# Vol 14, No 2 (2010)

**Year:**2010**Articles:**42**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1257

### In Memory of Anatoliy A. Kilbas

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):6-9

6-9

### Some aspects of initial value problems theory for differential equations with Riemann-Liouville derivatives

#### Abstract

Some subjects of the well-formed initial value problems for ordinary differential equa- tions with Riemann-Liouville derivatives are discussed. As an example the simplest linear homogeneous differential equation with two fractional derivatives is cited. It's shown, that the requirement of the highest derivative summability inﬂuence the value of the lowest derivative order or the initial values in Cauchy type conditions. The speciﬁc class of functions, allowing the non-summability of the highest derivative, is intro- duced. The correctness of the modiﬁed Cauchy type problem and initial value problems with local and nonlocal conditions is substantiated.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):10-23

10-23

### On the stability of hybrid homogeneous systems

#### Abstract

The hybrid system consisting of the family of subsystems with homogeneous right-hand sides and a switching law is considered. It is assumed that the zero solution of each subsystem is asymptotically stable. By the use of the Lyapunov functions method, the classes of admissible switching laws are determined under which the corresponding hybrid system is also asymptotically stable. The region of asymptotic stability of the zero solution is investigated.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):24-32

24-32

### On a homogenous thermoconvection model of the non-compressible viscoelastic kelvin-voight fluid of the non-zero order

#### Abstract

The homogeneous thermoconvection problem of the non-compressible viscoelastic Kelvin-Voight fluid of the non-zero order is considered. The conducted research is based on the results of the semilinear Sobolev type equations theory, because the first initial value problem for the corresponding system of the differential equations in private derivatives is reduced to the abstract Cauchy problem for the specified equation. The concepts of the p-sectorial operator and the resolving semigroup of operators of the Cauchy problem for the corresponding linear homogeneous Sobolev type equation are used. The existence and uniqueness theorem of the solution which is a quasi-stationary semi-trajectory is proved. The complete description of the phase space is obtained.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):33-41

33-41

### Equal-stress reinforcement the ring bending metal-composites plates working in conditions of steady creep

#### Abstract

The problem equal-stress reinforcements (ESR) is transverse the bending ring plates working in conditions of steady creeping of materials of all a component of a composition is formulated. The system of the resolving equations is analysed and boundary conditions at which smooth solutions of ESR-problem do not exist are certain. The opportunity of existence of several alternative solutions of ESR-problem is shown at the same input information and an opportunity of management by the gained ESR-projects. On a concrete instance the opportunity of origination of a border effect in bonding agent a bent ESR-plate is shown.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):42-54

42-54

### About residual stresses after rolling and exfoliation of two-ply stripe

#### Abstract

The residual stress definition problem was solved for stripe after rolling in one mill. Lengthwise residual stresses are basic cause of the defect initiation of stripe geometry of the buckles or corrugation type. Lengthwise residual stresses exist in the stripe with transversal residual stresses. These stresses induce the exfoliation of multi-layer stripes. The qualitative model for the two-ply stripe exfoliation under pressure of extension transversal residual stresses field was obtained subject to equality of the stripe materials stress-strain properties. The cause of the stepwise (unexpected) exfoliation was defined. The layer may be depressed essentially wanting the destruction (exfoliation) binding stripe.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):55-63

55-63

### The dirichlet problem in the 2d stationary anisotropic thermoelasticity

#### Abstract

In this article the Dirichlet problem for an anisotropic thermoelastic media is studied. It means, by definition, that a displacement vector and a stationary temperature are assigned at a boundary. This boundary value problem is reduced to a system of integral equations. Kernels of integral operators, entering into this system, are weakly regular in a bounded region with a Lyapunov boundary and Holder continuous boundary data. This boundary value problem keeps up the property of Fredholm solvability if a region and boundary data have weaker properties of smoothness.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):64-71

64-71

### Rabotnov's variational equation of a two-layer shell model and the critical buckling time of reinforced shells under creep

#### Abstract

Procedure of calculation of critical buckling time of shells under creep with use of Rabotnov's variation equation is presented. Comparison of critical times for the reinforced shells is calculated with use of three-layer model with experimental data and results obtained by other authors is also presented.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):72-78

