# Vol 18, No 4 (2014)

**Year:**2014**Articles:**16**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1246

### Cauchy problem for the system of the general hyperbolic differential equations of the forth order with nonmultiple characteristics

#### Abstract

We consider the Cauchy problem for the hyperbolic differential equation of the forth order with nonmultiple characteristics. We generalize this problem from the similar Cauchy problem for the hyperbolic differential equation of the third order with nonmultiple characteristics which solution was constructed as an analogue of D'Alembert formula. We obtain the regular solution of the Cauchy problem for the hyperbolic differential equation of the forth order with nonmultiple characteristics in an explicit form. This solution is also an analogue of D'Alembert formula. The existence and uniqueness theorem for the regular solution of the Cauchy problem for the hyperbolic differential equation of the forth order with nonmultiple characteristics is formulated as the result of the research. In the paper we consider the Cauchy problem for the system of the general hyperbolic differential equations of the forth order with nonmultiple characteristics.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(4):7-15

7-15

### On one generalization of Bessel function

#### Abstract

In this paper the generalized Bessel function $J_{\mu ,\omega } ( x )$ is introduced. The function $J_{\mu ,\omega } ( x )$ is given as one solution of the following differential equation: $$ x^2{y}''+x{y}'+\left( {x-\mu ^2} \right)\left( {x+\omega ^2} \right)y=0, \quad \mu , \omega \notin \mathbb Z. $$ The representation of the $J_{\mu ,\omega } ( x )$ by the power series is given. The theorem on integral representations of the function $J_{\mu ,\omega } ( x )$ is established. The main properties of the function $J_{\mu ,\omega } ( x )$ are studied. The integral transforms of Bessel type with the function $J_{\mu ,\omega } ( x )$ is constructed. Formula of inversion of this transform is received.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(4):16-21

16-21

### On a class of nonlocal problems for hyperbolic equations with degeneration of type and order

#### Abstract

Nonlocal problems for the second order hyperbolic model equation were studied in the characteristic area. The type and order of equations degenerate on the same line $y = 0$. Nonlocal condition is given by means of fractional integro-differentiation of arbitrary order on the boundary. Nonlocal condition connects fractional derivatives and integrals of the desired solution. For different values of order operators of fractional integro-differentiation within the boundary condition the unique solvability of the considered problems was proved or non-uniqueness of the solution was estimated.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(4):22-32

22-32

### On the solvability of nonlocal problem with generalized operators M. Saigo for Bitsadze-Lykov equation

#### Abstract

A nonlocal boundary value problem for the equation of moisture transfer was studied in the field, which is the union of two characteristic triangles. The novelty of the formulation of the problem lies in the fact that the boundary conditions include operators of generalized of fractional integro- differentiation in the sense of M. Saigo. The uniqueness of the solution of the problem was proved using the extremum principle for hyperbolic equations. Properties of operators of generalized fractional integro-differentiation in the sense of M. Saigo were used in the proof. Existence of a solution is equivalent reduced to the solvability of a characteristic singular integral equation with Cauchy kernel for which the smoothness of the right-hand side was studied.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(4):33-41

33-41

### Solution of the contact problem on indentation of rectangular punch in an elastic roughnesses half-space in the presence of coulomb friction

#### Abstract

The numerical solution of the static three-dimensional contact problem of the indentation of a rectangular stamp with a flat base in an elastic rough half-space in the presence of Coulomb friction and previously unknown adhesion and slip zones is obtained. Accounting for surface roughness in this problem is carried out based on the spherical model of microroughnesses by introducing the nonlinear terms describing surface microroughnesses crushing and shearing to the expression of relative displacement of the interacting bodies. The influence of the values of the friction coefficient and the parameters of the microscopic irregularities on the size and shape of the zone of adhesion and the distribution of the tangential contact stresses are analyzed. It is shown that the inclusion of surface microroughness shear forming roughness can lead to a substantial increase in the size of the zone of adhesion.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(4):42-52

42-52

### Three-dimensional surface wave in half-space and edge waves in plates with mixed boundary conditions on the front edge

