# Vol 26, No 2 (2022)

**Year:**2022**Articles:**11**URL:**https://journals.eco-vector.com/1991-8615/issue/view/5153**DOI:**https://doi.org/10.14498/vsgtu/v226/i2

## Full Issue

## Other Manuscript Types (Biographies, Letter to the Editor, Commentary, and etc.)

### On the 60th anniversary of Prof. Yuri N. Radayev

#### Abstract

February 10, 2022 the famous scientist in mechanics of solids and applied mathematics, teacher, organizer of science and higher education in Russia Yuri N. Radayev is celebrating his 60th anniversary. Yuri N. Radayev is known as a prominent scientist in the field of mechanics and applied mathematics. The principal directions of his academic activity are the Mathematical Theory of Plasticity, Fracture Mechanics, the Theory of Cracks and Microdamages, Coupled Hyperbolic Thermoelasticity and Thermomechanics, Micropolar Elasticity, Mechanics of Granular Solids, Mechanics of Growing Solids. In this biographical background we discuss the scientific and educational work of Prof. Yuri N. Radayev, give an information on his achievements and a list of his main publications.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(2):207-221

## Differential Equations and Mathematical Physics

### Convergence of approximate solutions by heat kernel for transport-diffusion equation in a half-plane

#### Abstract

In this paper, by using the heat kernel and the transport operator on each step of time discretization, approximate solutions for the transport-diffusion equation on the half-plane ${\mathrm{\mathbb{R}}}_{+}^{2}$ are constructed, and their convergence to a function which satisfies the transport-diffusion equation and the initial and boundary conditions is proved. These approximate solutions can be considered as a deterministic version of (the approximation of) the stochastic representation of the solution to parabolic equation, realized by the relationship between the heat kernel and the Brownian motion. But as they are defined only by an integral operator and transport, their properties and their convergence are proved without using probabilistic notions. The result of this paper generalizes that of recent papers about the convergence of analogous approximate solutions on the whole space ${\mathrm{\mathbb{R}}}^{n}$. In case of the half-plane, it is necessary to elaborate (not trivial) estimates of the smoothness of the approximate solutions influenced by boundary condition.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(2):222-258

### Dezin problem analog for a parabolic-hyperbolic type equation with periodicity condition

#### Abstract

In this paper we consider an inhomogeneous second-order parabolic-hyperbolic mixed type equation, represented as one-dimensional heat equation in the parabolic part and the one-dimensional wave equation in the hyperbolic part. For the equation, an analog of the Dezin problem is investigated, which means to find a solution to the equation that satisfies inner-boundary condition, relating the value of the desired function on the equation type change line to the value of the normal derivative on the hyperbolicity region boundary, and inhomogeneous periodicity nonlocal boundary conditions. A substitution is given that allows us to reduce the problem to an equivalent one and, without losing generality, restrict ourselves to investigate the problem with homogeneous conditions for an inhomogeneous equation.

The solution is constructed as the Fourier series on the orthonormal system of eigenfunctions of the corresponding one-dimensional spectral problem. A criterion for the solution uniqueness to the problem is established.

In case when the uniqueness criterion is violated, an example of a nontrivial solution to a homogeneous problem is given, and a necessary and sufficient condition for the existence of a solution to an inhomogeneous problem is obtained.

In justifying the solution existence, the problem of small denominators in the sum of the series with respect to the ratio of the rectangle sides in the hyperbolic part of the domain. An estimate of the denominator separation from zero under certain conditions with respect to the problem parameters is obtained. This estimate allows us to substantiate the uniform convergence of the series and their derivatives up to the second-order inclusive under certain conditions for given functions.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(2):259-272

### An initial boundary value problem for a partial differential equation of higher even order with a Bessel operator

#### Abstract

In present paper, an initial-boundary value problem is formulated in a rectangle for a higher even order partial differential equation with the Bessel operator. Applying the method of separation of variables to the considered problem a spectral problem is obtained for an ordinary differential equation of higher even order. The self-adjointness of the last problem is proved, which implies the existence of the system of its eigenfunctions, as well as the orthonormality and completeness of this system. The uniform convergence of some bilinear series and the order of the Fourier coefficients, depending on the found eigenfunctions, is investigated. The solution of the considered problem is found as the sum of the Fourier series with respect to the system of eigenfunctions of the spectral problem. The absolute and uniform convergence of this series, as well as the series obtained by its differentiating, have been proved. The uniqueness of the solution of the problem is proved by the method of spectral analysis. An estimate is obtained for the solution of the problem which implies the continuous dependence of the solution on the given functions.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(2):273-292

