**Publisher**Общество с ограниченной ответственностью Международная академическая издательская компания "Наука/Интерпериодика"**Country**Россия**Web**https://elibrary.ru/title_about.asp?id=33583

Aksenov A.V., Polyanin A.D., Flegontov A.V.

The article describes the biographical data of Professor Valentin Fedorovich Zaitsev. His scientific and teaching activities are briefly described.DOI: 10.1134/S2304487X18040028

Krutova I. Yu., Opryshko O.V.

The kinetic energy of upward swirling air flows, such as tornadoes or tropical cyclone has been calculated. There are several tornado classes, the characteristics of which are collected in the Fujita scale. Tornadoes of different classes and tropical cyclone can occur at latitudes of π/3, π/4, π/6 because of the effect of the Coriolis force far from the equator, as S.P. Bautin explains in his works. A method for approximate analytical and numerical calculation of the kinetic energy formed because of the formation of an ascending swirling flow has been presented. For a set of gas-dynamics equations, one specific characteristic Cauchy problem with initial data on the horizontal plane z = 0, which is a contact characteristic of multiplicity two, is considered. The gas-dynamic parameters are calculated using the fourth-order Runge–Kutta method, when solving a system of ordinary differential equations. A formula is given for calculating the kinetic energy of a special flow in the field under consideration. Gas density, circumferential, radial, and vertical components of the velocity vector have been calculated. The kinetic energy for a special steady flow of air of various intensities has been calculated using automated mathematical calculations.

Shmigirilov Yu. G., Frankovskiy B.A.

The electrical characteristics of the underwater discharge of an impact power generator have been experimentally studied in order to obtain a mathematical model of the nonlinear resistance of the discharge channel. The study has been performed with the impact power generator included in an experimental mobile device to excite pressure shock waves. A high-voltage electric breakdown has initiated a discharge of a low-voltage impact generator. The applicability of existing mathematical models to the electric discharge of a high-voltage capacitor bank and an impact generator has been analyzed. The discharge process of the generator includes three stages: (i) the breakdown of the electrode gap by a high voltage of the capacitor bank and the formation of a conductive channel, (ii) the short-term period of joint operation of a shock generator and a capacitor bank on an arc discharge, and (iii) the main stage of the generator discharge. The existing approximations of the time variation of the channel resistance can be used in calculations of the high-voltage electric discharge of the capacitor bank, but they are inapplicable to the discharge of the shock generator. The results of the study indicate that there is no universal mathematical model of the electrical resistance of the channel for three stages of the discharge. In the detailed description of transient processes, it is necessary to use approximating formulas of electrical resistances in accordance with the stage of the electric discharge. In engineering calculations, it is possible to neglect the transient processes of discharging a capacitor bank and to consider only the discharge of an impact generator. From the results of our studies, an empirical formula describing changes in the active electrical resistance of the channel during the discharge of the generator has been obtained for such calculations.

Kozhanova M. Yu., Litvinenko O.V., Lyapkov A.A., Golubenko I.S.

The experimental data on the change in the physical-mechanical properties of a thermosetting polymer with a unique complex of technically valuable properties are considered and analyzed. The study of mechanical properties has revealed the dose dependence of the tensile strength of radiomodified polydicyclopentadiene synthesized by the PolyHIPE technology. The study of the modification of the physical–mechanical properties of the polymer has showed an inhomogeneity in the strain dependence of the voltage change observed in the samples irradiated by electrons in the entire range of doses. This effect is the most pronounced at 50 kGy. When studying the strain–force dependence of the sample irradiated with a dose of 10 kGy, a plateau in the region of 0.04–0.12 mm has been found, which is most likely due to the inhomogeneities of the macrostructure. A comparative analysis of the strain–elongation dependences for the samples irradiated with a dose of 50 kGy and unirradiated samples has been carried out. The reduction of the strength limit by 42% under irradiation has been established. The growth of the tensile stress in the dose range from 20 to 40 kGy has been determined experimentally to the value for the unirradiated sample. A family of the strain–force curves has been presented for irradiation with different doses. The curve for irradiation with a dose of 40 kGy characterizes the change in the deformation properties of the material, despite a general decrease in the strength characteristics of the material.

Polyanin A.D., Shingareva I.K.

A singular point whose position is unknown a priori exists in Cauchy problems with blow-up solutions (for this reason, the application of standard fixed-step numerical methods for solving such problems can lead to significant errors). We have described a method for numerical integration of blow-up problems for nonlinear systems of ordinary differential equations of the first order $({{x}_{m}})_{t}{''} = {{f}_{m}}(t,\;{{x}_{1}},\; \ldots,\;{{x}_{n}})$, $m = 1,\; \ldots,\;n$, based on the introduction a new nonlocal independent variable ξ, which is related to the original variables t and ${{x}_{1}},\; \ldots,\;{{x}_{n}}$ by the equation $\xi _{t}{''} = g(t,\;{{x}_{1}},\; \ldots,\;{{x}_{n}},\;\xi )$. With a suitable choice of the regularizing function g, the proposed method leads to equivalent problems whose solutions are represented in a parametric form and do not have blowing-up singular points. Therefore, the transformed problems admit the use of standard fixed-step numerical methods. Several test problems are formulated for systems of ordinary differential equations that have monotonic and non-monotonic blow-up solutions, which are expressed in terms of elementary functions. The comparison of exact and numerical solutions of test problems has shown a high efficiency of numerical methods based on nonlocal transformations of a special kind. It has been shown that nonlocal transformations in combination with the method of lines can be successfully used to integrate initial-boundary value problems, described by nonlinear partial differential equations, which have blow-up solutions.