72-78

### The essentially over define nonclassical problem for body with crack

#### Abstract

One of the basic problems of the fracture mechanics is the initiation and propagation of the crack in a body. In spite of the variety of approaches to the modeling of such processes, the problem is still actual now. The problem of control is a modern one when for real surface data the different tapes of concentrators should be estimated. Here the problem of a crack finding is investigated for the asymptotic information in the tip a crack with the help of the integral of solidness. The problem is seen as essential to define the problem of the theory of elasticity. When for the boundary are given over define condition one to find out the surface fracture. The needed relations and numerical calculations are made.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):79-84

79-84

### Elastic equilibrium state of thick-walled heavy transversally-isotropic spheres fixed on the interior surface

#### Abstract

Using decomposition of hoop and radial components of displacement vector to the trigonometrical and generalized power series, the new precise analytical solution to problem on equilibrium state of thick-walled heavy transversally-isotropic central-symmetric body, which is fixed on the interior surface and is subject to the action of uniform external lateral pressure, is obtained. This can set a pattern for precise solutions in particular cases of the relations for displacements, stresses and strains at the points inside thick-walled heavy isotropic sphere, the interior surface of which is fixed, while the exterior one being under the uniform pressure. The estimation of an initial strength of solid-cast reinforced concrete sphere is carried out on the basis of a multicriteria approach taking into account real damage mechanisms (i. e. damage from tension or compression in radial, hoop and axial directions, and from transversal and antiplane shear) of anisotropic central-symmetric bodies.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):85-95

85-95

### Reinforcement of planar structures along orthogonal curvilinear trajectories

#### Abstract

The resolving equations for linear orthotropic non-homogeneous elasticity problem, including the deformation compatibility equation, are obtained in cases of bipolar, elliptic, parabolic, hyperbolic and cardioidal coordinate systems for planar constructions extreme deformations detection in the context of planar problem. The type of obtained partial differential equations system for deformations tenser components is examined using the determinantal method.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):96-104

96-104

### Asymptotic analysis of hydraulic fracture crack process growth

#### Abstract

In this work the self-similar solution of a problem of propagation of hydraulic fracture crack is presented. It is proposed that a crack is focused vertically and research of of its development process is analyzed within the coupled problem of the hydrodynamics theory (flow of viscous fluids) and the linear theory of elasticity (fracture crack growth). Asymptotics of the self-similar solution at the long and small distances from tip of a crack were obtained. Simple approximations of the solution in all length range are offered.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):105-116

105-116

### Reliability estimation of stochastic heterogeneous rod constructional elements by the use of parametric failure criterion

#### Abstract

Methodology of calculation of probability of no-failure working life with one structural parameter was investigated on the basis of extension of Rabotnov 's determinate creep equations. Residual working life method using conditional probability of no-failure operation for specific product is introduced.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):117-124

117-124

### Parametric identification of the differential operators for the systems with turbulent friction on the base of finite-difference equations

#### Abstract

Numerical technique for determination of the differential operator paramaters for the systems with turbulent friction is considered. Mean-square estimation of coefficients of stochastic difference equations describing observation results of the impulse characteristic of the system form the basis of this method. The results of numerical analytical treatments of recommended symbolic models and algorithms for calculation on their basis are presented. These results confirm high efficiency of the developed method for the parametric identification on the base of finite-difference equations.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):125-133

125-133

### The estimation of linear regression is based on the generalized least modules method

#### Abstract

The generalized least modules method is shown in this paper. It can be applied to find estimations of parameters of the linear regression model that is based on experimental data. The theorems of existence and finding of solution are proved. The consistency of estimator is proved as well. The results of investigation of regression parameters are demonstrated here. Monte-Carlo method was used for this investigation.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):134-142

134-142

### Mathematical modelling of glass-metalic envelope three-layer cylindrical shell process formation

#### Abstract

The results of cylindrical layered silicate glass based material shell manufacturing process mathematical modeling are presented in the article. Problems of cooling of a com-posit shell and axisymmetric deformation in the course of its formation are considered. The received results testify the possibility of manufacturing of three-layer cylindrical glass-metalic envelope shells which can be used for creation of strong cases for deep-water technical devices. Limiting values of residual pressure on interface surfaces of metal coverings with a glass layer which it is essential less strength of connected materials of a composit cover are received.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):143-151

