# Vol 16, No 2 (2012)

**Year:**2012**Articles:**24**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1252

### Properties of the integral curve and solving of non-autonomous system of ordinary differential equations

#### Abstract

In this paper, we consider non-autonomous system of ordinary diﬀerential equations. For a given non-autonomous system, we introduce the distribution probability-density function of representative points of the ensemble of Gibbs, possessing all the characteristic properties of the probability-density function, and satisfying the partial differential equation of the ﬁrst order (Liouville equation). It is shown that such distribution probability-density function exists and represents the only solution of the Cauchy problem for the Liouville equation. We consider the properties of the integral curve and the solutions of non-autonomous system of ordinary diﬀerential equations. It is shown that under certain assumptions, the motion along trajectories of the system is the maximum of the distribution probability-density function, that is, if all the required terms are satisﬁed, an integral curve of non-autonomous system of ordinary differential equations at any given time is the most probable trajectory. For the linear non-autonomous system of ordinary differential equations, it is shown that the motion along the trajectories is carried out in the mode of distribution probability-density function and the estimate of its solutions is found.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(2):7-17

7-17

### On uniqueness of the second boundary value problem solutions for the third order composite type equation in unbounded domains

#### Abstract

In the paper the second boundary value problem for the third order composite type equations is investigated. We established Saint-Venants type energy estimates for weak solutions of the problem on Sobolev classes. The obtained estimates are used to prove uniqueness theorems in the classes of functions growing at inﬁnity. These uniqueness classes depend on the geometrical characteristics of the domain. Moreover, energy estimates allowing us to investigate behavior of solution in the neighborhood of singular points were obtained.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(2):18-25

18-25

### The Sobolev-type equations of the second order with the relatively dissipative operator pencils

#### Abstract

Of concern is the Cauchy problem for the Sobolev-type equation of the second order. We introduce the deﬁnition of relatively dissipative operator pencils, generalize the notion of dissipativity and relative dissipativity of operators. The connection with the theory of accretive operators is established. According to the Keldysh ideology, the original problem is reduced to the Cauchy problem for the Sobolev-type equation of the ﬁrst order and the results for the investigated problem are obtained.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(2):26-33

26-33

### Structure of the essential spectrum of a model operator associated to a system of three particles on a lattice

#### Abstract

We consider a model operator H associated to a system of three particles interacting via nonlocal pair potentials on a three dimensional lattice. The existence conditions of the eigenvalues of a corresponding Friedrichs model are found and the structure of the essential spectrum of H is studied.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(2):34-43

34-43

### Construction of Mikusinski operational calculus based on the convolution algebra of distributions. Basic provisions

#### Abstract

The main provisions of the operational calculus based on the convolution algebra of distributions D+ and D− that extends this method to the negative values of the argument are given. The relation between the proposed method and the classical operational calculus built on the Laplace transform is provided.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(2):44-52

44-52

### To the calculation of equilibrium parameters and stability of torsion process of circular bars made of weakening material

#### Abstract

The problem of torsion of a bar with circular cross-section is considered. The bar is made of the material with a stress-strain diagram having the falling branch describing the softening state. It is shown that the bar has several possible equilibrium positions during the deformation process. The Newton-Kantorovich method is employed to deﬁne the stress-strain state in all equilibriums positions. The deﬁnition of stress and strain in both steady and unsteady equilibrium is carried out using the method of simple iterations. It is established that the divergence of simple iterations corresponds to the moment of stability loss.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(2):53-64

53-64

### Experimental analytical method for quasi-homogeneous material characteristics determination based on elasto-plastic analysis of experimental data

#### Abstract

The possibility of material mechanical characteristics estimation based on solving of the elasto-plastic problems for plane with a hole is studied. The proposed experimentalanalytical method for the material characteristics determination depends on the analysis of circular hole contour displacement and the sizes of inelastic strains zones near it. It is shown, that three problems can be solved for the material mechanical characteristics estimation according to the assignment of experimental data. One of such problems is considered relating to the rock mechanics. The analysis of this problem solution is made and the scope of its applicability is noted. The validity of similar analysis using for the characteristics determination both of homogeneous and quasihomogeneous material is presented.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(2):65-71

65-71

### Evaluation of reliability of axisymmetric stochastic elements of constructions under creepage on the basis of theory of runs

#### Abstract

Probabilistic methods of evaluating of strength reliability of construction axisymmetric elements of working in creep conditions were considered. Rheological properties of the material were described by the random function of one variable (of radius r). Evaluation of reliability of stochastically inhomogeneous axisymmetric structural elements was considered by deformation criterion. The restriction was imposed on random displacement, which depends on the time and spatial coordinates. It was assumed that the random ﬁeld of function w(r, t) has Gaussian distribution, for which some simpliﬁcations were made. Maximum valid value of displacement was deterministic (value was taken from the model problems). The calculation of probability of failure-free operation of microinhomogeneous thick-walled cylinder with given parameters under internal pressure q was considered as an example. The probability of failure-free operation and service life of thick-walled cylinder obtained in this work by the deformation criterion and the results obtained by other authors was compared. It was shown that this approach can be used for fail-safe constructions, which critical level of runs is a rare event.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(2):72-77