Aksenov A.V., Druzhkov K.P.

The system of equations of two-dimensional shallow water over a rough bottom is considered. An overdetermined system of equations for finding the allowed symmetries is obtained. The consistency of this overdetermined system of equations is investigated. A general form of the solution of this system is obtained. The kernel of the symmetry operators is found. Cases are presented where kernel extensions of symmetry operators exist. The corresponding classifying equations are given. The results of the group classification have indicated that the system of equations of two-dimensional shallow water over a rough bottom cannot be linearized by point change of variables in contrast to the system of equations of one-dimensional shallow water in the cases of horizontal and inclined bottom profiles.

Savyolova Т.I., Tolmacheva N.S.

Electron backscattering diffraction experiments are widely used in metal science to study the structure and texture of materials. The analysis of the adequacy of measuring the characteristics of materials is a topical task. To do this, mathematical models of the sample are created using the results of electron backscattering diffraction measurements. In this work, we have developed a mathematical model of the grain structure of an elliptic polycrystalline sample and derive calculation formulas. Random variables are the minor semi-axis, the coefficient of elongation (the major-to-minor semi-axis ratio), and the angle of rotation of the ellipse. These parameters have the exponential and uniform distributions, respectively. Numerical examples of the lengths of the chords of ellipses are obtained using the Monte Carlo method. Results of a numerical experiment are presented in the form of histograms. The choice of the simulation parameters is based on the electron backscattering diffraction data for some steels.

Kozlov V.K., Kudryashov N.A., Chmykhov M.A.

A mathematical model of natural convection in the field of gravity in the Boussinesq approximation has been presented. This model contains the continuity equation, where density variations are ignored, the Navier–Stockes equation, and the equation for heat flow. The Rayleigh–Benard convection in a rectangular box with different types of boundary conditions has been investigated. The equations solved numerically by an original solver with the use of the object-oriented programming language OpenFOAM. Solver is based on the Pressure-Implicit with Splitting of Operators (PISO) algorithm and finite volume method. À modified PISO algorithm implementation has been presented. It has been changed for calculating conservation of mass. The Prandtl, Grashof, and Rayleigh numbers have been examined. The solver has been tested for the Rayleigh–Benard convection between parallel planes with different temperatures, steady convection in a horizontal fluid layer, and a natural convection flow in a square box enclosed by non-isothermal wall. Results have been visualized with the use of open source application ParaView.

Salomatin A.A.

Information on the development of a high-performance unit for a diode-pumped laser power supply has been reported. This device operates from the external dc power supply, which has a function of output voltage adjustment by an RS-232 (or RS-485) digital interface or an analog interface. The unit is based on the classic scheme of a linear current stabilizer on a current adjustable operational amplifier. The adjustment is performed by changing the reference voltage supplied from a digital-to-analog converter. The scheme involves the optical separation of the digital and power sections. The unit is also equipped with an emergency shutdown scheme. The unit has functions of output current stabilization and adjustment up to 10 A, allows obtaining 6-μs-duration edges with an efficiency of more than 95%. A high efficiency is achieved by adjusting the output voltage of a power supply by a communication interface. The adjustment is performed so that the drain-to-source voltage of the power transistor is minimal and enough to switch on the transistor. The device guarantees current stabilization in both the pulsed and continuous modes. The device scheme allows duplication of power channels. The main ideas, functions, and technical implementation of the device, as well as experimental results, have been reported.

Popkov V.G., Savyolova T.I.

The normal distribution on the rotation group SO(3) of the three-dimensional Euclidean space is used more often than other standard distributions in quantitative texture analysis and applied crystallography when the distribution function of grain orientations is reconstructed from pole figures obtained experimentally with X-ray or neutron methods. To this end, pole figures are approximated by normal distributions and their projections on the sphere S2. Among the approaches to determining the normal distribution in the study of textures of polycrystalline materials, the normal distribution that satisfies the central limit theorem on the rotation group is the most correct. The canonical normal distribution satisfies this theorem and can be computed by Fourier expansion or by the analytic approximation method. To achieve a given accuracy, the Fourier series method requires large computational powers, while the analytic approximation method has weaker requirements. In this paper, we give sections of the orientation distribution function for the canonical normal distribution obtained by the analytic approximation method for various parameters. Examples of pole figures are computed. The error of the analytic approximation method calculations in the norms of the spaces C(SO(3)) and L2(SO(3)) for various parameters is estimated. It is concluded that an approximating formula can be applied to calculate the canonical normal distribution in a certain range of parameters.