## Mechanics of Solids

### Orientation nature of the damage-memory effect under triaxial cyclic nonproportional compression of a sandstone

#### Abstract

The paper describes the mechanisms and conditions for the damage-memory effect (Kaiser effect) in rocks subjected to a three-dimensional nonproportional cyclic loading with changes in the rocks' shape and orientation of the Lamé-ellipsoid. The experiments with the cubic samples taken from polymictic sandstone were conducted on Triaxial Independent Loading Testing System with continuous recording of an acoustic emission signals. The results of a nonproportional triaxial compression under the developed protocol, it is 9-cycle loading program, have shown that a dominate mechanism of the damage-memory effect in each ensemble of cracks (vectored differently) is the development of micro-cracks of opening fracture mode oriented subnormally to the minimum main stress. It was found that the Kaiser damage-memory effect is detected not so much to the fact of opening cracks, friendly oriented, as to a discrete growing (increase of length) of already existing and newly emerging micro-cracks. The obtained results can be considered as a trigger for models development oriented to strain and destruction of rocks, taking into account the anisotropic nature of damage accumulation.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(2):293-310

## Mathematical Modeling, Numerical Methods and Software Complexes

### The characteristic Cauchy problem of standard form for describing the outflow of a polytropic gas into vacuum from an obligue wall

#### Abstract

The initial-boundary value problem for the system of equations of gas dynamics, the solution of which describes the expansion of a polytropic gas into vacuum from an oblique wall in the space of self-similar variables *x*/*t*, *y*/*t* in the general inconsistent case, is reduced to the characteristic Cauchy problem of standard form in the space of new independent variables ϑ, ζ. Equation ϑ=0 defines the characteristic surface through which the double wave adjoins the well-known solution known as the centered Riemann wave. Equation ζ=0 means that an oblique wall is chosen for the new coordinate axis, on which the impermeability condition is satisfied. For this new initial-boundary value problem, in contrast to the well-known solution of a similar problem obtained by S. P. Bautin and S. L. Deryabin in the space of special variables, the theorem of existence and uniqueness for the solution of the system of equations of gas dynamics in the space of physical self-similar variables in the form of a convergent infinite series was proved. An algorithm is described to build the series coefficients.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(2):322-338

### Evaluation of influence of turbulence models on the vortex formation processes modeling in wind power

#### Abstract

The paper studies the results of mathematical modeling of the external flow of Siemens 3D model SWT–3.6–120 (B52 air foil) horizontal axis wind turbine (HAWT), using the Navier–Stokes equations averaged by Reynolds (RANS) closed by k−ε, k−ω Shear Stress Transport (SST) and Eddy Viscosity Transport (EVT) turbulence models. The task of correct determination of the wind speed vector deviation angle over the nacelle of the HAWT is required by operation of the yawing system, which determines in turn the efficiency of the entire turbine. The Struhal number was chosen as a comparison criterion, defined for the transverse flow around the cylinder, describing the frequency of the formation of vortex structure behind the butt part of the blade of the HAWT. The calculated area consists of 3 million tetrahedral volumes with prismatic layer on the surface of the nacelle, using local grinding. The place of flow direction parameters registration is located at a height of 3 m above the nacelle and at a distance of 8 m from the blade shank, which corresponds to the standard location of the weather vane. The analysis of the obtained results showed that the k−ε and EVT turbulence models describe the flow parameters over the HAWT nacelle in almost the same way, but the EVT model represents just one differential equation, thereby it is preferable by the computational cost criterion. Also, one of the advantages of one-parameter turbulence model (EVT model) is a smaller number of closing semi-empirical constants, the analysis of which allows the expanding of the engineering techniques scope for the modeling of turbulent processes in solving the practical problems related to the design of control systems for the wind turbines, increasing their efficiency.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(2):339-354

## Short Communications

### Inverse source problem for an equation of mixed parabolic-hyperbolic type with the time fractional derivative in a cylindrical domain

#### Abstract

This article is devoted to the study of an inverse source problem for a mixed type equation with a fractional diffusion equation in the parabolic part and a wave equation in the hyperbolic part of a cylindrical domain. The solution is obtained in the form of Fourier–Bessel series expansion using an orthogonal set of Bessel functions. The theorems of uniqueness and existence of a solution are proved.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(2):355-367

### Periodic solutions for an impulsive system of integro-differential equations with maxima

#### Abstract

A periodical boundary value problem for a first-order system of ordinary integro-differential equations with impulsive effects and maxima is investigated. The obtained system of nonlinear functional-integral equations and the existence and uniqueness of the solution of the periodic boundary value problem are reduced to the solvability of the system of nonlinear functional-integral equations. The method of successive approximations in combination with the method of compressing mapping is used in the proof of one-valued solvability of nonlinear functional-integral equations. We define the way with the aid of which we could prove the existence of periodic solutions of the given periodical boundary value problem.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(2):368-379