# Vol 24, No 1 (2020)

**Year:**2020**Articles:**12**URL:**https://journals.eco-vector.com/1991-8615/issue/view/2520

## Full Issue

### Quantum evolution as a usual mechanical motion of peculiar continua

#### Abstract

Quantum particles are considered as continuous media having peculiar properties. These properties are formulated so that all main quantum mechanics postulates can be strictly derived from them. A deterministic description of the process of position measurement is presented. The mechanism of occurrence of randomness in the measurement process is shown and the Born rule is derived. A realistic interpretation of the wave function as a component of a peculiar variable force acting on the apparatus is introduced, and the wave equation is derived from the continuity equation of the peculiar continuum. The deterministic view on the phenomena of the microcosm allows us to eliminate the limitations caused by the uncertainty principle and to describe dynamically those processes that cannot be considered using conventional quantum mechanics.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(1):7-21

7-21

### The energy transfer velocity by a plane monochromaticelectromagnetic wave through a layer of matter

#### Abstract

Stationary problems for the diffraction (tunneling) of a plane electromagnetic wave through a layer of matter with dielectric properties, as well as a quantum particle tunneling through a rectangular potential barrier are considered. It is shown that there are no superluminal motions, and the transit time is always longer when the wave passes the structure at the speed of light.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(1):22-40

22-40

### Asymptotic estimates of the difference of products of Bessel functions by the integral of these functions

#### Abstract

In the study of direct and inverse problems of finding the right-hand side of degenerate equations of mixed type with different boundary conditions, the problem arises of establishing asymptotic estimates for the differences of the products of cylindrical functions by the integral of these functions. Previously, on the basis of the established new formula for finding the finite binomial sum, the differences between the products of cylindrical functions and a definite integral of these functions are calculated through a generalized hypergeometric function. Using the asymptotic formula for large values of the argument for the generalized hypergeometric function, asymptotic estimates are established for large values of the parameter for the indicated differences of the Bessel functions of the first and second kind, as well as for modified Bessel functions.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(1):41-55

41-55

### Analytical solution of elastostatic problems of a simply connected body loaded with nonconservative volume forces: theoretical and algorithmic support

#### Abstract

The possibility of constructing a full-parametric analytical solution of the stress-strain state problem for the body caused by the influence of volumetric forces is studied. In the general case of Cesaro, the displacements at each point of the body are determined through the volume forces by an integral expression with a singular nucleus. Therefore, with an arbitrary shape of the body, its elastic state can be constructed only numerically. A strict analytical solution is written in the classical version, corresponding to the potential forces. These forces are traditional objects of mechanics, but their list is quite limited. The current level of development of science and technology in the world requires the use of forces of an arbitrary nature, which can be generated both at the level of molecular interaction, and the interaction of electromagnetic fields inside the body. They certainly are not conservative. In addition, the use of perturbation methods in solving nonlinear elastostatic problems and thermoelasticity problems creates, at each iteration of the asymptotic approximation, artificially generated volume forces of a polynomial nature or forces fairly accurately approximated by polynomials. The ability to write out strict or highly accurate private decisions during the iteration provides an invaluable service to the calculator. New method of constructing a strict solution of the problem about the corresponding elastic state of the body for a very wide range of forces, approximated by polynomials from spatial coordinates or, even for a narrower class- polynomial forces, is formed. It is based on the isomorphism of Hilbert spaces of forces of this kind and their corresponding elastic states (sets of displacements, deformations, stresses).The existence theorem of isomorphic countable bases of these spaces is proved, and algorithms for their filling are constructed. The particular solution of the problem about the elastic field from polynomial forces is constructed by decomposition of a given load on an orthonormal basis, written simply in the final form, and in the analytical form. The correction from the particular solution is made to the boundary conditions of the homogeneous elasticity problem for the body, after which its solution is constructed. Computational approaches, oriented to computer algebra, provide analytical form of solution. A convenient variant of this approach is the method of boundary states (MBS), which has a number of advantages over widely used numerical (finite elements, boundary elements, finite differences, etc.) and one significant drawback: the MBS computational complex has not received a final completion. The advantages of MBS are briefly stated and its laconic description is given. The use of the MBS approach makes it possible to write out a full-parametric form of solutions for bodies of arbitrary geometric shape. MBS is used to construct a solution of the problem of linear-elastic flattened spheroid, loaded with a self-balanced system of volumetric forces. The solution was constructed for two variants of loading, namely potential, non-potential forces. The analytical version of the solution is given only for the displacement field (other characteristics of the elastic state are easily written out through the defining relations).Certain interest is the graphic illustration of stress fields, made at fixed values of parameters.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(1):56-73

