# Vol 24, No 4 (2020)

**Year:**2020**Articles:**12**URL:**https://journals.eco-vector.com/1991-8615/issue/view/3466

## Full Issue

### Non-local problems with an integral condition for third-order differential equations

#### Abstract

The paper is devoted to the study of the solvability of nonlocal problems with an integral variable $t$ condition for the equations $$u_{tt}+(\alpha\frac{\partial}{\partial t}+\beta)\Delta u=f(x,t)$$($\alpha$, $\beta$ are valid constants, $\Delta$ is Laplace operator by spatial variables). Theorems are proved for the studied problems existence and non-existence, uniqueness and non-uniqueness solutions (having all derivatives generalized by S. L. Sobolev included in the equation).

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(4):607-620

607-620

### Existence of solutions to quasilinear elliptic equations in the Musielak–Orlicz–Sobolev spaces for unbounded domains

#### Abstract

The paper considers the existence of solutions of the Dirichlet problem for nonlinear elliptic equations of the second order in unbounded domains. Restrictions on the structure of quasilinear equations are formulated in terms of a special class of convex functions (generalized $N$-functions). Namely, nonlinearities are determined by the Musilak–Orlicz functions such that the complementaries functions obeys the condition $ \Delta_2 $. The corresponding Musielak–Orlicz–Sobolev space does not have to be reflexive.This fact is a significant problem, since the theorem for pseudomonotone operators is not applicable here. For the class of equations under consideration, the proof of the existence theorem is based on an abstract theorem for additional systems. An important tool which allowed to generalize available results on the existence of solutions of the considered equations for bounded domains to the case of unbounded domains is an embedding theorem for Musielak–Orlicz–Sobolev spaces.Thus, in this paper, we find conditions on the structure of quasilinear equations in terms of the Musielak–Orlicz functions sufficient for the solvability of the Dirichlet problem in unbounded domains.In addition, we provide examples of equations which demonstrate that the class of nonlinearities considered in the paper is wider than non-power nonlinearities and variable exponent nonlinearities.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(4):621-643

621-643

### On alternating and bounded solutions of one class of integral equations on the entire axis with monotonic nonlinearity

#### Abstract

The paper is devoted to the study of the existence and analysis of the qualitative properties of solutions for one class of integral equations with monotonic nonlinearity on the entire line. The indicated class of equations arises in the kinetic theory of gases. The constructive theorems of the existence of bounded solutions are proved, and certain qualitative properties of the constructed solutions are studied. At the end of the paper, specific applied examples of these equations are given.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(4):644-662

644-662

### The influence of the dimensions of the surface hardening region on the stress-strain state of a beam with a notch of a semicircular profile

#### Abstract

The influence of a size of the surface-plastic hardening region on the stress-strain state of a beam with a notch of a semicircular profile is investigated. The problem is reduced to a boundary value problem of fictitious thermoelasticity with the initial (plastic) deformations modeled by temperature anisrotropic deformations in an inhomogeneous temperature field. The solution is based on the finite element method. For model calculations, experimental data on the distribution of residual stresses in a smooth beam made of EP742 alloy after ultrasonic mechanical hardening were used as initial information. A variative numerical analysis of the effect of the notch radius and the size of the hardening zone of the beam face on the distribution of the components of the residual stress tensor in the smallest section from the bottom of the concentrator is carried out. It is shown that when the hardening zone is more than 16–20 % of the entire face area, the stress-strain state in the smallest section is practically stabilized. It was established that if the radius of the semicircular notch is less than the thickness of the hardened layer (the material compression area), an increase (in modulus) of the normal longitudinal component of the residual stress tensor occurs, and if the radius of the notch is greater than the thickness of the hardened layer, then a decrease (in modulus) of this value is observed in comparison with a similar component for a smooth reinforced beam for all values of the hardening zone more than 16–20 % of the entire face area of the beam. An experimental verification of the developed numerical method based on the finite element method for a beam with a fully hardened face is performed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(4):663-676

