# Vol 26, No 4 (2020)

**Year:**2020**Articles:**7**URL:**https://journals.ssau.ru/est/issue/view/499

## Articles

### TRICOMI PROBLEM FOR MULTIDIMENSIONAL MIXED HYPERBOLIC-PARABOLIC EQUATION

#### Abstract

It is known that in mathematical modeling of electromagnetic fields in space, the nature of the electromagnetic process is determined by the properties of the media. If the medium is non-conducting, then we obtain multidimensional hyperbolic equations. If the medium’s conductivity is higher, then we arrive at multidimensional parabolic equations. Consequently, the analysis of electromagnetic fields in complex media (for example, if the conductivity of the medium changes) reduces to multidimensional hyperbolic-parabolic equations. When studying these applications, one needs to obtain an explicit representation of solutions to the problems under study. Boundary-value problems for hyperbolic-parabolic equations on a plane are well studied; however, their multidimensional analogs have been analyzed very little. The Tricomi problem for the above equations has been previously investigated, but this problem in space has not been studied earlier. This article shows that the Tricomi problem is not uniquely solvable for a multidimensional mixed hyperbolic-parabolic equation. An explicit form of these solutions is given.

**Vestnik Samarskogo Universiteta. Estestvennonauchnaya Seriya / Vestnik of Samara University. Natural Science Series**. 2020;26(4):7-14

### SYMMETRIC FINITE REPRESENTABILITY OF ℓp IN ORLICZ SPACES

#### Abstract

It is well known that a Banach space need not contain any subspace isomorphic to a space ℓp (1 6 p < ∞) or c0 (it was shown by Tsirel’son in 1974). At the same time, by the famous Krivine’s theorem, every Banach space X always contains at least one of these spaces locally, i.e., there exist finite-dimensional subspaces of X of arbitrarily large dimension n which are isomorphic (uniformly) to ℓnp for some 1 6 p < ∞ or cn0 . In this

case one says that ℓp (resp. c0) is finitely representable in X. The main purpose of this paper is to give a characterization (with a complete proof) of the set of p such that ℓp is symmetrically finitely representable in a separable Orlicz space.

**Vestnik Samarskogo Universiteta. Estestvennonauchnaya Seriya / Vestnik of Samara University. Natural Science Series**. 2020;26(4):15-24

### A NONLOCAL PROBLEM FOR A HYPERBOLIC EQUATION WITH A DOMINANT MIXED DERIVATIVE

#### Abstract

In this article, we consider the Goursat problem with nonlocal integral conditions for a hyperbolic equation with a dominant mixed derivative. Research methods of solvability of classical boundary value problems for partial differential equations cannot be applied without serious modifications. The choice of a research method of solvability of a nonlocal problem depends on the form of the integral condition. In the process of developing methods that are effective for nonlocal problems, integral conditions of various types were identified [1]. The solvability of the nonlocal Goursat problem with integral conditions of the first kind for a general equation with dominant mixed derivative of the second order was investigated in [2]. In our problem, the integral conditions are nonlocal conditions of the second kind, therefore, to investigate the solvability of the problem, we propose another method, which consists in reducing the stated nonlocal problem to the classical Goursat problem, but for a loaded equation. In this article, we obtain conditions that guarantee the existence of a unique solution of the problem. The main instrument of the proof is the a priori estimates obtained in the paper.

**Vestnik Samarskogo Universiteta. Estestvennonauchnaya Seriya / Vestnik of Samara University. Natural Science Series**. 2020;26(4):25-35

### ABOUT SOLVABILITY OF ONE PROBLEM WITH NONLOCAL CONDITIONS FOR HYPERBOLIC EQUATION

#### Abstract

In this article we consider a nonlocal problem with integral condition of the second kind for hyperbolic equation. The choice of a method for investigating problems with nonlocal conditions of the second kind depends on the type of nonintegral terms. In this article we consider the case when the nonintegral term is a trace of required function on the boundary of the domain. To investigate the solvability of the problem we use method of reduction for loaded equation with homogeneous boundary conditions. This method proved to be effective for defining a generalized solution, to obtain apriori estimates and to prove existence of unique generalized solution of the given problem.

### COMPUTER SIMULATION OF CRACK GROWTH. MOLECULAR DYNAMICS METHOD

#### Abstract

The aim of the study is to determine the stress intensity factors using molecular dynamics (MD) method. In the course of the study, a computer simulation of the propagation of a central crack in a copper plate was carried out. The simulation was performed in the LAMMPS (Large-scale Atomic / Molecular Massively Parallel Simulator) software package. A comprehensive study of the influence of geometric characteristics (model dimensions, crack length), temperature, strain rate and loading mixing parameter on the plate strength, crack growth and direction was carried out. The article proposes a method for determining the coefficients of the asymptotic expansion of M. Williams’ stress fields. The analysis of the influence of the choice of points on the calculation of the coefficients and the comparison of the results obtained with the analytical solution are carried out.

### Photoelastic study of a double edge notched plate for determination of the Williams series expansion

#### Abstract

In this work, digital photoelasticity method is applied for assessment of the crack tip linear fracture mechanics parameters for a plate with double edge notches and different other crack configurations. The overarching objective of the study is to obtain the coefficients of the Williams series expansion for the stress and displacement fields in the vicinity of the crack tip by the digital photoelasticity technique for the double edge notched plate. The digital image processing tool for experimental data obtained from the photoelasticity experiments is developed and utilized. The digital image processing tool is based on the Ramesh approach but allows us to scan the image in any direction and to analyse the image after any number of logical operations. In the digital image processing isochromatic fringe analysis, the optical data contained in the transmission photoelastic isochromatics were converted into text file and then the points of isochromatic fringes with minimum light intensity were used for evaluating fracture mechanics parameters. The multi-parameter stress field approximation is used. The mixed mode fracture parameters, especially stress intensity factors (SIF) are estimated for specimen configurations like double edge notches and inclined center crack using the proposed algorithm based on the classical over-deterministic method. The effects of higher-order terms in the Williams expansion were analysed for different cracked specimens. It is shown that the higher order terms are needed for accurate characterization of the stress field in the vicinity of the crack tip. The experimental SIF values estimated using the proposed method are compared with analytical / finite element analysis (FEA) results, and are found to be in good agreement.

### QUANTUM DYNAMICS OF THE CUBIT SYSTEM IN EXTERNAL FIELDS

#### Abstract

A system of two dipole-dipole interacting two-level elements (qubits) in external fields is considered. It is shown that using the coherent states (CS) of the dynamic symmetry group of the SU(2)×SU(2) system, the time evolution can be reduced to the "classical" dynamics of the complex parameters of the CS. The trajectories of the CS are constructed and the time dependences of the probability of finding qubits at the upper levels are calculated.