# Vol 26, No 4 (2022)

**Year:**2022**Articles:**10**URL:**https://journals.eco-vector.com/1991-8615/issue/view/5155**DOI:**https://doi.org/10.14498/vsgtu/v226/i4

## Full Issue

## Differential Equations and Mathematical Physics

### Boundary value problems for Sobolev type equations of fractional order with memory effect

#### Abstract

Boundary value problems are studied for a one-dimensional Sobolev type integro-differential equation with boundary conditions of the first and third kind with two fractional differentiation operators $\alpha $ and $\beta $ of different orders. Difference schemes of the order of approximation $O({h}^{2}+{\tau}^{2})$ for $\alpha =\beta $ and $O({h}^{2}+{\tau}^{2-\mathrm{max}\{\alpha ,\beta \}})$ are constructed for α≠β. Using the method of energy inequalities, a priori estimates are obtained in the differential and difference interpretations, from which the existence, uniqueness, stability, and convergence of the solution of the difference problem to the solution of the original differential problem at a rate equal to the order of approximation of the difference scheme follow. Numerical experiments were carried out to illustrate the results obtained in the paper.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(4):607-629

### An analogue of the Tricomi problem for a mixed type of quasilinear equation with two lines of degeneracy

#### Abstract

The paper proves the unique solvability of an analog of the Tricomi problem for a quasilinear equation of mixed type with two lines of degeneracy. The class **R _{1}** of generalized solutions in the hyperbolic part of the domain is introduced. The uniqueness of the solution is proved by the method of energy integrals. The existence of a solution is proved by the method of integral equations. The boundary value problem is reduced to an equivalent system of integral equations, the solvability of which is proved using the Schauder principle. As a result, the application of the Schauder principle resulted in the global solvability of the problem under study without any restrictions on the size of the area of the region under consideration and on the value of the given functions.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(4):630-649

### Vibrations of plate with boundary “hinged attachment” conditions

#### Abstract

In the paper, the initial problem for the equation of vibrations of a rectangular plate with boundary conditions of the “hinged attachment” type is studied. An energy inequality is established, from which the uniqueness of the solution of the stated initial-boundary problem follows. The corresponding existence and stability theorems for the solution of the problem in the classes of regular and generalized solutions are proved. The existence of a solution to the problem posed is carried out by the method of spectral analysis and it is constructed as the sum of an orthogonal series over a system of eigenfunctions corresponding to a two-dimensional spectral problem, which is constructed by the method of separation of variables. A complete substantiation of the convergence of the constructed three-dimensional series in the class of regular solutions of the considered equation is given. The generalized solution is defined as the uniform limit of the sequence of regular solutions of the initial boundary value problem.In the paper, the initial problem for the equation of vibrations of a rectangular plate with boundary conditions of the “hinged attachment” type is studied. An energy inequality is established, from which the uniqueness of the solution of the stated initial-boundary problem follows. The corresponding existence and stability theorems for the solution of the problem in the classes of regular and generalized solutions are proved. The existence of a solution to the problem posed is carried out by the method of spectral analysis and it is constructed as the sum of an orthogonal series over a system of eigenfunctions corresponding to a two-dimensional spectral problem, which is constructed by the method of separation of variables. A complete substantiation of the convergence of the constructed three-dimensional series in the class of regular solutions of the considered equation is given. The generalized solution is defined as the uniform limit of the sequence of regular solutions of the initial boundary value problem.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(4):650-671

### An initial-boundary problem for a hyperbolic equation with three lines of degenerating of the second kind

#### Abstract

In present paper, a hyperbolic partial differential equation of the second kind degenerates on the sides and on the base of the rectangle has been considered. For the considered equation, an initial-boundary problem with non-local boundary conditions has been formulated. The uniqueness, existence, and stability of the solution to the stated problem were investigated. The uniqueness of the solution to the problem was proved by the method of energy integrals. The existence of obtained solution was investigated by the Fourier method based on the separation of variables. To do this first, a spectral problem for an ordinary differential equation was investigated, which arises from the formulated problem in virtue of the separation of variables. It was proved that this spectral problem can have only positive eigenvalues. Then, Green's function of the spectral problem has been constructed, and equivalently reduced to an integral Fredholm equation of the second kind with a symmetric kernel. Hence, on the basis of the theory of integral equations, it has been concluded that there is a countable number of eigenvalues and eigenfunctions of the spectral problem. Using the properties of constructed Green's function of the spectral problem, some lemmas, using to prove the existence of a solution to the problem on the uniform convergence of some bilinear series, were proved. Lemmas on the order of the Fourier coefficients of a given function have also been proved. The solution of the considered problem is derived as the sum of a Fourier series with respect to the system of eigenfunctions of the spectral problem. The uniform convergence of this series and the series obtained from it by term-by-term differentiation is proved using the lemmas mentioned above. At the end of the paper, two estimates are obtained for the solution to the problem, one of which is in the space of square summable functions with weight, and the other is in the space of continuous functions. These inequalities imply the stability of the solution in the corresponding spaces.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(4):672-693