663-676

### The coupled non-stationary thermo-electro-elasticity problem for a long hollow cylinder

#### Abstract

A new closed solution of the coupled non-stationary thermo-electro-elasticity problem for a long piezoelectric ceramic radially polarized cylinder is constructed while satisfying the boundary conditions of thermal conductivity of the 1st and 3rd kind on its front surfaces. The case when the rate of change of the temperature field does not affect the inertia characteristics of the elastic system is considered.This makes it possible to include linear equations of equilibrium, electrostatics, and thermal conductivity with respect to the radial component of the displacement vector, electric potential, and also the function of changing the temperature field in the initial calculated relations of the problem under consideration.In the calculations, the classical Fourier law of thermal conductivity is used.To solve the problem, the mathematical apparatus of incomplete separation of variables is used in the form of a generalized biorthogonal finite integral transformation based on the multicomponent relation of the eigenvector functions of two homogeneous boundary value problems. An important point in the procedure of the structural algorithm of this method is the selection of the adjoint operator, without which it is impossible to solve non-self-adjoint linear problems of mathematical physics.The constructed calculated relationships make it possible to determine the stress-strain state, temperature and electric fields induced in a piezoelectric ceramic element under an arbitrary temperature external influence. An analysis of the numerical results makes it possible to determine the cylinder wall thickness at which the electric field leads to a redistribution of the temperature field. It is established that the rate of change in the volume of a piezoceramic body under external temperature influence does not significantly affect the temperature field.The developed calculation algorithm finds its application in the design of non-resonant piezoelectric temperature sensors.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(4):677-691

677-691

### Mathematical modeling of the asteroids' motion belonging to the Apollo and Aten groups

#### Abstract

This article evaluates the accuracy of solutions to differential equations of motion, taking into account relativistic effects obtained on the basis of a new principle of interaction, using the example of studies of the evolution of the orbits of five asteroids.A numerical integration of equations of the asteroids' motion with the initial data referred to different points in time is carried out. Based on a comparison of the results of the study, certain patterns are revealed. At time intervals in the absence of rapprochement of the asteroid with the Earth less than 0.1 au it is possible to apply with equal efficiency the differential equations given in the paper. The loss of accuracy of numerical integration is directly dependent on the magnitude of the rapprochement of the asteroid width the Earth. Due to the fact that in the right sides of the equations of motion we have differences of the coordinates of the asteroid and the planet, with sufficient proximity, the relative accuracy of the coordinates is many times greater than the relative accuracy of the difference. For the studied asteroids, when they approach the Earth, the relative error of the difference in the coordinates of the asteroid and the Earth is approximately 227 to 44900 times higher than the limiting relative error of the coordinates of the asteroid itself. Predicting the motion of Apophis after its close approach to the Earth based on the solution of the equations of motion by modern methods leads to large errors, the reduction of which is possible only by improving the initial data of the elements of the orbits of the asteroid. About the possibility of close approach of Apophis with the Earth on a time interval from April 14, 2029 to January 1, 2100 it can be argued with a certain degree of probability. The results of the research can be generalized to all asteroids of Apollo and Aten groups.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(4):692-717

692-717

### A method for increasing the order of approximation to an arbitrary natural number by the numerical integration of boundary value problems for inhomogeneous linear ordinary differential equations of various degrees with variable coefficients by the matrix method

#### Abstract

The paper includes the well-known matrix method of numerical integration of boundary value problems for inhomogeneous linear ordinary differential equations with variable coefficients, which provides retaining an arbitrary number of Taylor series expansion members of the sought-for solution or, equally, using the Taylor polynomial of arbitrary degree.The difference boundary value problem approximating the differential boundary value problem is divided into two subtasks: the first subtask includes difference equations, in the construction of which the boundary conditions of the boundary value problem were not used. The second subtask includes difference equations, in the construction of which the boundary conditions of the problem were used.Based on the earlier results, the method of increasing the order of approximation of the second subtask per unit, and, consequently, of the entire difference boundary problem as a whole is obtained and tested. The earlier findings are as follows:a) the order of approximation of the first and second subtasks is proportional to the degree of the Taylor polynomial used;b) the order of approximation of the first subtask depends on the parity or oddness of the degree of the Taylor polynomial used. It turned out that when using the degrees of the Taylor polynomial which are equal to $2m{-}1$ and $2m$, the approximation orders of these two subtasks are the same;c) the order of approximation of the second subtask coincides with the order of approximation of the first subtask, if the second subtask does not contain the specified values of any derivatives included in the boundary conditions;d) the presence in the second subtask of at least one derivative value of varying degrees included in the boundary conditions leads to a decrease in the order of approximation per unit in both the second subtask and the entire difference boundary value problem in general.The theoretical conclusions have been confirmed by numerical experiments.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(4):718-751

718-751

### On the Neuber theory of micropolar elasticity. A pseudotensor formulation

#### Abstract

The present paper deals with a pseudotensor formulation of the Neuber theory of micropolar elasticity. The dynamic equations of the micropolar continuum in terms of relative tensors (pseudotensors) are presented and discussed. The constitutive equations for a linear isotropic micropolar solid is given in the pseudotensor form. The final forms of the dynamic equations for the isotropic micropolar continuum in terms of displacements and microrotations are obtained in terms of relative tensors. The refinements of Neuber's dynamic equations are discussed. Those are also considered in the cylindrical coordinate net.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(4):752-761