143-151

### Mathematical modelling of processes occuring in power cylinders of hydraulic press while it's decompression

#### Abstract

The main ways of decompression used in modern control systems and mathematical modelling of processes occuring in power cylinders under hydraulic press during its decompression are considered. Mechanism of fluid column oscillations in power cylinders is described. Oscillating effects in power cylinders while decompression is analyzed, its mathematical description is given. By means of numerical analysis on the basis of hydraulic forging press with active force of 60 MN peculiarities of typical process of power-cylinders decompression are shown. Practical recommendations concerning different decompressing valves using and its influence on maintenance properties of presses are given.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):152-158

152-158

### About one method of obtaining of the exact analytical decision of the hyperbolic equation of heat conductivity on the basis of use of orthogonal methods

#### Abstract

On the basis of use of a method of division of variables and an orthogonal method of Bubnov-Galyorkin the exact analytical decision of the hyperbolic equation of heat conductivity for an infinite plate under boundary conditions of the first sort is obtained. It is shown that having warmed up (or cooled) a body it is defined by movement of front of a shock thermal wave on which there is a breakage temperature curve (temperature jump). The front of a thermal wave divides investigated area on two subareas - revolted where the temperature changes from wall temperature (a boundary condition of the first sort) to the temperature at the front waves and not revolted where the temperature is equal to reference temperature.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):159-169

159-169

### A composite heat state modelling based on universal decomposition composites rule

#### Abstract

The mathematical model and numerical method to solving the problems of heat state of majority composites in conditions of an appreciably unsteady high-thermal stressing are offered. Its base on the identification of the decomposition composites rule. This rule has been obtained and let to exclude chemical kinetics in a region of the decomposition in problems about heat state of the composition. Analysis of the numerical results proves this approach to modeling of heat state composites to be a proper one.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):170-178

170-178

### Dynamic problem for the spirally-multilayered non-uniform cylinder taking into account forces of viscous resistance

#### Abstract

The structural algorithm of a method of final integral transformations constructs closed solution of the task on axes symmetrical oscillation of the spirally-multilayered non-uniform cylinder with the account of forces of viscous resistance. The cylinder is formed by winding of preliminary tense metal tape on elastic pliable tube. The spectrum of circular frequencies free axes symmetrical oscillations is analyzed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):179-189

179-189

### Untied axisymmetris dynamic problem of the return piezoeffect for the radially polarised cylinder

#### Abstract

The non-stationary problem of the return piesoeffect for anisotropic piesoceramic radially polarized cylinder is considered untied axisymmetris at action on its external radial surfaces of the electric potential which is any function on axial co-ordinate and time. The new closed decision is developed by a decomposition method to its own vector-functions in the form of structural algorithm of final transformations. The received ratios allow to define frequencies of own fluctuations, the is intense-deformed condition of an element, and also potential and intensity of induced electric field.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):190-203

190-203

### Properties of graphs of problems network planning and management

#### Abstract

Properties of a task graph of an optimum investment are considered the additional limited resource for reduction of a critical way of the network project at non-uniform marking change of arches graph of the project. The algorithm construction of the count of the project on the set matrix of precedence of works is offered. The algorithm reduction lists of technological precedence of works of the project to lists of direct precedence by correct ordering is developed. Examples of addition fictitious works are resulted and reception minimization of necessary number of fictitious works for simplification of the count the project is specified. It is shown that, on an extreme measure, with addition of fictitious works, columns of the project it will be constructed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):204-211

204-211

### Simulation of fractal surface of films formed by polymerization in gas discharge

#### Abstract

The paper describes a technique for modeling the structure of surfaces of polymers synthesized by plasma vapor deposition (polymerization). The method is realized by using computer technology and allows to simulate geometry of the surfaces for a given initial distribution of structural elements and to calculate the fractal dimension of the model. The fractal characteristics of surfaces of polymers experimentally synthesized in the discharge of fluorine and silicone monomers are found.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):212-221

212-221

### Experimental study of the effect induced by air shot-blasting processing, thermal exposition and high cycle fatigue tests on physical and mechanical condition hardening layer of cylindrical samples of alloys V95 and D16T

#### Abstract

The influence of multicycle fatigue actions and thermal exposition, previously hardened in the rate of the air shot-blasting, on such characteristics of cylindrical samples made from alloys V95 and D16T as roughness and microhardness was researched. In fact, surface plastic hardening significantly affects the growth of microhardness in comparison with samples unhardened. Microhardness decreases to values corresponding to sample unhardened after thermal exposition. Surface hardening increases the roughness layer and further thermal exposition and fatigue tests have no effect on these parameters. Results of experimental measurements are presented in tables and graphs.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):222-228