#### Abstract

In the ﬁrst part of this paper the propagation of sinusoidal three-dimensional surface waves is investigated for an isotropic elastic half-space with mixed boundary conditions. It is assumed that the boundary is ﬁxed in one of the tangential directions and traction free in the other directions. The exact dispersion relation is derived which shows the existence and uniqueness of the three-dimensional surface wave. The speed of this wave depends on the angle of propagation and lies between the shear wave speed and Rayleigh wave speed. The graphs of this dependence are presented for various values of Poisson ratio. In the second part of this paper the three-dimensional edge waves in plates with mixed boundary conditions on the edge are investigated. The faces of the plate are assumed to be traction free. Both symmetric and antisymmetric solutions of three-dimensional dynamic equations of elasticity are considered. It is assumed that the edge is ﬁxed in one of the tangential directions and traction free in the normal and the other tangential direction. Asymptotic analysis is performed, which shows that there is an inﬁnite spectrum of higher order edge waves in such plates. The results of numerical calculations based on the modal expansion method are presented to conﬁrm asymptotic analysis. In addition, by the numerical investigation the fundamental edge wave was found in the symmetric case (the edge is ﬁxed in the tangential direction transversally to the faces). The phase velocity of this wave tends to some limit value depending on the Poisson ratio as the wave number increases. In the antisymmetric case the ﬁrst higher order wave has the same limit value. The dispersion curves are presented for various values of Poisson ratio.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(4):53-64

53-64

### Approximate analytical solution of the problem for the tube with elliptic outer contour under steady-state creep condition

#### Abstract

The boundary value problem of steady-state creep for thick-walled outer elliptic contour’s tube under internal pressure is considered. The approximate analytical solution of this problem for the state of plane deformation by the method of small parameter including the second approach is under construction. The hypothesis of incompressibility of material for creep strain is used. As a small parameter the value of ﬂattening factor of the ellipse for external contour is used. Analysis of analytical solution is executed depending on the steady-state creep nonlinearity parameter and ﬂattening factor of ellipse that is ratio of the difference of the semi-major and semi-minor axis to the semi-major axis which is outer radii of the unperturbed thick-walled tube. It is shown that with increasing of value of ﬂattening factor to 0.1 of outer radii of tube tangential stresses in weakest section at θ = π/2 increase by 1.7-1.8 times. The results of computations are presented in tabular and graphic form.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(4):65-84

65-84

### On strong and weak discontinuities of the coupled thermomechanical field in micropolar thermoelastic type-II continua

#### Abstract

The present study is devoted to problem of propagating surfaces of weak and strong discontinuities of translational displacements, microrotations and temperature in micropolar (MP) thermoelastic (TE) type-II continua. First part of the paper is concerned to discussions of the propagating surfaces of strong discontinuities of ﬁeld variables in type-II MPTE continua. Constitutive relations for hyperbolic thermoelastic type-II micropolar continuum is derived by the ﬁeld theory. The special form of the ﬁrst variation of the action integral is used in order to obtain 4-covariant jump conditions on wave surfaces. Three-dimensional form of the jump conditions on the surface of a strong discontinuity of thermoelastic ﬁeld are derived from 4-covariant form. Problems of propagation of weak discontinuities in type-II MPTE continua are discussed too. Geometrical and kinematical compatibility conditions due to Hadamard and Thomas are used to study possible wave surfaces of weak discontinuities. It is shown that the surfaces of weak discontinuities can propagate exist without weak discontinuities of the temperature ﬁeld.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(4):85-97

85-97

### The method of solution of the elastic-plastic boundary value problem of tension of strip with stress raisers with allowance for local domains of softening plasticity of material

