University proceedings. Volga region. Physical and mathematical sciences

Background. The numerical method for solving hypersingular integral equations on a segment that arise in many problems of mathematical physics is considered. Materials and methods. Galerkin method is used to solve hypersingular equations with basic functions – Chebyshev polynomials of the 2^nd kind. The projection method is considered in special classes of functions. Results and conclusions. The convergence of Galerkin method for solving hypersyngular equations in special classes of functions is proved. An estimate of the convergence rate of Galerkin method is obtained.

Background. The oscillating motions of dynamic systems are investigated, namely, motions that are not bounded and, in addition, have the property that they do not tend to infinity as time tends to plus infinity. Such motions play an important role in various problems of mathematical physics, celestial mechanics, thermodynamics and astrophysics. Materials and methods. New concepts related to the oscillation of the set of all solutions of a system of differential equations are introduced into consideration: the concept of equioscillation in the limit of the set of all solutions and partial analogues of this concept. Results. Based on the principle of comparison of Matrosov with Lyapunov vector functions and the connection found by the author between Poisson boundedness and oscillability of solutions, sufficient conditions for equioscillability in the limit of the set of all solutions, as well as partial analogues of these conditions, are obtained. The work continues the author’s research on the study of the boundedness and oscillation of the sets of all solutions of differential systems using Lyapunov functions and Lyapunov vector functions. Conclusions. The obtained theoretical results can be used to analyze complex dynamic systems in various fields of science.

Background. The purpose of the work is a numerical study of the features of resonant (due to the excitation and propagation of plasmonic excitations) polarization magneto- optical (MO) effects and methods for controlling the polarization of THz radiation during scattering on graphene nanoribbon gratings in an external magnetic field. Materials and methods. Graphene is a 2D material with unique optical and electronic properties. It serves as a platform for new THz applications and microminiature systems with new capabilities. Excitation of surface magnetoplasmons-polaritons with a dispersion law changed due to the application of an external magnetic field significantly enhances MO effects in graphene structures. For the first time, a numerical study of polarization MO effects was carried out by automated modeling methods using the CST MWS software package based on the solution (using the finite element method in the frequency domain) of the electrodynamic problem of TEM wave diffraction on a graphene nanoribbon grating with the application of a perpendicular external magnetic field and analysis of the diffracted field characteristics in the THz range. Results. The results of modeling 3D e-Field scattering patterns of a normally incident p-polarized TEM wave on a cell of a graphene nanoribbon grating in a perpendicular external magnetic field at the frequencies of plasmon resonance (at B0 = 0) and magnetoplasmon resonances for different values of B0 (2, 4, 7, 10 T) were obtained. Based on the analysis of the results of calculating the ratio of the horizontal and vertical components Ex/Ey of the diffracted field and the axial ratio AR at the cross-section points (φ = 0°) of the main lobe of the 3D e-Field scattering patterns, the polarization type of the scattered THz radiation was studied and the Faraday rotation angle of the polarization plane of the transmitted wave and the Kerr rotation angle describing the rotation of the polarization axis of the reflected wave were calculated. Conclusions. It follows from the results of the numerical study that at the diffraction of a normally incident TEM-wave with p-polarization on a cell of the graphene nanoribbon grating with the application of a perpendicular external magnetic field at the frequencies of magnetoplasmon resonances frequency-tunable MO-effects are observed: 1) rotation of the polarization plane of a linearly polarized wave transmitting through a magnetized graphene grating, when the wave vector of the incident wave is parallel to the vector of the external magnetic field intensity (Faraday effect), while the Faraday rotation angle depends on the value of the external magnetic field; 2) changes in the orientation of polarization and ellipticity of the reflected wave (polar Kerr MO-effect), a linearly polarized THz wave, reflected from magnetized graphene, becomes elliptically polarized; in this case, the major axis of the polarization ellipse rotates by a certain angle with respect to the plane of polarization of the incident THz radiation proportional to the magnitude of the external magnetic field.

Background. A theory of disk galaxies with a violation of cylindrical symmetry is being constructed, which solves the problem of the spiral structure of galaxies within the framework of a general density wave model. Materials and methods. The theory is based on the classical hydrodynamics of a continuous self-gravitating medium with a quasiclassical gravitational field containing a description of the effect of hidden mass (dark matter). Results. The equations of the spatial distribution of the parameters of the medium and the gravitational field, as well as the equations of the evolution of scale factors, are derived under the conditions of dynamic equilibrium of a rotating medium with a violation of cylindrical symmetry. An analogy is established between the effect of hidden mass and the gravitational permeability of the medium. The description of the interaction of the disk structure with the external environment is considered. Conclusions. The theory allows us to build models of disk galaxies with a spiral structure in the presence of a hidden mass effect. Such models include descriptions of the intrinsic evolution of disk galaxies in the presence of external influences.

Background. In the modern production of power electronics devices, sintering technology is used for low-temperature connection of their structural elements using special pastes based on ultrafine silver powders. The key problem in the manufacture of such pastes is the agglomeration of silver particles in the powder. The purpose of the work was to develop methods for modifying the granulometric composition of silver powders, leading to an increase in the proportion of submicron particles in them. Materials and methods. The work investigated ultrafine silver powder of domestic production. The granulometric composition of the powder was determined by laser diffraction using a SALD-3101 granulometer. For measurements, suspensions of the studied powder in isopropyl alcohol with concentrations of 50 μg/ml and 75 mg/ml were prepared. Suspensions were subjected to ultrasound treatment at a frequency of 35 kHz in a PSB-5735-05 ultrasonic bath. Results. It was found that the original silver powder consisted of two fractions of particles with sizes from 0.5 to 2.0 μm and from 10 to 500 μm. As a result of ultrasonic (US) treatment of a lowconcentration powder suspension (50 μg/ml) for 80 minutes, the volume fraction of submicron fraction particles increased from 18 to 60%. Long-term (4 hours or more) US treatment of a highly concentrated suspension (75 mg/ml) did not result in the appearance of silver particles smaller than 1 μm in it. Such particles appeared in the suspension upon addition of myristic acid as a dispersant, followed by US-treatment. Dilution of the suspension with isopropyl alcohol to a concentration of 50 μg/ml, followed by US-treatment, made it possible to increase the volume fraction of silver particles with sizes less than 1 μm to 70%. Conclusions. A method has been developed for modifying the granulometric composition of ultrafine silver powders, which makes it possible to increase the volume fraction of submicron particles to 70%.