222-228

### Integral representations and asymptotic expansion formulas of Mittag-Leffler-type function of two variables

#### Abstract

Special function generalizing Mittag-Leffler-type function for two variables is considered. Integral representations for this function in different variation range of arguments for a certain value of parameters is obtained. Asymptotic formulas and asymptotic properties of this function for large arguments is established. Theorems for these formulas and these properties are provided.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):229-236

229-236

### Properties of inversion operator of the Abel matrix equation

#### Abstract

Generalization of integral-differential Riemann-Liouville operator on the matrix order-is reviewed and its properties are studied. Theorem of the composition of operators of the matrix of integration and differentiation can be proved. The necessary and sufficient conditions for the unique solvability of the matrix Abel equation in a special class of functions are obtained.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):237-243

237-243

### Numerical method of value boundary problem decision for 2d equation of heat conductivity with fractional derivatives

#### Abstract

In this work a solution is obtained for the boundary problem for two-dimensional thermal conductivity equation with derivatives of fractional order on time and space variables by grid method. Explicit and implicit difference schemes are developed. Stability criteria of these difference schemes are proven. It is shown that approximation order-by time equal but by space variables it equal two. A solution method is suggested using fractional steps. It is proved that the transition module, corresponding to two half-steps, approximates the transition module for given equation.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):244-251

244-251

### A Mixed Problem for One 3D Space Analogue of Hyperbolic Type Equation

#### Abstract

It is well known that differential equations with an operator are used for study of the processes connected with appearances of vibration and other mechanics problems, and also play an essential role in the theory of approximation and mapping. In the present work a unique solution for the mixed problem of the full hyperbolic equation of the third order with constant factors, in a three-dimensional Euclidean space, was obtain with the Riemann method, which then becomes considerably simpler at the expense of integral representation of one of boundary conditions. Owing to this it can be used for statement and a solution of new boundary value problems.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):252-257

252-257

### Observers for the quasi-linear systems

#### Abstract

The problem of quasi-linear systems stabilization in the case of incomplete feedback is considered. While synthesising a stabilizing control, estimation of a current state vector-is applied. To design this estimation, observers are constructed. The full-dimensions observer and the Luenberger's observer are studied and sufficient conditions of its existence are obtained.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):258-262

258-262

### The boundary value problem for the loaded equation of mixed parabolic-hyperbolic type in rectangular area

#### Abstract

In work necessary and sufficient conditions of uniqueness of the decision of a regional problem for the loaded equation mixed parabolic hyperbolic type in rectangular area are established. The problem decision is constructed in the form of the number sum on own functions of a corresponding one-dimensional problem on own values.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):263-267

263-267

### Estimations of a differential operator in spectral parameter problems for elliptic equations with discontinuous nonlinearities

#### Abstract

The basic boundary value problems for semilinear equations of elliptic type with a spectral parameter and discontinuous nonlinearity are considered in a bounded domain with a sufficiently smooth boundary. The parameter values for which the corresponding problem has the nonzero solution are called eigenvalues. The existence of eigenvalue problem solutions for equations of elliptic type with discontinuous nonlinearities is considered in this paper. Estimations of the differential operator are obtained for these problems.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):268-271

268-271

### Modeling of residual stresses distribution into hardened flat details with V-type notches

#### Abstract

It is researched the residual stresses into the surface hardening details with V-type notches. It is established that depth of the notches exerts significant effect on the distribution of behavior of residual stresses distribution within notches region.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):272-275

272-275

### Structural-phenomenological models for predicting of the elastic properties for high-pore composites

#### Abstract

Structural-phenomenological approach for predicting of the elastic properties for high-pore composites is used. Nonlinear local-ergodic models for the elastic properties predicting are proposed. They give us narrower Voigt-Reiss results, than calculations for the linear models. Analytical solution for calculation of the correlate approximation is developed. The following approximations deal with the higher orders moment functions for the elastic properties. They do not practically affect the first approximation obtained, that is why the following approximations are not given.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):276-279

