# Vol 21, No 2 (2017)

**Year:**2017**Articles:**12**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1224

### Well-posedness of the Dirichlet and Poincaré problems for one class of hyperbolic equations in a multidimensional domain

#### Abstract

In early works the author studied the Dirichlet and Poincaré problems for multidimensional hyperbolic equations, which shows the well-posedness of these problems in cylindrical domains, significantly dependent on the height of the considered cylindrical domain. Here a multidimensional region inside a characteristic cone is considered, in which the Dirichlet and Poincaré problems have unique solutions for one class of hyperbolic equations.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(2):209-220

209-220

### Finite-difference method for solving Tricomi problem for the Lavrent’ev-Bitsadze equation

#### Abstract

In this paper we obtain an a priori estimate for solution of Tricomi problem for the Lavrent’ev-Bitsadze equation, from which the uniqueness of regular solution follows. Presented a numerical finite-difference method for solving the investigated problem. We obtain an a priori estimate for solution of the difference scheme, from which follows the second-order convergence.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(2):221-235

221-235

### A construction of analog of Fredgolm theorems for one class of first order model integro-differential equation with logarithmic singularity in the kernel

#### Abstract

The integral representations of the solution manifold for one class of the first order model integro-differential equation with logarithmic singularity in the kernel are constructed using arbitrary constants. The cases when the given integro-differential equation has unique solution are found. The analogue of Fredholm theorem is built for given integro-differential equation. The method of solving this problem can be used for the solving of higher order model and non-model integro-differential equations with singular coefficients.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(2):236-248

236-248

### On the “splitting” effect for multipoint differential operators with summable potential

#### Abstract

We study the differential operator of the fourth order with multipoint boundary conditions. The potential of the differential operator is summable function on a finite segment. For large values of spectral parameter the asymptotic behavior of solutions of differential equation which define the differential operator is found. The equation for eigenvalues of the studied operators is derived by studying the boundary conditions. The parameters of boundary conditions are selected in such a way that the main approach of the equation for eigenvalues has multiple roots. The author shows that for the studied operator the effect of “splitting” of multiple eigenvalues in the main approximation is observed. We derive all series of single eigenvalues of the investigated operator. The indicator diagram of the considered operator is studied. The asymptotic behavior of eigenvalues in all sectors of the indicator diagram is found. The obtained precision of the asymptotic formulas is enough for finding an asymptotics of eigenfunctions of the studied differential operator.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(2):249-270

249-270

### On a boundary-value problem with Saigo operators for a mixed-type equation

#### Abstract

The theory of mixed type equations is one of the most important parts of the theory of partial differential equations. This is due to the fact that equations of mixed type are connected with the problems of the theory of singular integral equations, integral transformations, and special functions. An actual continuation of the research in these fields will be the proof of the unique solvability of the inner-boundary problem. In the hyperbolic part of the domain, a condition is established that relates the generalized derivatives and fractional-order integrals to the Gauss hypergeometric function.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(2):271-277

271-277

### On the problem of optimal control in the coefficients of an elliptic equation

#### Abstract

In this paper we consider the optimal control problem for linear elliptic equations of the second order. Control functions are included in the coefficients of the equation for the state, including the coefficients of the highest derivatives. Space management is a product of Lebesgue and Sobolev spaces. The functional purpose is the sum of the integrals over the region and part of its border. The problems of correct statement of the problem in the weak topology of the space of controls are studied. It is proved that a set of optimal control problems is not empty, it is weakly compact and every minimizing sequence of the functional goals converges weakly in the space of controls to the set of optimal controls. The examples show that the solution of the problem can be not unique and minimizing sequence for the functional purpose can not have a limit in the strong topology of space management. Differentiability of proved Frechet functional is proved and the expression for its gradient is found. A necessary condition for optimality in the form of variational inequalities.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(2):278-291

278-291

### Application of the perturbation method for the determination of stress-strain state of a thick-style two-layer anisotropic shaft of non-circular cross section with elastoplastic torsion

#### Abstract

The present work is devoted to the problem of elastoplastic torsion of the two-layer slightly anisotropic non-circular cross section shaft. The cross section is a doubly connected region. The shaft is oriented in a cylindrical coordinate system so that the Z axis is directed along the axis of the shaft. The influence of mass forces is not taken into account. Let the rod twist about the Z axis by equal and opposite pairs of forces. Suppose that the lateral surface of the rod is free of loads. The value of the moment is such that for some parts of the cross section the material passes into a plastic state and plastic zones are formed. The propagation of plastic flow comes from the outer contour inside the section. Suppose that the value of the torque is such that the plastic region entirely covers the outer contour of the cross section, and there is an elastoplastic boundary that is located between the inner contour and the interface of the layers. It is considered as an anisotropic material that in particular cases is in the kinematic properties of the anisotropy and anisotropy according to Hill. Each of the layers has its own anisotropy parameters. With using perturbation method, stress-strain state and elastoplastic boundary at first approximate is defined.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(2):292-307

292-307

### On the method of orthogonal projections in the theory of elasticity

#### Abstract

The method of orthogonal projections applied to the task of determining the stresses in the elastic deformable bodies, which allowed us to relax the requirements to the smoothness of the functions defining external forces and to the components of the tensor of the initial strains, which cause the appearance of balanced self-stresses. Examples of the calculation of quench stresses in a circular cylinder and residual stresses after shrinkage of the binder in composite cylinders made by winding are given.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(2):308-325

