University proceedings. Volga region. Physical and mathematical sciences
ISSN (print): 2072-3040
Founder: Penza State University
Editor-in-Chief: Krevchik Vladimir Dmitrievich, Doctor of Physics and Mathematics. Sc., professor
Frequency / Access: 4 issues per year / Open
Included in: Higher Attestation Commission List, RISC
Registration: the journal is registered by the Federal Service for Supervision in the Sphere of Telecom, Information Technologies and Mass Communications. Registration certificate: ПИ № ФС77-26984 from 19.01.2007.
Periodicity: 4 issues per year Number of copies: 1000 copies.
Scientific areas (subject groups):
1.1.2. Differential Equations, Dynamical Systems and Optimal Control
1.1.6. Computational Mathematics
1.3.3. Theoretical Physics
1.3.6. Optics
1.3.8. Condensed matter Physics
1.3.11. Semiconductor Physics
1.3.15. Physics of the Atomic Nucleus and Elementary Particles
The journal publishes original articles describing results of fundamental and applied research in physics and mathematics, as well as survey articles by leading experts in the journal’s subject area.
最新一期
编号 1 (2025)
MATHEMATICS
Measurement of the near electromagnetic field and restoration of inhomogeneity parameters in a dielectric body
摘要
Background. The inverse problems of electromagnetic sensing, aimed at determining the internal parameters of the object from external measurements of the electromagnetic field, are incorrectly posed and computationally complex. Non-linearity and instability of solutions require the use of special regularization methods. The development of effective non-iterative methods for solving such problems, especially for three-dimensional objects, remains an urgent task for various fields such as medical imaging, geophysics and non-destructive testing. The purpose of the work is to develop and analyze a non-iterative method for solving the inverse electromagnetic scattering problem for determining the dielectric constant of a limited three-dimensional object based on near-field measurements. Materials and methods. The work is based on solving the direct problem of diffraction of a monochromatic electromagnetic wave on a limited volume diffuser using a singular integro- differential equation of an electric field. A two-step non-iterative method is proposed to solve the inverse problem. It is based on the measurement of the near field scattered by an object, and can be applied for solutions from finite-dimensional spaces of piecewise constant functions. Results. A method for solving the inverse problem of electromagnetic scattering is implemented. The results of solving the direct and inverse problems are presented. A comparison of transmission coefficients for several experiments is obtained. Conclusions. The developed non-iterative method for solving the inverse problem of electromagnetic scattering provides determination of the dielectric constant of a limited threedimensional object based on near-field measurements. The method demonstrates its effectiveness and can be applied in various fields requiring non-invasive determination of object parameters.
3-12
Differential symmetry operators first order canonical differential equations
摘要
Background. Symmetries play an important role in mechanics and theoretical physics. The main models in these sciences are differential equations and systems of equations. Therefore, the study of symmetries of differential equations has not only theoretical, but also practical meaning. Canonical second-order differential equations are one of the basic equations of mathematical physics. The article sets the task of describing first-order differential symmetry operators of canonical equations and Lie algebras formed by such operators. Materials and methods. The introduction to this work is devoted to a brief overview of the general theory of differential substitutions of dependent variables. Such substitutions generate differential symmetry operators, and first-order operators, in particular, form Lie algebras with respect to the commutator. Paragraph 2 describes in general terms the concepts that are directly used in this article. Canonical equations and Laplace invariants are introduced. Results. In section 3, we formulate and prove a theorem on necessary and sufficient conditions under which a linear differential operator of the first order is an operator of differential symmetry of a canonical equation. In section 4, the theorem is used to describe the set of differential symmetry operators of the Euler-Poisson equations. In section 5, we establish the general form of the commutator of first-order differential symmetry operators and prove that the Lie algebra of first-order differential symmetry operators of the Euler-Poisson equations is isomorphic to the Lie algebra of second-order matrices. Section 6 contains the differential symmetry operators of canonical equations with constant coefficients, as well as canonical equations of the form. The Lie algebras of such operators turn out to be solvable four-dimensional Lie algebras with a one-dimensional center. Conclusions. The results obtained seem to be quite significant. But the main result is Theorem 1, which can be used to describe Lie algebras of differential symmetry of first-order operators in other interesting cases not covered in this paper.
