## Vol 21, No 3 (2017)

**Year:**2017**Articles:**12**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1225

Articles

To the 60th Anniversary of Professor Alexander Vladimirovich Manzhirov

#### Abstract

On May, 24, 2017 the 60th jubilee of Prof. A.V. Manzhirov was celebrated at Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences. Prof. A.V. Manzhirov is known as a prominent scientist in the field of mechanics and applied mathematics. The principal directions of his academic activity are Mechanics of Growing Solids, Theory of Creep and Viscoelasticity, Biomechanics, Contact Mechanics, Tribology, Integral Equations and their numerous applications. The present dedication is devoted to the Prof. A.V. Manzhirov’s scientific biography and contains the list of his selected publications.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(3):401-416

Delta-problems for the generalized Euler-Darboux equation

#### Abstract

Degenerate hyperbolic equations are dealing with many important issues for applied nature. While a variety of degenerate equations and boundary conditions, successfully matched to these differential equation, most in the characteristic coordinates reduced to Euler-Darboux one. Some boundary value problems, in particular Cauchy problem, for the specified equation demanded the introduction of special classes in which formulae are simple and can be used to meet the new challenges, including Delta-problems in squares that contain singularity line for equation coefficients with data on adjacent or parallel sides of the square. In this short communication the generalized Euler-Darboux equation with negative parameters in the rectangular region is considered.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(3):417-422

Spectral characteristics of a nonlocal problem for two linear systems of partial differential equations

#### Abstract

We study the boundary-value problem for a linear system of differential equations written in the form of differential-operator equations $$ aD_t u(t)+bBu(t)=f(t) $$ with nonlocal boundary conditions at $t$. Such a boundary value problem for a linear system of differential equations (including partial derivatives), we shall call nonlocal. The purpose of the article is to study the spectral characteristics of differential operators generated by the nonlocal task for the two linear systems of differential equations considered in a bounded region of finite-dimensional Euclidean space.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(3):423-436

Introduction to the generalized theory of non-equilibrium Cahn-Hilliard phase transitions (Thermodynamic problems in continuum mechanics)

#### Abstract

The occurrence of convective currents and their development from regular forms with the subsequent transition to irregular turbulent currents draw attention to the fact that they are responsible for the efficiency of many technological processes of heat and mass transfer. Such technological processes are basic in the chemical, petrochemical, power, metallurgical and other industries. Convective flows arise in liquids and gases in the gravitational field in the presence of spatial inhomogeneity of the density created by the inhomogeneity of the temperature and the concentration of components arising during, for example, chemical reactions or other causes. With increasing temperature difference, the resting liquid loses its stability, which then leads to the appearance of a convective flow (Rayleigh-Bénard instability). A further increase in the temperature difference leads to an instability of the primary convective flow, and the hydrodynamic crisis leads to a heat transfer crisis. The paper reconstructs the early stage of the Rayleigh-Bénard convective instability considered as a nonequilibrium phase transition with the spinodal decomposition (diffusion separation) mechanism.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(3):437-472

The problem with Saigo operators for a hyperbolic equation that degenerates inside the domain

#### Abstract

A nonlocal problem is investigated for a degenerate hyperbolic equation $$ |y|^{m} u_{xx}-u_{yy}+a |y|^{\frac{m}{2}-1} u_{x}=0 $$ in a domain bounded by the characteristics of this equation. The boundary condition for this problem contains a linear combination of generalized fractional integro-differentiation operators with a hypergeometric Gauss function in the kernel. The uniqueness of the solution is proved using the Tricomi method. The existence of a solution is equivalent to the solvability of a singular integral equation with a Cauchy kernel.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(3):473-480

Fractal and mechanical microand nanorange properties of sylvite and halite crystals

#### Abstract

This article involves the treatment of micro- and nanorange scanning and indentation data for salt rock crystals obtained with help of the scanning microscope Dimension Icon using the mathematical models. It also describes the basic methods of fractal analysis. It shows the effectiveness of the method of minimal covering which is chosen to research the fractal properties of salt rock crystal surfaces. The article includes the algorithm of this method and the description of its generalization for the two-dimensional case. The values of fractal index and multifractal parameters have been calculated on the basis of the minimal covering method. The article also involves the anisotropy effects for fractal properties, comparison of fractal behavior on different scale levels. It gives the values of hardness for different parts of the crystals and studies the correlation between hardness and fractal index and describes the character of the influence of fractal dimension on roughness.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(3):481-495

Integro-differential equations the second boundary value problem of linear elasticity theory. Message 1. Homogeneous isotropic body

#### Abstract

The system of equations of the second boundary value problem of the linear theory of elasticity for homogeneous isotropic bodies is reduced to two separate integro-differential equations of Fredholm type, which allowed to apply for their research the theorem of Fredholm. The spectral radii of the corresponding operators are determined and the existence and uniqueness of the solution of the second boundary value problem are proved. It is also established that the decision of the second integro-differential equation can be found by successive approximations and presented convergent with a geometric rate close to Neumann. The method application is illustrated on the example of calculation of residual stresses in a quenched cylinder.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(3):496-506