## Mechanics of Solids

### Numerical analysis of nonlinear vibrations of a plate on a viscoelastic foundation under the action of a moving oscillating load based on models with fractional derivatives

#### Abstract

**Aim.** In the present paper, nonlinear vibrations of an elastic simply supported plate on a viscoelastic foundation under the action of a moving oscillating load are studied in the case of the internal resonance 1:1 accompanied by the external resonance. The properties of the viscoelastic foundation are given via the generalized Fuss–Winkler model with the damping term described by the standard linear solid model with the Riemann–Liouville fractional derivatives. The external load is presented by linear viscoelastic oscillator based on the Kelvin–Voigt model with a fractional derivative in the case when the viscosity of the oscillator is considered to be small value. The dynamic behavior of the plate is described by a set of nonlinear ordinary differential equations of the second order in time with respect to generalized displacements. **Methods.** To solve the resulting set of equations, the method of multiple time scales is used in combination with the method of expansion of the fractional derivative in a Taylor series. **Results.** Resolving equations for determining of the nonlinear amplitudes and phases of force driven vibrations of the plate are obtained. The governing set of equations allows one to control not only the damping properties of the environment and the foundation by changing the fractional parameters, but also to control the damping parameters of the external load. **Conclusion.** Numerical analysis has shown that in the system “a plate on a viscoelastic foundation + a moving oscillating load”, energy transfer between the interacting vibration modes is observed. A comparison of the results of numerical studies for various values of the external load is presented, and the dependence of the amplitudes of nonlinear vibrations on the values of the fractional parameters of the environment and the foundation is also shown.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(4):694-714

### Creep and long-term fracture of a narrow rectangular membrane inside a rigid low matrix with proportional dependence on the transverse pressure on time

#### Abstract

In this work, we studied the creep and long-term fracture of a narrow rectangular membrane in confined conditions (inside a rigid low matrix) with a proportional dependence on the magnitude of transverse pressure on time.

Deformation of the membrane is considered as a sequence of three stages. At first stage, the membrane is deformed under free conditions until it touches the transverse side of the rigid matrix. At second stage, the membrane is deformed when it touches the transverse wall of the matrix until it touches its longitudinal walls. At third stage, the membrane is already deformed while simultaneously touching the longitudinal and transverse walls of matrix.

The study is carried out under two types of contact conditions: 1) ideal sliding of the membrane along the walls of the matrix; 2) sticking of the membrane to the walls of the matrix.

The analysis of the gradual long-term fracture of the membrane is carried out using the kinetic theory of creep by Yu. N. Rabotnov, while the parameter of material damage in this problem has a scalar character.

The obtained equations are used to analyze the creep and long-term fracture of a membrane made of 2.15Cr-1Mo steel, which is deformed under variable transverse pressure at a temperature of 600°C until its destruction.

As a result of solving the system of constitutive and kinetic equations, the values of the damage parameter accumulated during each stage of deformation, as well as the time to fracture of the membrane, are obtained. In the case of membrane fracture at the first stage of deformation, the time to fracture at the first stage does not depend on the type of contact conditions, and in the case of membrane fracture at the second and third stages of deformation, the time to fracture in the case of ideal slip is not less than in the case of sticking.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(4):715-737

## Mathematical Modeling, Numerical Methods and Software Complexes

### Comparison of the orbital elements of major planets, the Moon and the Sun using various mathematical models on the time interval with 1600 to 2200