13-28
On a strict mathematical definition of the brachistochrone’s shape taking into account the thermal effect in the contact zone
摘要
Background. Current research on the properties of brachistochrones as lines with extreme properties indicates that this work is relevant because it takes into account for the first time the influence of thermal effects on the shape of the brachistochrone. The purpose of the study is an analytically rigorous solution to the problem. Matherials and methods. The main method for solving the problem is the moving basis method, which has proven itself well in solving many problems related to the study of various properties of brachistochrones. Results. A rigorous analytical solution of the formulated problem is given, taking into account the thermal effect, which was taken into account by introducing a dissipative function. Conclusions. Thanks to the algorithm proposed in the article, a general methodological approach has been formulated that is useful in solving such problems related to taking into account the thermal properties of materials.
29-44
On bifurcations of a periodic orbit tangent to switching lines at two points
摘要
Background. Dynamic systems defined by discontinuous piecewise smooth vector fields on a plane are natural mathematical models of relay systems in automatic control theory. Periodic trajectories describe self-oscillations. Although a significant number of works have been devoted to the study of the birth of periodic trajectories, the description oftypical bifurcations is far from complete. The purpose of this research is to study bifurcations of periodic trajectories similar to bifurcations of double and triple cycles of a smooth dynamic system. Materials and methods. The method of point mappings and other methods of the qualitative theory of differential equations are applied. Results. A generic twoparameter family of piecewise smooth vector fields on a plane is considered. It is assumed that for zero values of the parameters the field has a periodic trajectory Γ touching the switching lines at two singular points of the fork type and not containing other singular points. In this case, both components into which Γ divides the plane intersect with the separatrices of the forks that are not contained in Γ. Three cases are considered. In the first case, Γ is stable and bifurcates similarly to a triple cycle, in the second case, Γ is stable, but its bifurcations consist only in changing the number of sections of sliding motions on it, and in the third case, Γ is semistable and bifurcates similarly to a double cycle. Conclusions. Several possible scenarios for the birth and rebirth of periodic trajectories of a piecewise smooth dynamic system when its parameters change are indicated.
45-57
Iterative methods for solving quadratic Volterra integral equations of the first kind
摘要
Background. The paper is devoted to the numerical study of integral equations of the first kind with quadratic nonlinearity, which are part of the generalized Volterra integropower series and describe dynamical systems with one input and one output. Such equations are widely used in the modeling of stationary systems with constant dynamic characteristics during the transfer process. Materials and methods. The proposed iterative numerical methods are based on the preliminary linearization of the integral operator according to the modified Newton-Kantorovich scheme and the use of the regularization parameter to ensure stability to fluctuations in the input data. To solve linear equations ateach iteration, the method of successive approximations is used in combination with the approximation of the exact solution by a polynomial spline constructed for each segment on the zeros of Legendre polynomials. The Gauss compound quadrature formula is used to calculate integrals. Results and conclusions. A number of iterative numerical schemes for solving quadratic Volterra integral equations are proposed. The convergence theorems of the modified Newton-Kantorovich method are formulated. Numerical results confirming the convergence of the methods are presented.
58-69
PHYSICS
Non-covalent interaction of boron and nitrogen atoms
摘要
Background. The calculation of the interaction between boron and nitrogen atoms is interesting from the point of view of predicting physical properties and creating new dielectric materials and carbon-free nanomaterials. The purpose of the work is to calculate non-covalent (dispersion) interaction for pairs of atoms B-B, N-N and B-N based on the first principles of quantum mechanics. The calculation is carried out in practice for the first time. Materials and methods. The article uses the density functional theory (DFT) in the electron gas approximation. The Coulomb, kinetic, exchange, and correlation contributions to the interaction energy are taken into account. The electron density is given taking into account the shell structure of atoms in the Roothaan-Hartree-Fock approximation. An original numerical algorithm based on the use of quadrature formulas and CUDA computing parallelization technology is used to calculate improper integrals. Results. Radial electron density functions and corresponding potential curves are constructed over a wide range of interatomic distances. The parameters of potential wells and the constants of the dispersion interaction are calculated. The correctness of the Lorentz-Berthelot rules of thumb for combining potential parameters has been verified. Conclusions. The obtained values of the dispersion interaction constants for homoatomic pairs are consistent with the results known from the literature. Using first-principles calculations, it is possible to determine the parameters of model pair potentials, in particular the Sutherland potential. It is shown that the Lorentz- Berthelot rules do not work for the non-covalent interaction of boron and nitrogen atoms.