Solution of the boundary-value problem of torsion for solid and hollow cylindrical specimens made of the Steel 45 and AMG-6M alloy under short-term steady-state creep conditions

#### Abstract

We have developed a method for solving the boundary-value problem of torsion for solid and hollow cylindrical specimens under steady-state creep conditions. The definition of rheological model is carried out with experimental stationary creep curves under uniaxial tension in accordance with the modified method of least squares. Comparison of calculated characteristics of the stress-state with corresponding test data was made for shorttime creep of cylindrical specimens made of the Steel 45 or AMG-6M alloy. The dependencies for strain intensity at the characteristic point and torsion angle on time are obtained and compared with the data calculated by the method of characteristic point. The estimates of errors of deviation of calculated data from experimental values are given and there is good-enough correspondence between the experimental and calculated data. The calculated diagrams for shear stress along the radius at different time points are obtained during torsion for both solid and hollow cylinders.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(3):507-523

Refined model of elastic-plastic behavior of longitudinally reinforced curved wall-beam under dynamic loading

#### Abstract

An initial-boundary value problem is formulated to describe the dynamic behavior of flexible longitudinally reinforced wall-beams of the lesser curvature. Mechanical behavior of materials of composition of the beams is described by the equations of the theory of plasticity with isotropic hardening. The geometric nonlinearity of the problem is considered in the Karman approximation. The obtained equations and correlations allow with different degree of accuracy to determine the stress-strain state of the considered beams taking into account of their weakened resistance to the transverse shears. From the received relationships in the first approximation the equations, corresponding to the second variant of Timoshenko theory, are obtained. For the numerical integration of the problems the method of steps in time with the involvement of the central differences to approximate derivatives with respect to time, is used. The longitudinally reinforced straight and slightly curved beams-walls of relatively low height are considered. The dynamic response is investigated for the considered constructions depending on the action surface (concave or convex) of external pressure caused by the arrival of the air blast wave. It is found that at the time intervals exceeding a few tenths of fractions of a second, elastic-plastic behavior of flexible reinforced straight and curved wall-beams, determined according to the second variant of the Timoshenko theory, is significantly different from the inelastic dynamic response calculated according to the refined theory.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(3):524-545

The row-oriented form of the regularized Kaczmarz’s method

#### Abstract

This paper presents the new iterative method for solving the standard Tikhonov regularization problem. The basis of the method is the application the projection Kaczmarz algorithm to the augmented regularized normal system of equations. The use of the augmented regularized normal system of equations, instead the system of regularized normal equations, makes it possible to significantly reduce the spectral condition number of the original problem. The paper presents the row-oriented form of the regularized Kaczmarz algorithm. This form of the regularized Kaczmarz algorithm allows to solve problems in which the data are received sequentially (line by line). The proposed algorithm makes it possible to effectively calculate solutions of problems with sparse matrices of large and superlarge dimensions. The comparison’s results of the proposed row-oriented form of the algorithm with the column-oriented form of this algorithm are presented. By considering a certain classes of problems, the paper demonstrates that the proposed form of the regularized algorithm allows to reduce the number of iterations in comparison with the column-oriented form of the algorithm.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(3):546-555

Numerical method of estimation of parameters of the nonlinear differential operator of the second order

#### Abstract

The main problem of mathematical simulation is the problem of nonlinear estimation of parameters of the different physical systems. The article contains new numerical method of parameters estimation of the nonlinear differential operator of the second order with the dissipative force, proportional to n-motion speed level assessment. Mean square estimation of coefficients of the generalized regression model constructed taking into account the difference equations describing results of measurements of a pulse response of system is the cornerstone of the numerical method. Two landmark procedure of differentiated estimation of parameters of dynamic process realized in a method allow to provide high adequacy of the constructed model to data of an experiment. Application of the developed numerical method allows to increase significantly (several times) the accuracy of estimates of parameters of the nonlinear differential operator in comparison with the known methods due to elimination of the offset in estimates caused by use of approximation in case of simulation of an envelope of vibration amplitudes.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(3):556-580

Mathematical modelling of tissue formation on the basis of ordinary differential equations

#### Abstract

A mathematical model is proposed for describing the population dynamics of cellular clusters on the basis of systems of the first-order ordinary differential equations. The main requirement for the construction of model equations was to obtain a formal biological justification for their derivation, as well as proof of their correctness. In addition, for all the parameters involved in the model equations, the presence of biological meaning was guaranteed, as well as the possibility of evaluating them either during the experiment or by using models of intracellular biochemistry. In the desired model the intercellular exchange of a special signal molecules was chosen as the main mechanism for coordination of the tissue growth and new types selection during cell division. For simplicity, all signalling molecules that can create cells of the same type were not considered separately in the model, but were instead combined in a single complex of molecules: a ‘generalized signal’. Such an approach allows us to eventually assign signals as a functions of cell types and introduce their effects in the form of matrices in the models, where the rows are responsible for the types of cells receiving the signals, and the columns for the types of cells emitting signals.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(3):581-594