#### Abstract

An analysis of the accuracy of the orbital elements obtained according to the coordinates and components of the velocities, found using the coefficients of the Chebyshev polynomials of the DE405 planetary catalog, is carried out. We compared the elements of orbital elements in the time interval from 1600 to 2200 years found using the DE405 catalog and obtained by numerical integration of the equations of motion based on the interaction of moving material bodies with the surrounding space. On the example of the numerical integration of the Moon motion equations, the advantage of using the equations of motion based on the interaction of moving material bodies with the surrounding space is shown in comparison with relativistic equations. Based on a comparison of the elements of Mercury's orbits, found by coordinates obtained by solving equations based on the interaction of moving material bodies with the surrounding space, and obtained using the DE405 catalog, it is shown that the orbital elements practically coincide on a given interval time. The maximum discrepancy in the mean anomaly at the end of the integration interval is less than 1′′ (second). The discrepancies of the secular displacements of perihelions for Mercury, Venus, Earth + Moon and Mars were determined, the values of which for DE405 are respectively: 43.08′′, 8.4′′, 3.83′′ and 1.14′′. It is shown that the errors of the secular displacements of the perihelions of the planets Mercury, Venus, the barycenter of the Earth + Moon and Mars obtained using the DE405 catalog take the following values: 0′′, 6.06′′, 3.83′′ and 1.08′′. For the outer planets: Jupiter, Saturn, Uranus, Neptune and the dwarf planet Pluto, on the basis of the considered comparisons of various equations of motion, no discrepancies in the orbital elements were found. Based on the studies carried out, it is shown that the use of harmonic coordinates in relativistic equations when creating the DE405 catalog is justified only for Mercury and the outer planets: Jupiter, Saturn, Uranus, Neptune and the dwarf planet Pluto.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(4):738-763

### Solution of the Dorodnitsin-Ladyzhensky problem

#### Abstract

The article is devoted to a rigorous proof of the statement that entropy takes the maximum value on the surface of a body with a blunted nose, streamlined by a supersonic flow, in the presence of a plane of symmetry of the flow. This is obvious for bodies of rotation at zero angle of attack, and it is established by numerical calculations and experimentally at non-zero angles of attack. The proof boils down to the justification that the leading streamline (the current line crossing the shock along the normal) ends on the body. In other words, that the leading streamline and the stagnation line are coincide. Such a proof was obtained by G. B. Sizykh in 2019 for the general spatial case (not only for flows with a plane of symmetry). This rather complicated proof is based on the Zoravsky criterion, which only a narrow circle of specialists has experience using, and is based on the assumption of the continuity of the second derivatives of density and pressure. In this paper, for the practically important case of flows with a plane of symmetry (in particular, the flow around bodies of rotation at a non-zero angle of attack), an original simple proof is proposed, for which the continuity of only the first derivatives of the density and pressure fields is sufficient and it is not necessary to use the Zoravsky criterion.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(4):764-776

## Short Communications

### The non-uniaxial creep under complex loading

#### Abstract

Based on the model of incomplete reversibility of creep deformation, constitutive equations for the non-uniaxial stress state of metals under complex loading paths are proposed. The tensors of the viscoelastic, viscoplastic, and viscous components of the creep deformation are assumed to develop independently. The deformation kinetics is associated with the initial and deformation anisotropy. The measure of creep intensity for initially orthotropic materials is the equivalent stress introduced by Hill. In this case, the similarity of the stress and strain deviators is not required. The nature of the anisotropy of the deformation is associated with the value of the viscoplastic component of the deformation in the direction of the principal axes of the stress tensor. A superposition of the initial and deformation anisotropy is assumed. Samples made of 3KhV4SF tool steel and EI437B heat resistant alloy were tested, which are initially isotropic materials. The rheological coefficients of 3KhV4SF steel and EI437B alloy were calculated from the results of the uniaxial tension test samples at various levels of initial stresses. A comparative analysis of the forecast under complex loading according to the proposed equations with the test results was carried out.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(4):777-788

### Mathematical prediction of the probability of particle collisions during detonation spraying

#### Abstract

The paper presents methods of mathematical prediction of the probability of collision of particles of dissimilar materials in the process of detonation spraying of composite coatings. As a consequence of different properties of initial powder materials (mass, aerodynamic resistance), quality indicators of composite coatings are determined not only with the motion parameters of the particles but with their mutual position in the flow of the detonation products. In the case of the use of reactive components, the interaction of molten particles in the flow can lead to chemical reactions, formation of new materials on the substrate, heterogeneous structure of the coating, and deterioration of its strength and adhesive properties. A preliminary forecast of the probability of collision of particles before contact with the surface of the product makes it possible to conclude before conducting full-scale tests that high-quality coating indicators have been obtained.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(4):789-801