# Vol 26, No 3 (2020)

**Year:**2020**Articles:**5**URL:**https://journals.ssau.ru/est/issue/view/478

## Статьи

### ON THE SOLVABILITY OF SOME BOUNDARY VALUE PROBLEMS WITH INVOLUTION

#### Abstract

This article is devoted to the study of the solvability of some boundary value problems with involution.

In the space Rn, the map Sx=−x is introduced. Using this mapping, a nonlocal analogue of the Laplace operator is introduced, as well as a boundary operator with an inclined derivative. Boundary-value problems are studied that generalize the well-known problem with an inclined derivative. Theorems on the existence and uniqueness of the solution of the problems under study are proved. In the Helder class, the smoothness of the solution is also studied. Using well-known statements about solutions of a boundary value problem with an inclined derivative for the classical Poisson equation, exact orders of smoothness of a solution to the problem under study are found.

**Vestnik of Samara University. Natural Science Series**. 2020;26(3):7-16

### ON DECODING ALGORITHMS FOR GENERALIZED REED — SOLOMON CODES WITH ERRORS AND ERASURES

#### Abstract

The article is devoted to the decoding algorithms for generalized Reed — Solomon codes with errors

and erasures. These algorithms are based on Gao algorithm, Sugiyama algorithm, Berlekamp — Massey algorithm (Peterson — Gorenstein —Zierler algorithm). The first of these algorithms belongs to syndrome-free decoding algorithms, the others —to syndrome decoding algorithms. The relevance of these algorithms is that they are applicable for decoding Goppa codes, which are the basis of some promising post-quantum cryptosystems. These algorithms are applicable for Goppa codes over an arbitrary field, as opposed to the well-known Patterson decoding algorithm for binary Goppa codes.

**Vestnik of Samara University. Natural Science Series**. 2020;26(3):17-29

### PROBLEMS OF DIFFERENTIAL AND TOPOLOGICAL DIAGNOSTICS. PART 5. THE CASE OF TRAJECTORIAL MEASUREMENTS WITH ERROR

#### Abstract

Proposed work is the fifth work of the cycle on differential and topological diagnostics. The article gives an estimate of the errors method of direction fields in the case of not accurate trajectory measurements, but trajectory measurements with an error are limited by the modulus of a given a smooth function of time, and in case this error is a random variable distributed according to the normal law with fixed parameters. We show that in these more complex cases, you can specify the “best” number of required trajectory measurements, in

which the proposed algorithms of diagnostics will work constructively, and the malfunction will be determined unambiguously.

**Vestnik of Samara University. Natural Science Series**. 2020;26(3):30-39

### DETERMINATION OF THE COEFFICIENTS OF ASYMPTOTIC CRACK—TIP STRESS EXPANSION. MIXED MODE LOADING OF THE PLATE

#### Abstract

The aim of the study is to calculate the coefficients of M. Williams’ asymptotic expansion of stress and

displacement fields using the data of finite element modeling of a plate with an inclined central crack in a uniaxial tension field. In this work, we also simulated the loading of a half-disk with a vertical and oblique notch under conditions of three-point bending. The simulation was carried out in the multifunctional software SIMULIA Abaqus. The paper proposes an algorithm for calculating the coefficients. The program, written in the MAPLE computer algebra system, allows calculating any predetermined number of M. Williams expansion

coefficients (amplitude or scale factors) and uses the values of the stress tensor components at points in the vicinity of the crack and their coordinates as input. The analysis of the influence of the number of calculated coefficients on the accuracy of their determination is carried out. Recommendations on the choice of points for calculating the coefficients are given.

**Vestnik of Samara University. Natural Science Series**. 2020;26(3):40-62

### RESIDUAL STRESSES IN A THERMOELASTIC CYLINDER RESULTING FROM LAYER-BY-LAYER SURFACING

#### Abstract

The article investigates the residual stresses arising in a thermoelastic cylinder as a result of layer-by-layer deposition of material on its lateral surface. Residual stresses are defined as the limiting values of internal stresses developing during the technological process. Internal stresses are caused by incompatible deformations that accumulate in the body as a result of joining parts with different temperatures. For the analysis of internal stresses, an analytical solution of the axisymmetric quasi-static problem of thermoelasticity for a layer-by-layer growing cylinder is constructed. It is shown that the distribution of residual stresses depends

on the scenario of the surfacing process. In this case, the supply of additional heat to the growing body can significantly reduce the unevenness of the temperature fields and reduce the intensity of residual stresses. The most effective is uneven heating, which can be realized, for example, by the action of an alternating current with a tunable excitation frequency. This is illustrated by the calculations performed using the constructed

analytical solution.

**Vestnik of Samara University. Natural Science Series**. 2020;26(3):63-90