卷 24, 编号 2 (2024)
Mathematics
A highly accurate difference method for solving the Dirichlet problem of the Laplace equation on a rectangular parallelepiped with boundary values in C^(k,1)
摘要
A three-stage difference method for solving the Dirichlet problem of Laplace's equation on a rectangular parallelepiped is proposed and justified. In the first stage, approximate values of the sum of the pure fourth derivatives of the solution are defined on a cubic grid by the 14-point difference operator. In the second stage, approximate values of the sum of the pure sixth derivatives of the solution are defined on a cubic grid by the simplest $6$-point difference operator. In the third stage, the system of difference equations for the sought solution is constructed again by using the $6$-point difference operator with the correction by the quantities determined in the first and the second stages. It is proved that the proposed difference solution to the Dirichlet problem converges uniformly with the order $O(h^{6}(|\ln h|+1))$, when the boundary functions on the faces are from $C^{7,1}$ and on the edges their second, fourth, and sixth derivatives satisfy the compatibility conditions, which follows from the Laplace equation. A numerical experiment is illustrated to support the analysis made.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(2):162-172
162-172
New integral inequalities in the class of functions (h, m)-convex
摘要
In this article, we have defined new weighted integral operators. We formulated a lemma in which we obtained a generalized identity through these integral operators. Using this identity, we obtain some new generalized Simpson's type inequalities for $(h,m)$-convex functions. These results we obtained using the convexity property, the classical Hölder inequality, and its other form, the power mean inequality. The generality of our results lies in two fundamental points: on the one hand, the integral operator used and, on the other, the notion of convexity. The first, because the ''weight'' allows us to encompass many known integral operators (including the classic Riemann and Riemann – Liouville), and the second, because, under an adequate selection of the parameters, our notion of convexity contains several known notions of convexity. This allows us to show that many of the results reported in the literature are particular cases of ours.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(2):173-183
173-183
On potentiality, discretization, and integral invariants of the infinite-dimensional Birkhoff systems
摘要
In the study of the equations of motion of systems of various physical nature, there are problems in determining the qualitative indicators and properties of motion according to the known structure and properties of the equations under consideration. Such qualitative indicators for finite-dimensional systems are, in particular, integral invariants — integrals of some functions that retain their value during the system movement. They were introduced into analytical mechanics by A. Poincaré. In the future, the connection of integral invariants with a number of fundamental concepts of classical dynamics was established. The main purpose of this work is to extend some notions of the theory of integral invariants to broad classes of equations of motion of infinite-dimensional systems. Using a given Hamilton’s action, the equations of motion of potential systems with an infinite number of degrees of freedom are obtained, generalizing the well-known Birkhoff equations. A difference analog with discrete time is constructed for them. Based on it, a difference approximation of the corresponding integral invariant of the first order is found.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(2):184-192
184-192
Algorithmic search for integer Abelian roots of a polynomial with integer Abelian coefficients
摘要
In this work, we consider the operations over Abelian integers of rank $n$. By definition, such numbers are elements of the complex field and have the form of polynomials with integer coefficients from the $n$th degree primitive root of 1. In contrast, the degrees of such polynomials are not greater than Euler's totient function $\varphi(n)$. We provide an example to show that there are infinitely many Abelian integers inside any zero-centered circle on the complex plane. In this work, for considered operations we give in particular the algorithm of calculation of the inverse for the Abelian integer of rank $n$. It allows us to analyze not only the rings of such numbers but also the fields of Abelian integers. Natural arithmetics for such algebraic structures leads us to study the polynomials with integer Abelian coefficients. Thus, in the presented work we also investigate the problem of finding roots of such polynomials. As a result, we provide an algorithm that finds the integer Abelian roots of the polynomials over the ring of Abelian integers. This algorithm is based on the proposed statement that all roots of the polynomial are bounded by some domain. The computer calculations confirm the statistical truth of the statement.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(2):193-199
193-199
Global solvability of the inverse spectral problem for differential systems on a finite interval
摘要
The inverse spectral problem is studied for non-selfadjoint systems of ordinary differential equations on a finite interval. We provide necessary and sufficient conditions for the global solvability of the inverse problem, along with an algorithm for constructing its solution. For solving this nonlinear inverse problem, we develop ideas of the method of spectral mappings, which allows one to construct the potential matrix from the given spectral characteristics and establish conditions for the global solvability of the inverse problem considered.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(2):200-208
200-208
Mechanics
On a new approach to identifying inhomogeneous mechanical properties of elastic bodies
摘要
A new approach to solving the problem of identifying the variable characteristics of an inhomogeneous elastic isotropic body is presented. The most common formulations of problems on determining variable mechanical characteristics (the Lamé parameters and density are functions of coordinates) are presented. The inverse problem of identifying properties, due to its significant nonlinearity, is usually solved iteratively, with each iteration requiring the solution of a direct problem for some initial approximation and a system of the Fredholm integral equations of the first kind with smooth kernels to find corrections. This approach, in turn, requires specifying the displacement field in the area in which the loading occurs. An approach is proposed on the basis of which it is possible to carry out reconstruction by obtaining additional information about the displacement field in an area other than the loading area in a narrower search space. An example of such a reconstruction is presented in the problem of longitudinal vibrations of an inhomogeneous rod, where the amplitude-frequency response is specified at the internal point of the rod, and the loading is implemented at the end. The results of computational experiments on the reconstruction of the elasticity modulus and density in the form of two functions of the longitudinal coordinate are presented.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(2):209-221
