# Vol 24, No 2 (2020)

**Year:**2020**Articles:**10**URL:**https://journals.eco-vector.com/1991-8615/issue/view/2521

## Full Issue

### On a boundary value problem for a third-order parabolic-hyperbolic type equation with a displacement boundary condition in its hyperbolicity domain

#### Abstract

In the article, we investigate a boundary-value problem with a third-order inhomogeneous parabolic-hyperbolic equation with a wave operator in a hyperbolicity domain. A linear combination with variable coefficients in terms of derivatives of the sought function on independent characteristics, as well as on the line of type and order changing is specified as a boundary condition. We have established necessary and sufficient conditions that guarantee existence and uniqueness of a regular solution to the problem under study. In some cases, a solution representation is written out explicitly.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(2):211-225

211-225

### Group classification, invariant solutions and conservation laws of nonlinear orthotropic two-dimensional filtration equation with the Riemann–Liouville time-fractional derivative

#### Abstract

A nonlinear two-dimensional orthotropic filtration equation with the Riemann–Liouville time-fractional derivative is considered. It is proved that this equation can admits only linear autonomous groups of point transformations. The Lie point symmetry group classification problem for the equation in question is solved with respect to coefficients of piezoconductivity. These coefficients are assumed to be functions of the square of the pressure gradient absolute value. It is proved that if the order of fractional differentiation is less than one then the considered equation with arbitrary coefficients admits a four-parameter group of point transformations in orthotropic case, and a five-parameter group in isotropic case. For the power-law piezoconductivity, the group admitted by the equation is five-parametric in orthotropic case, and six-parametric in isotropic case. Also, a special case of power function of piezoconductivity is determined for which there is an additional extension of admitted groups by the projective transformation. There is no an analogue of this case for the integer-order filtration equation. It is also shown that if the order of fractional differentiation $\alpha \in (1,2)$ then dimensions of admitted groups are incremented by one for all cases since an additional translation symmetry exists. This symmetry is corresponded to an additional particular solution of the fractional filtration equation under consideration. Using the group classification results for orthotropic case, the representations of group-invariant solutions are obtained for two-dimensional subalgebras from optimal systems of symmetry subalgebras. Examples of reduced equations obtained by the symmetry reduction technique are given, and some exact solutions of these equations are presented. It is proved that the considered time-fractional filtration equation is nonlinearly self-adjoint and therefore the corresponding conservation laws can be constructed. The components of obtained conserved vectors are given in an explicit form.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(2):226-248

226-248

### Sobolev spaces and boundary-value problems for the curl and gradient-of-divergence operators

#### Abstract

We study boundary value and spectral problems in a bounded domain $G$ with smooth border for operators $\operatorname{rot} +\lambda I$ and $\nabla \operatorname{div} +\lambda I$ in the Sobolev spaces. For $\lambda\neq 0$ these operators are reducible (by B. Veinberg and V. Grushin method) to elliptical matrices and the boundary value problems and satisfy the conditions of V. Solonnikov's ellipticity. Useful properties of solutions of these spectral problems derive from the theory and estimates. The $\nabla \operatorname{div}$ and $ \operatorname{rot}$ operators have self-adjoint extensions $\mathcal{N}_d$ and $\mathcal{S}$ in orthogonal subspaces $\mathcal{A}_{\gamma }$ and $\mathbf{V}^0$ forming from potential and vortex fields in $\mathbf{L}_{2}(G)$. Their eigenvectors form orthogonal basis in $\mathcal{A}_{\gamma }$ and $\mathbf{V}^0$ elements which are presented by Fourier series and operators are transformations of series. We define analogues of Sobolev spaces $\mathbf{A}^{2k}_{\gamma }$ and $\mathbf{W}^m$ orders of $2k$ and $m$ in classes of potential and vortex fields and classes $ C (2k,m)$ of their direct sums. It is proved that if $\lambda\neq \operatorname{Sp}(\operatorname{rot})$, then the operator $ \operatorname{rot}+\lambda I$ displays the class $C(2k,m+1)$ on the class $C(2k,m)$ one-to-one and continuously. And if $\lambda\neq \operatorname{Sp}(\nabla \operatorname{div})$, then operator $\nabla \operatorname{div}+\lambda I$ maps the class $C(2(k+1), m)$ on the class $C(2k,m)$, respectively.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(2):249-274

249-274

### Creep and long-term strength of metals under unsteady complex stress states (Review)

#### Abstract

This article is an analytical review of experimental and theoretical studies of creep and creep rupture strength of metals under unsteady complex stress states published over the past 60 years.The first systematic studies of the creep of metals under complex stress conditions were published in the late 50s and early 60s of the 20th century in the Soviet Union (L. M. Kachanov and Yu. N. Rabotnov) and Great Britain (A. E. Johnson). Pioneering work on creep rupture strength first appeared in the USSR (L. M. Kachanov and Yu. N. Rabotnov). Subsequently, Yu. N. Rabotnov developed the kinetic theory of creep and creep rupture strength, with the help of which it is possible to efficiently describe various features of the creep process of metals up to fracture under various loading programs. Different versions of the kinetic theory use either a scalar damage parameter, or a vector parameter, or a tensor parameter, or a combination of them. Following the work of M. Kachanov and Yu. N. Rabotnov mechanics of continuum destruction began to develop in Europe, in Asia, and then in the USA.The hypothesis of proportionality of stress deviators and deviators of creep strain rates is accepted as the main connection between the components of stress tensors and creep strains. When modeling experimental data, the proportionality coefficient in this dependence takes different forms. The main problem in the development of this direction is the difficulty in obtaining experimental data with arbitrary loading programs.This review provides the main results of studies conducted by scientists from different countries. Except Yu. N. Rabotnov and L. M. Kachanov, also a significant contribution to the development of the direction of science made by Russian scientists N. N. Malinin, A. A. Ilyushin, V. S. Namestnikov, S. A. Shesterikov, A. M. Lokoshchenko, Yu. P. Samarin, O. V. Sosnin, A. F. Nikitenko, et al.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(2):275-318

