1500 K) temperatures in the range of high fields (10 MV/m–100 MV/m) the contribution of such effects to the polarization is amplified. The influence of the non-linearities to relaxation times for microscopic acts of transitions protons through the potential barrier is studied.

**Vestnik of Samara University. Natural Science Series**. 2017;23(3):71-83