56-73

### Elastic-plastic analysis of rotating solid shaft by maximumreduced stress yield criterion

#### Abstract

An elasto-plastic rotating solid cylinder under plane strain condition is investigated. The analysis is based on infinitesimal strain theory, maximum reduced stress yield criterion, its associated flow rule and perfectly plastic material behavior. It is assumed that angular velocity is monotonically increasing from 0 to the maximum value and then is monotonically reducing down to 0. In this investigation both loading and unloading phases are considered. It is assumed that angular velocity varies slowly with time, so angular acceleration can be neglected. Under above mentioned assumptions, there is only one non-trivial equilibrium equation in a cylinder. It is established that with increasing angular velocity four plastic regions appear in a cylinder. The last one forms at angular velocity which exceeds fully-plastic limit. Stresses image points of plastic regions lie on different sides and corners of yield surface. As the angular speed decreases, the whole cylinder behaves elastically again. At particular value of angular velocity secondary plastic flow may starts at the center of cylinder. Replasticization is possible only for sufficiently high maximum angular speed and the entire cylinder may be replasticized. Four secondary plastic regions may appear in the cylinder under unloading. The stresses image points in primary and secondary regions lie on opposite sides and corners of yield surface. In the present analysis it is assumed that the entire cylinder becomes replasticized just at stand-still. In this case only two secondary plastic regions emerge. Exact solutions for all stages of deformation are obtained. The systems of algebraic equations for determination of integration constants and border radii are formulated. The obtained results are illustrated by the distributions of stresses and plastic strains in the cylinder rotating at different speeds. Presented solutions are compared with known analytical solutions based on Tresca's criterion.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(1):74-94

74-94

### Numerical modeling of eccentric cylindrical shells partially filled with a fluid

#### Abstract

The paper is devoted to a numerical analysis of the dynamic behavior of horizontally oriented eccentric shells, interacting with a flowing fluid, which completely or partially fills the annular gap between them. The solution to the problem is developed in a three-dimensional formulation using the finite element method. When modeling elastic solids, we proceed from the assumption that their curved surface is accurately approximated by a set of plane segments, in which the strains are determined using the relations of the classical theory of plates. The motion of an ideal compressible fluid is described by the wave equation, which, together with the impermeability condition and the corresponding boundary conditions, is transformed using the Bubnov–Galerkin method. The mathematical formulation of the dynamic problem of thin-walled structures is based on the variational principle of virtual displacements. The assessment of stability is based on the calculation and analysis of complex eigenvalues of a coupled system of equations. The verification of the model is accomplished with reference to a quiescent fluid by comparing the obtained results with the known solutions. The influence of the size of the annular gap and the level of its filling with a fluid on the hydroelastic stability threshold of rigidly clamped shells is analyzed at different values of shells eccentricity. It has been shown that for eccentric shells, a decrease in the level of filling leads to an increase of the stability limits. The dependence of the critical flow velocity on the deviation of the inner shell from concentricity has been established.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(1):95-115

95-115

### A priori error estimates of the local discontinuous Galerkin method on staggered grids for solving a parabolic equation for the homogeneous Dirichlet problem

#### Abstract

In this paper, we present a priori error analysis of the solution of a homogeneous boundary value problem for a second-order differential equation by the Discontinuous Galerkin method on staggered grids. The spatial discretization is constructed using an appeal to a mixed finite element formulation. Second-order derivatives cannot be directly matched in a weak variational formulation using the space of discontinuous functions. For lower the order, the components of the flow vector are considered as auxiliary variables of the desired second-order equation. The approximation is based on staggered grids. The main grid consists of triangles, the dual grid consists of median control volumes around the nodes of the triangular grid. The approximation of the desired function is built on the cells of the main grid, while the approximation of auxiliary variables is built on the cells of the dual grid. To calculate the flows at the boundary between the elements, a stabilizing parameter is used. Moreover, the flow of the desired function does not depend on auxiliary functions, while the flow of auxiliary variables depends on the desired function. To solve this problem, the necessary lemmas are formulated and proved. As a result, the main theorem is formulated and proved, the result of which is a priori estimates for solving a parabolic equation using the discontinuous Galerkin method. The main role in the analysis of convergence is played by the estimate for the negative norm of the gradient. We show that for stabilization parameter of first order, the $L^2$-norm of the solution is of order $k+{1}/{2}$, if stabilization parameter of order $h^{-1}$ is taken, the order of convergence of the solution increases to $k+1$, when polynomials of total degree at least $k$ are used.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(1):116-136

116-136

### Numerical integration by the matrix method and evaluation of the approximation order of difference boundary value problems for non-homogeneous linear ordinary differential equations of the fourth order with variable coefficients