72-77

### An investigation of residual stress relaxation processes which arises in plate circle hole surface layer in vibrocreep conditions

#### Abstract

Development and implementation of calculation method of residual stress relaxation in the surface hardened layer of plate hole under vibrocreep conditions with combine static and cyclic loading are presented. Based on the ideas of decomposition and aggregation, the calculation of the kinetics of relaxation of residual stresses in the hole surface layer is reduced to gluing solutions of three boundary value problems. While solving the ﬁrst boundary value problem the three-dimensional distribution of residual stress ﬁelds and plastic strains in hole hardened layer of plate is deﬁned. Solving of the second boundary value problem deﬁnes the stress-strain state of a hole hardened layer of plate during vibrocreep without taking into account the surface hardened layer. For the third boundary value problem the relaxation of residual stresses in surface hardening layer, deformable in the hard loading at given values of the strain tensor components, which are determined by the second boundary value problem solving, is investigated. The numerical analysis of the cyclic component amplitude value inﬂuence on residual stress relaxation process is completed. The numerical analysis presents a residual stress relaxation rate increasing according to cyclic component value.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(2):78-84

78-84

### Analysis of a nonlinear dynamic model of the Couette flow for structured liquid in a flat gap

#### Abstract

Parametric investigation of structured liquid Couette ﬂow in a plain gap is presented. Bifurcation conditions of steady non-uniform solutions are deﬁned from unsteady uniform ones in the nonmonotonic region of rheological curve. Relevant bifurcation diagrams are plotted. The coincidence between the steady-state solution and the solution of unstable problem is noted.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(2):85-92

85-92

### A sufficient condition for stability of the calculation of parameters of aperiodic processes based on second order difference equations

#### Abstract

The stability problem for the calculation of parameters of the second order damping aperiodic processes is considered. The numerical method of the second order aperiodic process parameters determination, based on iterative procedure of diﬀerence equation coefficients calculation, is described. The inequalities allowing to provide the stability of the difference equation according to the considering aperiodic process parameters limits of variation, known a priori, are obtained. The theorem on the sufficient condition of stability of the normal equations system under the solving of problem of difference equation coefficients mean-square estimation is formulated and proved. The obtained results have the great practical importance and can be used for the selection of discretization period of experimental curve, describing the second order observed aperiodic process in the system output.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(2):93-102

93-102

### Numerical solution of axisymmetric problem of the theory of elasticity on the basis of continuum graph model

#### Abstract

A numerical method for analysis of the stress strain state of elastic media based on a discrete model in form of directed graph is suggested. To analyze a deformable body using the graph approach, we partition a solid body on elements and replace each element by its model in the form of an elementary cell. The matrices, presenting several structure elements of the graph, and the equations, describing the elementary cells, contribute to deriving the constitutive equations of the intact body. Numerical examples are presented.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(2):103-114

103-114

### A representation in terms of hypergeometric functions for the temperature field in a semi-infinite body that is heated by a motionless laser beam

#### Abstract

We have considered an analytical expression for the temperature ﬁeld of a semi-inﬁnite body that is heated by a circular heat source located at the free surface. Unsteady temperature ﬁeld is expressed in terms of the Appell and the Srivastava hypergeometric functions. We have studied some special areas in heated body where a non-stationary temperature ﬁeld is expressed in terms of the Kampé de Fériet function. The obtained expressions have allowed to carry out the separation of the stationary and non-stationary parts of temperature ﬁeld from each other. Calculations of the steady temperature ﬁelds generated by circular or Gaussian sources have been accomplished. Signiﬁcant quantitative differences in these ﬁelds were not found.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(2):115-123

115-123

### Designing of dynamic calculation model of press with pump-accumulator drive working stroke

#### Abstract

Peculiarities of hydraulic press ram moving on diﬀerent stages of the working stroke on the basis of experimental oscillogram are considered. Rated scheme of the working stroke with pointing out the assumptions of the hard hydraulic drive model is adduced. Dynamic calculation models of the ﬁrst and second stages of the working stroke of hydraulic press with pump-accumulator drive are designed. Description of the coefﬁcients of differential Ricatti equation is given, its analytical decision concerning real forging press is adduced. Presented moving metal-and-ﬂuid masses, viscous hydraulic resistance of the line accumulator - press and the active press force on the working stroke are considered in detail. Its inﬂuence on ram moving on the working stroke is revealed. The inﬂuence of constructive characteristic and shutting time of the control valve on the dynamics of ram decelerating on the working stroke is analyzed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(2):124-131

124-131

### On the stability of a class of essentially nonlinear difference systems

#### Abstract

The problem of the zero solution stability for a certain class of essentially nonlinear difference systems is studied. Theorems on the stability by the inhomogeneous approximation are proved. Systems of triangular form are considered as systems of nonlinear approximation. Conditions under which perturbations do not destroy stability of the zero solution are formulated in the form of the inequalities establishing relation between orders of perturbations and homogeneity of functions, entering into the system of nonlinear approximation.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(2):132-143