276-279

### Substatiation of boundary conditions when during interaction of a string with bending hardness with roller basis

#### Abstract

The paper considers the problem of vibrations in flexible link with bending hardness when one of the basis is a roller or a pulley. When vibrating the touch point link moves along the basis. Movement of touch point depends on the vibrating process that makes the problem non-linear. Using decomposition function describing vibrations in the number of touch points, boundary conditions are received when link and basis are interesting. Rating of dependence of frequency vibrations on amplitude is received. Stated, that at low amplitude vibrations boundary conditions can be written in the form of equality of angle to zero and displace of flexible link. Error when using such boundary conditions increases with increasing amplitude.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):280-284

280-284

### Longitudinal impact on rigid barrier homogeneous and stepped bar is lowered longitudinal stiffness

#### Abstract

Mathematical modelling of longitudinal elastic central blow of non-uniform rod system about a rigid barrier is carried out, at not keeping communications by the analytical decision of the wave differential equation by a method of Dalambera. The rod system consists of a homogeneous core of constant cross-section section and the step non-uniform core having two homogeneous sites of various length and the area of cross-section sections. Ties with a rigid barrier and between cores are not witholding.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):285-290

285-290

### Additional boundary condition in heat conductivity problems with variables medium physical properties

#### Abstract

The field of obtaining approximated analytical solutions of mathematical physics boundary problems is developed. It is based on additional boundary conditions, allowing to define any item number of model (approximated) analytical solutions and to obtain solutions with the required accuracy degree. Physical meaning of additional boundary conditions lies in implementation of the basic differential equation and its different order derivatives at boundary points of the field. It leads to implementation of differential equation within the given field with number of approximations dependant on the accuracy.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):291-296

291-296

### Behavior analysis of mechanical systems based on known statistical data

#### Abstract

A method of constructing the cone of possible solutions for the known statistics. These data are indications of devices that fix the position of a point after a certain period of time. Thus we obtain the database which helps to construct a differential inclusion. From a certain point differential inclusion can be made manageable, and to build a cone of possible solutions so that the controlled process minimizes a given functional quality. A model example was developed, which according to statistics contains a cone of possible solutions and management with the functional qualities, proportional to the kinetic energy.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):297-301

297-301

### Influence of dislocation density of copper electrode on the kinetics of anode process in acidic CuSO4 solution

#### Abstract

Deformed copper samples were burned at different temperatures. This procedure formed different dislocation density in the samples. Then such prepared samples were used as the anodes in acidic C11SO4 solution. The voltage for electrolyse was 10 mV. The experiment has shown that the increase of the dislocation density in the anode leads to the increase of the density of anode current. It was discovered that the density of anode current could become 2,5 times bigger. This can be explained by the fact that dislocations decrease the potential barrier during the transfer of copper ions from the grid into the electrolyte.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):302-304

302-304

### Numerical integration of celestial bodies equations with taking into account regularization and use osculating elements of major planets

#### Abstract

The regular differential equations for a problem of perturbed motion are received. The computing algorithm and the program of numerical integration of the equations of celestial motion by a method of Everhart taking into account regularization and use osculating of elements of major planets is developed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):305-308

305-308

### Projection regularization algorithm for solving linear algebraic system of large dimension

#### Abstract

The iterative projection algorithm for solving ill-posed systems of linear algebraic equations is examined. This algorithm is based on transforming the regularized normal equations to the equivalent augmented regularized normal system of equations. The proposed algorithm can effectively solve ill-posed problems of large dimension.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):209-312

209-312

### On one property of the weighed sums of equal powers as matrix products

#### Abstract

Finite sum of weighted sum of equal powers is represented in matrix form. This is expressed by the presence of two matrix components, the first of which does not contain, and the second contains the specified weights. It is significant that if the maximum power grounds in the amounts of this type has a value of no less important exponent related to its term, the first matrix component is the same for all amounts that meet the specified condition. Revealed common property of the matrix representation used to obtain the modified final sum from the increased number of weights.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):313-317

313-317

### Calculation block of initial stress-strain state for constructions of a program complex Strelax

#### Abstract

The developed software package for calculating residual stresses relaxation in the surface hardened layer of structural elements under creep conditions is announced. Block of software package for calculating the fields of residual stresses and strains in the hardened layer of constructions (cylindrical samples) after strain hardening is described. The calculation is produced for one or two components of the experimentally measured residual stresses.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2010;14(2):318-321

318-321