# Vol 23, No 1 (2017)

**Year:**2017**Articles:**6**URL:**https://journals.ssau.ru/est/issue/view/247

## Full Issue

## Статьи

### PROBLEM WITH TIME-DEPENDENT BOUNDARY CONDITIONS FOR HYPERBOLIC EQUATION

#### Abstract

In this paper, we consider a problem for hyperbolic equation with standard initial data and nonlocal dynamic conditions. Such conditions may arise when a thick short bar fixed by point forces and springs. The existence and uniqueness of the problem are proved. The proof is mainly based on a priori estimates and Galerkin procedure.

**Vestnik of Samara University. Natural Science Series**. 2017;23(1):7-14

### ON SOLVABILITY OF NONLOCAL PROBLEM FOR THIRD-ORDER EQUATION

#### Abstract

In this paper nonlocal problem with integral conditions for partial differential equation of the third order is considered. The existence of a unique classical solution is proved in rectangular domain. The proof is carried out by the method of auxiliary problems. At first the problem for a new function for partial differential equation of the first order is considered. Then the solvability of integral analogue of Goursat problem for hyperbolic equation of the second order is proved by equivalent reduction of the problem to the Volterra integral equation of the second kind.

**Vestnik of Samara University. Natural Science Series**. 2017;23(1):15-20

### PROBLEM WITH DYNAMIC BOUNDARY CONDITIONS FOR A HYPERBOLIC EQUATION

#### Abstract

We consider an initial-boundary problem with dynamic boundary condition for a hyperbolic equation in a rectangle. Dynamic boundary condition represents a relation between values of derivatives with respect of spacial variables of a required solution and first-order derivatives with respect to time variable. The main result lies in substantiation of solvability of this problem. We prove the existence and uniqueness of a generalized solution. The proof is based on the a priori estimates obtained in this paper, Galyorkin’s procedure and the properties of Sobolev spaces.

**Vestnik of Samara University. Natural Science Series**. 2017;23(1):21-27

### ALPHA-MATRIX AND GRAPH-GENERATED GRAMMARS

#### Abstract

In this paper we consider the extension of graph-generated grammars based on their matrix representations. We study two classes of graph-generated grammars associated with the vertex and edge marking of graphs. We define alpha-matrices over a semiring of languages specified by finite alphabet A and then define the corresponding matrix algebras. These concepts are then used for constructive representation of graph-generated languages and research of their equivalence. We define a matrix-generated grammars as a natural superclass of graph-generated grammars. All the proofs are illustrated by examples.

**Vestnik of Samara University. Natural Science Series**. 2017;23(1):28-40

### CASES OF INTEGRABILITY CORRESPONDING TO THE PENDULUM MOTION IN FOUR-DIMENSIONAL SPACE

#### Abstract

In this article, we systemize some results on the study of the equations of motion of dynamically symmetric fixed four-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 four-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. We also show the nontrivial topological and mechanical analogies.

**Vestnik of Samara University. Natural Science Series**. 2017;23(1):41-58

### EXPERIMENTAL DETERMINATION OF COEFFICIENTS OF A MULTIPARAMETER DECOMPOSITION OF FIELD OF CRACK TIP STRESSES: PHOTOELASTICITY METHOD

#### Abstract

The purpose of this study is multiparameter asymptotic analysis of the stress field in the immediate vicinity of the crack tip in a linearly elastic material and construction of complete asymptotic expansion of M. Williams stress field in the vicinity of the crack tip. Multiparametric analysis of the stress field is based on the polarization-optical methods of mechanics of a deformable solid (the method of photoelasticity). Digital processing of the results of optoelectronic measurements performed on a series of samples with cracks and notches is carried out. Different classes of samples from optically sensitive materials, in particular a sample with two collinear cracks under conditions of normal detachment, were considered. A set of programs has been prepared that makes it possible to determine the scale (amplitude) multipliers of complete asymptotic expansion of M.Villiams for the stress field at the crack tip. Using the basic law of photoelasticity, first five coefficients of complete asymptotic expansion of M. Williams are calculated. The results of the experiments are compared with the available analytical solution. It is shown that the results of processing optoelectronic measurements are in good agreement with the analytical solution obtained for an infinite plate with two collinear cracks.

**Vestnik of Samara University. Natural Science Series**. 2017;23(1):59-68