275-318

### Exact solutions to generalized plane Beltrami–Trkal and Ballabh flows

#### Abstract

Nonstationary plane flows of a viscous incompressible fluid in a potential field of external forces are considered. An elliptic partial differential equation is obtained, with each solution being a vortex flow stream function described by an exact solution to the Navier–Stokes equations. The obtained solutions generalize the Beltrami–Trkal and Ballabh flows. Examples of such new solutions are given. They are intended to verify numerical algorithms and computer programs.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(2):319-330

319-330

### Research of a retrial queueing system with exclusion of customers and three-phase phased by follow-up

#### Abstract

In this paper, we consider a retrial queueing system (RQ-system) which receives to the input a Poisson flow with a given intensity. If at the time of customer the server is busy, the displacement of customer standing on the server takes place. Customers that do not have time to be successfully serviced go into orbit, in order to, after an accidental exponential delay, again turn to the server for maintenance. It is shown that the limiting characteristic function of the number of customers in the orbit and the states of the server converges to a three-dimensional Gaussian distribution. The mean vector and covariance matrix are obtained for this distribution. A stationary probability distribution of the server states is also found.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(2):331-342

331-342

### Stochastic calculation of curves dynamics of enterprise

#### Abstract

The article proposes mathematical models of the stochastic dynamics of the single-factor manufacturing enterprises development through internal and external investments. Balance equations for such enterprises are formulated, describing random processes of continuous increase in output and growth of production factors. The interaction of proportional, progressive and digressive depreciation with internal and external investments is investigated. Equations are obtained to determine the equilibrium state of the enterprise and the limiting values of the factors of production are calculated. The cases of the stable progressive development of the enterprise, the suspension of its work during the re-equipment of production and the temporary crisis of production shutdown during equipment replacement are considered. The algorithm for the numerical solution of stochastic differential equations of enterprise development is constructed in accordance with the Euler–Maruyama method. For each implementation of this algorithm, the corresponding stochastic trajectories are constructed for the random function of the production factor. A variant of the method for calculating the expectation of a random function of a factor of production is developed and the corresponding differential equation is obtained for it. It is shown that the numerical solution of this equation and the average value of the function of the production factor calculated from two hundred realizations of stochastic trajectories give almost identical results. Numerical analysis of the developed models showed good compliance with the known statistical data of the production enterprise.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(2):343-364

343-364

### Couette flow of hot viscous gas

#### Abstract

A new exact solution is found for the equations of motion of a viscous gas for a stationary shear flow of hot (800–1500 K) gas between two parallel plates moving at different speeds (an analog of the incompressible Couette flow). One of the plates was considered thermally insulated. For the dependence of the coefficient of viscosity on temperature, the Sutherland formula is adopted. Unlike other known exact solutions, instead of a linear association between the viscosity and thermal conductivity coefficients, a more accurate formula was used to calculate the thermal conductivity coefficient, having the same accuracy in the temperature range under consideration as the Sutherland formula (2 %). Using the obtained exact solution, the qualitative effect of compressibility on the friction stress and the temperature, and velocity profiles were investigated. It is shown that the compressibility of the gas leads to an increase in the friction stress, if one of the plates is thermally insulated. The new exact solution was compared with the known exact solution (Golubkin, V.N. & Sizykh, G.B., 2018) obtained using the Sutherland formula for the viscosity coefficient and the Reynolds analogy for the thermal conductivity coefficient. It was found that both solutions lead to the same conclusions about the qualitative effect of compressibility on the friction stress and on the temperature and velocity profiles. However, the increase in friction stress caused by compressibility of the gas turned out to be underestimated twice when using the Reynolds analogy. This shows that the assumption of a linear relationship between the coefficients of viscosity and thermal conductivity can lead to noticeable quantitative errors.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(2):365-378

365-378

### $\alpha$-Differentiable functions in complex plane

#### Abstract

In this paper, the conformable fractional derivative of order $\alpha$ is defined in complex plane. Regarding to multi-valued function $z^{1-\alpha}$, we obtain fractional Cauchy–Riemann equations which in case of $\alpha=1$ give classical Cauchy–Riemann equations. The properties relating to complex conformable fractional derivative of certain functions in complex plane have been considered. Then, we discuss about two complex conformable differential equations and solutions with their Riemann surfaces. For some values of order of derivative, $\alpha$, we compare their plots.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(2):379-389

379-389

### An undamped oscillation model with twodifferent contact angles for a spherical dropletimpacting on solid surface

#### Abstract

In order to further elucidate the dynamic theory of droplet oscillating on solid surface, a new handling method of contact angle of the droplet during the process of the oscillation was founded, which is based on the spherical model. The influence of gravity on the contact angle andspreading radius was discussed. Thus, an equation between the spreading radius of the dropletand time flow was founded. The results of theoretical calculation were compared with smoothednumerical results.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2020;24(2):390-400

390-400