752-761

### A new class of non-helical exact solutions of the Navier–Stokes equations

#### Abstract

The paper presents a new class of exact solutions for the Navier–Stokes equations. These solutions describe unsteady three-dimensional in velocities and two-dimensional in coordinates for a viscous incompressible fluid flow. The procedure for constructing an exact solution generalizes Trkal's method proposed for studying screw flows. The new class of exact solutions allows to describe non-hecical flows (the velocity vector forms a nonzero angle with the vorticity vector) and fluid flows existing in a finite time.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(4):762-768

762-768

### Static thermal stability of a shallow geometrically irregular shell made of orthotropic temperature-sensitive material

#### Abstract

A flat orthotropic geometrically irregular shell of constant torsion, whose thermomechanical parameters are linearly dependent on temperature, is considered. When the temperature reaches a certain value, the change in the shape of the equilibrium occurs abruptly, which causes a change in the initial geometry of the shell. These temperatures are called critical. For practice, the relationships connecting the critical temperatures with the geometrical and thermomechanical parameters of the geometrically irregular shell are of considerable interest. The solution of the problems of static thermal stability of geometrically irregular shells usually begins with an analysis of their initial momentless state. Tangential forces caused by shell heating are defined as solutions of a system of singular differential equations of momentless thermoelasticity. These efforts are contained in the Brian or Reissner forms in the equations of static thermal stability and the further solution of the problem essentially depends on their structure. In this paper, the solution of singular momentless thermoelasticity is found by elementary functions. Using the method of displacement functions, the equations of moment thermoelasticity, written in the components of the displacement field, are reduced to a single singular differential equation in partial derivatives of the eighth order depending on the temperature, which is assumed to be constant. The solution is written as a double trigonometric series. The coefficients of the series, based on the Galerkin procedure, are determined as solutions to a linear homogeneous algebraic system of equations. From the equality to zero of the determinant of this system, an algebraic equation of the fifth degree is obtained for the relative critical temperature. The smallest positive real root of which is the desired temperature. A quantitative analysis of the influence of the geometrical and thermomechanical parameters of the geometrically irregular shell on the value of the critical temperature is carried out.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(4):769-779

769-779

### The invariant of stagnation streamline for a stationary vortex flow of an ideal incompressible fluid around a body

#### Abstract

In this study, using the Euler equations we investigate the stagnation streamline in the general spatial case of a stationary incompressible fluid flow around a body with a smooth convex bow. It is assumed that in some neighborhood of the stagnation point everywhere, except for the stagnation point, the fluid velocity is nonzero; and that all streamlines on the surface of the body in this neighborhood start at the stagnation point. Here we prove the following three statements. 1) If on a certain segment of the vortex line the vorticity does not turn to zero, then the value of the fluid velocity in this segment is either identically equal to zero or nonzero at all points of the segment of the vortex line (velocity alternative). 2) The vorticity at the stagnation point is equal to zero. 3) On the stagnation streamline, the vorticity is collinear to the velocity, and the ratio of the vorticity to the velocity is the same at all points of the stagnation streamline (invariant of the stagnation streamline). On the basis of the obtained results, it is concluded that if in the free stream the velocity and vorticity are not collinear, a stationary flow around the body is impossible. However, the question of vorticity at the stagnation point in plane-parallel flows remains open, because the accepted assumption that the velocity of the fluid differs from zero in some neighborhood of the stagnation point everywhere, except for the stagnation point itself, excludes plane-parallel flows from consideration.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(4):780-789

780-789

### A method for replicating exact solutions of the Euler equations for incompressible Beltrami flows

#### Abstract

In the paper, Beltrami flows or helical flows are flows in which the vorticity and velocity vectors are collinear, and the proportionality coefficient between these vectors is nonzero and is the same at all points of the flow. A method is proposed that allows using known helical solutions to obtain new helical solutions of the Euler equations for an incompressible fluid. Some of these new solutions cannot be obtained by the known methods of replicating solutions by shifting and rotating the coordinate system, symmetry, scaling, cyclic permutation of the velocity and coordinate components, vector summation. The new replication method is applied to such parametric families of exact solutions in which the proportionality coefficient between velocity and vorticity remains unchanged for different values of the parameter. The essence of the method is that for such families the derivative of the velocity with respect to the parameter is also the helical velocity. The sequential differentiation of the speed of a new solution with respect to a parameter gives an endless chain of new exact solutions.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(4):790-798

790-798