308-325

### Analysis of creep curves produced by the linear viscoelasticity theory under cyclic stepwise loadings

#### Abstract

Basic qualitative properties of the creep curves generated by the linear integral constitutive relation of viscoelasticity (with an arbitrary creep compliance) under cyclic piecewise-constant uni-axial loadings (with an arbitrary asymmetry stress ratio) are studied analytically. General formulas and a number of exact two-sided bounds are obtained for maximal, minimal and ratcheting strain values during each cycle, for their sequences limits, for the rate of plastic (non-recoverable) strain accumulation and for cyclic creep curve deviation from the creep curve at constant stress which is equal to the cycle mean stress. Their dependence on loading cycle parameters and creep compliance properties are analyzed. Monotonicity and convexity intervals of cyclic creep curves, sequences of maximal and minimal strain values and ratcheting strain sequence, their evolution with cycle number growth and conditions for their boundedness, monotonicity and convergence are examined. The linear viscoelasticity theory abilities for simulation of ratcheting, creep acceleration, cyclic hardening or softening and cyclic stability under symmetric cyclic loadings are considered. The analysis carried out revealed the importance of convexity restriction imposed on a creep compliance and the governing role of its derivative limit value at infinity. It is proved that the limit value equality to zero is the criterion for non-accumulation of plastic strain, for memory fading and for asymptotic symmetrization of cyclic creep curve deviation from the creep curve at the mean stress. The qualitative features of theoretic cyclic creep curves are compared to basic properties of typical test creep curves of viscoelastoplastic materials under cyclic multi-step uni-axial loadings in order to elucidate the linear theory applicability scope, to reveal its abilities to provide an adequate description of basic rheological phenomena related to cyclic creep and to develop techniques of identification and tuning of the linear constitutive relation. In particular, it is proved that the linear constitutive relation with an arbitrary (increasing convex-up) creep compliance function provides the absence of ratcheting and cyclic softening under symmetric cyclic multi-step loadings and the absence of creep acceleration whenever a symmetric cyclic loading is added to a constant load.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(2):326-361

326-361

### On the solution of fluid flow and heat transfer problem in a 2D channel with backward-facing step

#### Abstract

The stable stationary solutions of the test problem of hydrodynamics and heat transfer in a plane channel with the backward-facing step have been considered in the work for extremely high Reynolds numbers and expansion ratio of the stream $ER$. The problem has been solved by numerical integration of the 2D Navier--Stokes equations in `velocity-pressure' formulation and the heat equation in the range of Reynolds number $500 \leqslant {\sf Re} \leqslant 3000$ and expansion ratio $1.43 \leqslant ER \leqslant 10$ for Prandtl number ${\sf Pr} = 0.71$. Validity of the results has been confirmed by comparing them with literature data. Detailed flow patterns, fields of stream overheating, and profiles of horizontal component of velocity and relative overheating of flow in the cross section of the channel have been presented. Complex behaviors of the coefficients of friction, hydrodynamic resistance and heat transfer (Nusselt number) along the channel depending on the problem parameters have been analyzed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(2):362-375

362-375

### Modeling of freezing processes by an one-dimensional thermal conductivity equation with fractional differentiation operators

#### Abstract

We have studied the Stefan problem with Caputo fractional order time derivatives. The difference scheme is built. The algorithm and the program for a numerical solution of the Stefan problem with fractional differentiation operator are created. For the given entry conditions and freezing ground parameters we have obtained the space-time temperature dependences for different values of parameter α. The functional dependences of the interface motion for the generalized Stefan conditions depending on the value of α are estimated. Finally we have found that the freezing process is slowed down during the transition to fractional derivatives.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(2):376-387

376-387

### The Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. Editorial Policy

#### Abstract

The Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences is the periodical scientific edition published by Samara State Technical University since 1996. Samara State Technical University is one of eleven leading consolidated universities in Russia. Having centuryplus long history now it is a top research and educational center in Volga region. For a long time the journal was an edition where the new scientific results of Russian scientific schools had been published. Now the journal is focused on both Russian and foreign scientists, working in the priority research areas of Samara State Technical University because the main purpose of the journal is an open dissemination of scientific knowledge among Russian and foreign scientists. Since 2011 the journal is a quarterly printed edition (four issues a year); issue size-200 p.; language of articles-Russian, English. The journal is published in printed and electronic version. The editorial board takes and estimates the manuscripts irrespective of race, gender, nationality, heritage, citizenship, occupation, employment, residence, political, philosophic, religious and any other views of the author. The contributed article should be a completed scientific research. It shouldn’t have been published, or be in process of publication in other editions. The manuscript should contain novel scientific results in the priority research areas of Samara State Technical University, including “Differential Equations and Mathematical Physics”, “Mechanics of Deformable Solids”, “Mathematical Modeling, Numerical Methods and Software Systems”. The journal is published at the expense of publisher. All materials are publishing free of charge, the author’s fee is not provided. All materials of the electronic version are freely available. The target audience of the journal are the scientists working in the following areas: “Differential Equations and Mathematical Physics”, “Mechanics of Deformable Solids”, “Mathematical Modeling, Numerical Methods and Software Systems”. The journal is indexed in the Russian Science Citation Index database on the Web of Science platform. The journal is indexed in VINITI abstracts databases. The issue details are publishing in ULRICH’S Periodical Directory. The journal articles are indexed in Scholar.Google.com, zbMATH, CyberLeninka.ru, Math-Net.ru. The journal is integrated in CrossRef and FundRef search systems.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(2):388-397

388-397