#### Abstract

The use of the second degree Taylor polynomial in approximation of derivatives by finite difference method leads to the second order approximation of the traditional grid method for numerical integration of boundary value problems for non-homogeneous linear ordinary differential equations of the second order with variable coefficients.The study considers a previously proposed method of numerical integration using matrix calculus which didn’t include the approximation of derivatives by finite difference method for boundary value problems of non-homogeneous fourth-order linear ordinary differential equations with variable coefficients. According to this method, when creating a system of difference equations, an arbitrary degree of the Taylor polynomial can be chosen in the expansion of the sought-for solution of the problem into a Taylor series.In this paper, the possible boundary conditions of a differential boundary value problem are written both in the form of derived degrees from zero to three, and in the form of linear combinations of these degrees. The boundary problem is called symmetric if the numbers of the boundary conditions in the left and right boundaries coincide and are equal to two, otherwise it is asymmetric.For a differential boundary value problem, an approximate difference boundary value problem in the form of two subsystems has been built. The first subsystem includes equations for which the boundary conditions of the boundary value problem were not used; the second one includes four equations in the construction of which the boundary conditions of the problem were used.Theoretically, the patterns between the order of approximation and the degree of the Taylor polynomial were identified.The results are as follows:a) the approximation order of the first and second subsystems is proportional to the degree of the Taylor polynomial used;b) the approximation order of the first subsystem is two units less than the degree Taylor polynomial with its even value and three units less with its odd value;c) the approximation order of the second subsystem is three units less than the degree Taylor polynomial regardless of both even-parity or odd-parity of this degree, and the degree of the highest derivative in the boundary conditions of the boundary value problem.The approximation order of the difference boundary value problem with all possible combinations of boundary conditions is calculated.The theoretical conclusions are confirmed by numerical experiments.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(1):137-162

137-162

### The splitting of Navier–Stokes equations for a class of axisymmetric flows

#### Abstract

In the framework of the Navier–Stokes equations, unsteady axisymmetric flows of a homogeneous viscous incompressible fluid, in which the axial and circumferential velocities depend only on radius and time are considered, and the radial velocity is zero. It is shown that the velocity of such flows is the sum of the velocities of two flows of a viscous incompressible fluid: axial flow (radial and circumferential velocities are zero) and circumferential flow (radial and axial velocities are zero). Axial and circumferential movements occur independently, without exerting any mutual influence. This allows us to split the boundary value problems for the type of flows under consideration containing three unknown functions (pressure, circumferential and axial velocities) into two problems, each of which contains two unknown functions (pressure and one of the velocity components). In this case, the sum of pressures of the axial and circumferential flow will be the pressure of the initial flow. The discovered possibility of splitting allows using known solutions to replenish the “reserves” of axial and circumferential exact solutions. These solutions, in its turn, can be summed in various combinations and, as a result, give the velocities and pressures of new exact solutions of the Navier–Stokes equations.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(1):163-173

163-173

### On extension of the domain for analytical approximate solution of one class of nonlinear differential equations of the second order in a complex domain

#### Abstract

In previous research the authors have implemented the investigation of one class of nonlinear differential equations of the second order in the neighborhood of variable exceptional point. The authors have proven the following: the existence of variable exceptional point, theorem of the existence and uniqueness of solution in the neighborhood of variable exceptional point. The analytical approximated solution in the neighborhood of variable exceptional point was built. The authors researched the influence of disturbance of variable exceptional point on an approximated solution. The results obtained for the real domain have been extended to the complex domain $|z|<|\tilde z^*|\leqslant |z^*|$, where $z^*$ is precise value of variable exceptional point, $\tilde z^*$ is approximate value of variable exceptional point. In the present paper, the authors have carried out the investigation of analytical approximate solution of the influence of disturbance of variable exceptional point in the domain $|z|>|\tilde z^*|\geqslant |z^*|$, giving special attention to change of direction of movement along the beam towards the origin of coordinates of a complex domain. These researches are actual due to the variable exceptional point pattern (even fractional degree of critical pole). The received results are accompanied by the numerical experiment and complete the investigation of analytical approximated solution of the considered class of nonlinear differential equations in the neighborhood of variable exceptional point depending on the direction of movement along the beam in a complex domain.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(1):174-186

174-186

### The nonlocal problem for a non-stationary third order composite type equation with general boundary condition

#### Abstract

We consider a nonlocal boundary value problem for non-stationary composite type equation of the third order. The values of function and its derivatives up to the second order on the boundary are given as a linear combination. The initial conditions are nonlocal. We prove the unique solvability for this problem. In proving the problem solution uniqueness we use the method of energy integrals and the theory of quadratic forms. For the problem solution construction we use the potential theory and Volterra integral equations. Some asymptotic properties of the fundamental solutions of the equation are studied.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(1):187-198

187-198

### Integro-differential equations of the second boundary value problem of linear elasticity theory. Communication 2. Inhomogeneous anisotropic body

#### Abstract

In communication 1, the integro-differential equations of the second boundary value problem of the theory of elasticity for a homogeneous isotropic body were considered. The results obtained are extended to boundary value problems for the general case of an inhomogeneous anisotropic body. It is shown that the integro-differential equations found are also Fredholm type equations. The existence and uniqueness of their solution is proved, the conditions under which the solution can be found by the method of successive approximations are determined. An example of calculating the residual stresses in an inhomogeneous quenched cylinder is given.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(1):199-208

199-208