132-143

### Recurrent methods of construction of cumulative functions of risk

#### Abstract

The estimation problem of the enterprise total losses related to the improper performance of liability is considered. The total losses distribution function and density for the aggregate contracts are constructed by the recurrent methods, proposed by N. de Pril and H. Panjer. The numerical experiments based on ﬁve groups of contract portfolio are carried out. The results analysis showing the advantages and disadvantages of calculating algorithms is made. In particular, the values of means at risk obtained by H. Panjer method are superior to the corresponding values founded by N. de Pril method, and the maximum value of loss occurrence probability in the distribution density realized by H. Panjer method is less than the one, gotten by N. de Pril method. The results can be used for the further development of N. de Pril and H. Panjer approaches for estimation of total losses of risk aggregate.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(2):144-151

144-151

### Bifurcations in a multicomponent Walras-type system under lack of information

#### Abstract

We consider a Walras-type equilibrium model with a supply function speciﬁed by A. A. Shananin. In order to investigate a stability of market equilibrium under lack of information we develop a distributed model where price becomes discrete space variable. A bifurcation analysis allows us to examine asymptotic behavior of solutions to the model. We show that in a certain range of parameter values the system becomes spatially unstable and exhibits a qualitatively diﬀerent bifurcation type.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(2):152-157

152-157

### Sampling of expert assessment pair functions

#### Abstract

The pair functions, used in relation to the value peer review object, which is involved in the implementation of a manufacturing or research purposes, are under consideration. These functions have a consistent character of increasing with respect to the quality criterion of the object, and form a mathematical model for evaluation. The presence of uniqueness displayed by readout functions of this species under the time factor conditions is proved. Traces of such manifestation are related to the assignment of certain signs of the private increments of the functions corresponding to the changes in the estimated object quality criterion, as well as the time delay since its acquisition in comparison with the time of issuance of expert information.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(2):158-163

158-163

### Research of efficiency of algorithms of method Everhart with high order of approximating formulas

#### Abstract

The modiﬁed algorithm for the numerical integration of the equations of celestial motion by the Everharts method is developed. The study of the effectiveness of the algorithm for approximating high-order formulas is carried. High efficiency of the method is shown on the example of joint integration of the equations of motion of major planets, the Moon, the Sun and the small bodies of Solar system.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(2):164-173

164-173

### Boundary control problem for the telegraph equation

#### Abstract

In the paper we consider the boundary control problem for the telegraph equation. We study the case of the short period of control, when the initial and ﬁnal data determine the solution in two regions, having the common part. It means, the control problem has the solution only for the special way related initial and ﬁnal conditions. We give these relations for two intervals of control time changing and construct solutions for two Cauchy problems in the regions bounded by the characteristics of the equation. This construction allows to ﬁnd data on characteristics and to solve two Goursat problems. Finally, the substitution of necessary values of spatial coordinate in the obtained expressions gives the required boundary control functions.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(2):174-177

174-177

### Investigation of destruction and decarburization kinetics of a thick-walled tubes under hydrogen corrosion

#### Abstract

The behavior of the thick-walled pipeline under hydrogen corrosion is studied. The model example of thick-walled tube made of St. 20 steel with a region of local heating is considered. The internal and external pressures of hydrogenous medium are used as loading factors in the model example. The greatest eﬀect of the value of internal pressure of hydrogenous medium on time to fracture is noted. Also, the faster decarburization in the region of high temperature is pointed out.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(2):178-182

178-182

### On the numerical solution of the Dirihlets problem for the Poissons equation with fractional order derivatives

#### Abstract

Difference approximation for the Caputo fractional derivative of the 4−β, 1 < β≤2, order is obtained in the work. The difference schemes for solving the Dirichlet's problem for the Poisson's equation with fractional derivatives are developed. The right part and initial data stability of difference problem and its convergence are proved.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(2):183-187

183-187

### Obtaining exact analytical solutions of the thermoelasticity problem for multilayer cylindrical structures

#### Abstract

This article is told about obtaining exact analytical solutions of the thermoelasticity problem for multilayer construction and also contains its algorithm when elastic properties of each layers were constant. As an example was solver speciﬁc problem for double layer hollow cylinder with set load at the inner surface and rigidly ﬁxed at the external surface. Thermal state of each layer was set as a function of spatial variable.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(2):188-191

188-191

### Comparative analysis of mathematical models for estimating the impact probability of asteroid Apophis

#### Abstract

Methods for estimating the impact probability of asteroid Apophis were implemented. The probability of a collision was assessed using two methods: the Monte Carlo method, and the second one, utilizing a ratio between the interval of values of orbital elements, leading to a collision, and the conﬁdence intervals of orbital elements. It was established that the variation in the semimajor axis makes the main contribution to the estimation of the impact probability. The expected date and the estimate of the probability of collision of asteroid Apophis and Earth were computed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(2):192-196

192-196