70-80
Investigation of the electronic and optical properties of thin-film coatings based on SnO₂:Sb
摘要
Background. Transparent conducting oxides (TCOs) based on tin dioxide (SnO₂) doped with antimony (Sb) are of significant interest for modern technologies due to their unique combination of high transparency in the visible range and good electrical conductivity. Such materials are widely used in optoelectronics, solar cells, and sensor devices. However, optimizing their properties requires a deep understanding of charge transport mechanisms, which can be achieved using the classical Drude theory. The purpose of this work is to apply the Drude theory to analyze the electronic and optical properties of SnO₂:Sb thin films obtained by spray pyrolysis, as well as to investigate the influence of charge carrier concentration and mobility on conductivity and plasma frequency. Materials and methods. Thin films of SnO₂:Sb were deposited on glass substrates using the spray pyrolysis method. The charge carrier concentration n was determined using the Hall effect, and the resistivity was measured using the four-point probe method. The classical Drude theory was used to analyze the electronic properties, allowing the calculation of carrier mobility, relaxation time, and plasma frequency. Optical properties were investigated using spectroscopy in the visible and near-infrared ranges. Results. The conductivity σ of SnO₂:Sb films varied in the range of 10³–10⁴ S/m depending on the antimony doping level. The carrier mobility ranged from 4.83∙10⁻⁴ to 15.91∙10⁻⁴ m²/(V·s). The plasma frequency was in the range of 1.19∙10¹⁴ to 7.94∙10¹⁴ rad/s, corresponding to wavelengths from 378 nm to 2520 nm. The SnO₂:Sb films demonstrated high transparency in the visible range (over 80%) for samples with low carrier concentration. With increasing carrier concentration, a shift in the plasma frequency toward the ultraviolet region was observed, leading to a decrease in transparency in the visible range. The drift velocity increased linearly with applied voltage and decreased with increasing distance between contacts. For samples with high mobility, the drift velocity reached values of 13.25∙10⁻⁴ m/s at U =5 mV and d =5 mm. Conclusions. The Drude theorywas successfully applied to analyze the electronic and optical properties of SnO₂:Sb thin films. It was established that the conductivity and transparency of the material can be optimized by varying the antimony doping level. It was shown that the drift velocity of charge carriers depends on mobility, voltage, and sample geometry. This opens up opportunities for designing devices with improved characteristics. The obtained results demonstrate the potential of using SnO₂:Sb in optoelectronic devices where a combination of high transparency and conductivity is required.
81-92
Research of heat transfer and magnetohydrodynamics in the flow of liquid between two coaxial cylinders
摘要
Background. Studies of viscous fluid flow between rotating cylinders (known as the Couette flow), both experimental and theoretical, are still relevant and are widely used in technical applications (heat exchangers, nuclear and chemical reactors, separators, astrophysics). This class of problems becomes more complicated when heat exchange takes place along with hydrodynamics. The complexity of such problems increases with a joint consideration of heat exchange and the flow of viscous conductive fluid between cylinders rotating with different angular velocities. Further research is needed to study and better understand such complex processes, which will serve to clarify the mathematical models of heat exchange and magnetohydrodynamics. The paper considers heat exchange and magnetohydrodynamics of fluid (for a given velocity field) between two rotating coaxial cylinders. The purpose of the work is to study the influence of angular velocities of cylinder rotation, Joule heat dissipation, internal heat sources/sinks, cylindrical layer thickness and magnetic Reynolds number on the temperature and magnetic induction fields of liquid in the cylindrical layer. Materials and methods. In dimensionless form, the problem of heat exchange and flow of electrically conductive liquid between two rotating cylinders is solved numerically in a cylindrical coordinate system. The control volume method (Patankar method) is used to solve the problem. Results. The influence of the velocity field, internal heat sources/sinks, Joule heat dissipation, cylindrical layer thickness on the temperature fields, radial and angular components of the magnetic induction of an electrically conducting liquid between two coaxial rotating cylinders is investigated. It is found that changing the direction of rotation of the cylinders leads to a change in the type of extremum of the angular component of magnetic induction. Reducing the magnetic Reynolds number increases the intensity of heat exchange in the liquid. Conclusions. The results obtained can be used both in the study of thermal and magnetohydrodynamics processes and in the design of power and chemical devices, separators, instruments and installations.
93-109