#### Abstract

The way of solution of the coupled boundary value problem of solid body deformation for the case of a plastically softening material is offered. The strain and stress fields obtained by the simulated undamaged construction behavior modeling under the action of fictitious forces are used as basic data for calculation. The equivalence of simulated undamaged medium strains and real medium strains is supposed. At each point of construction the damage parameter $\omega$ is calculated by means of constitutive relations of the endochronic plasticity theory. This damage parameter associates the components of the true stress tensor $\sigma_{ij}$ of simulated undamaged medium and the engineering stress tensor $\sigma^0_{ij}$ of real medium by $\sigma^0_{ij}=\sigma_{ij}/(1+\omega)$. Using the tensor $\sigma^0_{ij}$ we can calculate the generalized forces of real construction. The problems of tension of the plates weakened with centric circular hole and semicircular notches are solved and the necessary experiments are conducted. The strain and true stress fields are obtained by numerical calculation at the finite element analysis software and are used for the engineering stress of real construction computation according to the foregoing expression. Softening plasticity domains are plotted. It is found that at the moment before failure the stage of post critical deformation is implementing in the region of stress concentration, although the curve “total displacement - axial force” corresponds to the stage of plastic hardening.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(4):98-110

98-110

### On a stability of polar symmetrical deformation of bodies from softening materials

#### Abstract

Special case of continuum mechanical systems is considered. It is believed that deforming is carried out under conditions of polar symmetry of stresses and strains. Also it is assumed that material properties are described by Hencky model with softening under nonpositivity of volume deformation. Then union curve has region decreasing to zero. Aforementioned conditions are realized in such problems as expansion of spherical cavity in softening space and deforming of thick-walled spherical vessel by equable external pressure (it maybe bathyscaphe which is gradually submerged to the deep). Based on the Lagrange formalism integral quadratic functional is investigated. This functional is increment of total potential energy in the form of Lagrangian for mentioned problems. This study allows to formulate conditions of buckling for active loading which changes quasistatically. For considered problems sets of possible deformations are obtained. These possible deformations perturb the equilibrium position and do not break kinematic constraints. Obtained sets of possible deformations allow to write criterion of buckling of deformation process in explicit form for mentioned problems. It is established that only with sufficiently developed softening zone buckling of deformation process is possible.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(4):111-120

111-120

### The uniqueness of solution in the small sense of tasks of equally-stressed reinforcement of composite metal plates in conditions of steady-state creep

#### Abstract

The uniqueness of a solution in the small sense (in the sense of lack of inﬁnitely close solution) is proved for the boundary-value problem of equallystressed reinforcing metal composite plates in conditions of steady creep of materials of all phases of the composition, when in addition to static and kinematic boundary conditions and boundary conditions for the densities of the reinforcement, which is natural in such problems, on the contour of the plates an additional boundary conditions are speciﬁed for angles of reinforcement. In a large sense (in the sense of signiﬁcant differences solution) this problem can have multiple, but not inﬁnitely close, alternative solutions because of the nonlinearity of the static boundary conditions and equallystressed of reinforcement. The study of the problem of uniqueness of the solution of this problem is necessary when examining the issue of correctness setting of problems of equally-stressed of reinforcement.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(4):121-132

121-132

### On the question of optimization of the oil slick biodestruction in the bodies of water

#### Abstract

We propose the system of nonlinear partial differential equations of parabolic type that describes the processes of oil pollution of water surface and bacterial degradation of this contamination. Problem of the parametric optimization of the acceleration process of degradation of the oil pollution is solved using this model. Optimality criterion is to minimize the time spent on destruction of oil pollution. Coefficient, characterizing the rate of population growth of bacteria, is selected as control parameter. Research is conducted using linearization of functions from the right-hand side of an equation in the neighborhood of solutions having practical signiﬁcance. Also using the simpliﬁed model we obtain the formula that allows to predict the time for necessary destruction level progressing. Checking the results obtained for the linear equations is made using MATLAB. A comparison of experimental data and the calculated values is carried on to show the suggested model adequacy.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(4):133-143

133-143

### A method for the fast MOID computation for two confocal heliocentric orbits

#### Abstract

The paper is on the problem of classiﬁcation an asteroid as potentially hazardous (PHA), namely the estimation of the MOID parameter. Minimum Orbital Intersection Distance describes the minimal distance between two confocal heliocentric orbits. Analytical, numerical and hybrid methods used for the MOID estimation are reviewed. A brief description of the K. V. Kholshevnikov and G. F. Gronchi analytical methods, which are considered to be classical, is given. The task of calculating the MOID parameter for a large number of asteroids (more than 10,000) with a maximum calculating speed and the ability to parallelize the process is set. A numerical method based on geometrical considerations concerning the location of the bodies on their orbits is proposed. Let us consider two bodies A and E. Since only the minimum distance between two orbits is required, the information on the actual position of the bodies on their orbits is insigniﬁcant. The idea is to calculate one full revolution of the body A. For each position of body A the corresponding position of the body E is calculated under the following assumption. Consider a plane P , comprising the body A and the Sun. Therefore, plane P is perpendicular to the orbital plane of the body E. Of the two points at which the plane P intersects the orbit of the body E, E is considered to be at the point that is the nearest the body A. Thus, the position of the body E will depend on the position of the body A. As a result, from the geometric assumptions on the triangle formed by the Sun and two bodies, the distance between A and E is calculated. When one complete revolution of the body A with a certain step is calculated, we receive a set of the distances between two orbits, from which we can identify the areas of the local minima of the discrete representation of the distance function (the distance between the orbits of A and E). Then, the procedure of tuning is carried out to verify and precise the values of local minima of discrete representation of the distance function. As a result, the smallest value of the local minima is considered to be the estimation of the Minimum Orbital Intersection Distance (MOID) takes. Pros of the suggested method are as follows: high speed and adjustable calculation accuracy, the suitability to the use of parallel computing. Comparative tests of the described method were carried out. The results received are consistent with the classical G. F. Gronchi method.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(4):144-156

144-156

### Heat transfer simulation in stirring boundary layer using the semiempirical turbulence theory

#### Abstract

The dynamic and thermal boundary layer equations are derived for the stirring boundary layer using Prandtl semiempirical turbulence theory. Based on deﬁnition of the thermal perturbations front and supplementary boundary conditions the method of constructing an exact analytical solution of the boundary value problem simulating the formation of the thermal boundary layer in the dynamic boundary layer is obtained and applied to ﬁnd the exact analytical solutions of thermal boundary layer differential equation almost with a given degree of accuracy. The velocity distribution in stirring dynamic boundary layer and its thickness were taken by the well - known relations, found from experiments. The supplementary conditions fulﬁllment is equivalent to the fulﬁllment of the initial differential equation in the boundary point and in the thermal perturbations front. So, the more supplementary conditions we use the better fulﬁllment of the initial differential equation in the thermal boundary layer we have, because the range of thermal perturbations front changing includes the whole range of transverse spatial variable changing. Analysis of calculations results allows to conclude that the layer thickness within a stirring dynamic boundary layer more than twice less than thermal layer thickness in a laminar dynamic boundary layer. The study of the received in this paper criteria-based equation shows that the difference of heat transfer coefficients in the range 20000 ≤ Re ≤ 30000 of the Reynolds number on the experimental not exceed 7 %.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(4):157-169

157-169

### Complex time transformations peculiarities for wave function collapse description using quantum path integrals

#### Abstract

A quantum path integral was transformed into the real form using a complex representation of the time. Such procedure gives the possibility to specify measures for the sets of the virtual paths in continual integrals determining amplitudes of quantum states transitions. The transition amplitude is a real function of the complex time modulus. Negative time values correspond to the reverse sequence of events. The quantum evolution description in form of the virtual paths mechanical motion does not depend on the sign of the time, due to the reversibility of the classical mechanics laws. This allows to consider the negative half of the imaginary axis of the time for the path integral measure determination. In this case this integral has the form of Wiener's integral having the well-known measure. As the wave function collapse is irreversible effect, the causal chain of events cannot be changed. Thus, to describe the collapse the transformation of quantum path integrals have to be performed in upper half plane of the complex time. It is shown that the Wiener measure for the real continual integral can be continued analytically on this actual range of the complex time. This allows to use the quantum path integral for any actual range of the complex time.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(4):170-177

170-177

### On authentication codes based on orthogonal tables

#### Abstract

The authentication codes resistant to messages imitation and substitution are investigated. The case when the probabilities of imitation and substitution reach the lower limits has been highlighted. Such authentication codes are called optimal. We study constructions of optimal authentication codes based on orthogonal tables. The case of optimal authentication codes with optional uniform distribution on the set of keys is studied.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(4):178-186

178-186