209-221
Asymptotic theory of the hyperbolic boundary layer in shells of revolution at shock edge loading of the tangential type
摘要
The present work is devoted to the construction of asymptotically optimized equations of the hyperbolic boundary layer in thin shells of revolution in the vicinity of the dilation wave front at shock edge loading of the tangential type. These equations are derived by asymptotically integrating of the exact three-dimensional theory elasticity equations in the special coordinate system. This system defines the boundary layer region. The wave front has a complicated form, dependent on the shell curvature and therefore its asymptotical model is constructed. This geometrical model of the front defines it via the turned normals to the middle surface. Also, these turned normals define the geometry of the hyperbolic boundary layer applicability region. Constructed asymptotically optimised equations are formulated for the asymptotically main components of the stress-strain state: the longitudinal displacement and the normal stresses. The governing equation for the longitudinal displacement is the hyperbolic equation of the second order with the variable coefficients. The asymptotically main part of this equation is defined as the hyperbolic boundary layer in plates.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(2):222-230
222-230
Mathematical model of orthotropic meshed micropolar cylindrical shells oscillations under temperature effects
摘要
In the work the mathematical model of micropolar meshed cylindrical shells oscillations under the action of the vibrational and temperature effects is constructed. The shell material is an elastic orthotropic homogeneous Cosserat pseudocontinuum with constrained rotation of particles. The Duhamel – Neumann’s law was adopted. The mesh structure is taken into account according to the model of G. I. Pshenichnov, geometric nonlinearity according to Theodor von Karman theory. The equations of motion, boundary and initial conditions are obtained from the Ostrogradsky – Hamilton variational principle based on the Tymoshenko kinematic model. The constructed a mathematical model will be useful, among other things, in the study of the behavior of carbon nanotubes under various operating conditions.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(2):231-244
231-244
Analysis of elastic and elastoplastic models when interpreting nanoindentation results
摘要
One of the current and widely used non-destructive testing methods for monitoring and determining the elastic properties of materials is nanoindentation. In this case, to interpret the test results, a non-trivial task arises of constructing an adequate mathematical model of the indentation process. As a rule, in many cases, analytical formulas are used that are obtained from an elastic linear formulation of problems about the introduction of a non-deformable stamp into a homogeneous elastic half-space. Currently, the numerical formulation of the problem makes it possible to obtain and use a numerical solution obtained taking into account the complete plastic nonlinear behavior of the material. In this work, a study of contact problems on the introduction of a spherical and conical indenter into an elastoplastic homogeneous half-space was carried out. To verify the numerical solution, the problem of introducing a spherical and conical indenter into an elastic homogeneous half-space was also solved and compared with known analytical solutions. Issues of convergence and tuning of numerical methods, the influence of plasticity and the applicability of analytical solutions are explored. Problems are solved numerically using the finite element method in the Ansys Mechanical software package.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(2):245-253
245-253
Investigation of the influence of intersystem shunt characteristics on hemodynamic parameters and oxygen distribution
摘要
Various intersystem shunts are widely used methods of treatment of newborns with congenital heart disease and reduced pulmonary blood flow. Shunt placement is associated with a high risk of postoperative complications and mortality. It is possible to predict the development of some complications using mathematical modeling methods and adjust the treatment. In this study we investigated the system ''aorta – shunt – pulmonary artery''. Three kinds of shunt placement with three different diameters for three patients have been analyzed. To solve hemodynamic problems, 27 cases were investigated using the common hemodynamic indices (wall shear stress, time-averaged, oscillatory shear index, relative residence time, etc.). The dependence of pulmonary artery blood flow distribution on shunt location is shown; the preferred location differs for different patient geometries. The energy loss of 4 mm diameter shunts is almost 2 times larger that of 3 mm shunts. A patient-specific approach to the treatment of each child based on objective data can significantly reduce the number of pediatric deaths and increase the effectiveness of the rehabilitation process.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(2):254-274
254-274
Computer Sciences
Implementation of the identification and recognition system cognitive behavior of the observed
摘要
This article describes and analyzes the development of a system for identifying and recognizing the cognitive behavior of students to determine interest in facial expressions. The purpose of the study is to find suitable technologies for the implementation of this system. The definition of emotions will allow organizing control over the quality of the educational process, conducting statistics on the cognitive behavior of students during classes, and showing the level of interest of students in the material presented. The identification system will automatically determine and register the time of arrival and departure of students in real time. Based on the joint application of the Viola – Jones method and the nearest neighbors method using histograms of centrally symmetric local binary images, a system for face recognition in a real-time video sequence has been developed. The structure of the project is described and the software is developed in the Python programming language using the Keras open-source library. The developed system consists of two subsystems: an identification system and a cognitive behavior recognition system. The scientific novelty lies in an integrated approach to the development and research of algorithms for real-time face recognition and identification for solving applied problems.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(2):275-286
275-286
Queueing network model of a call center with customer retrials and impatient customers
摘要
The subject of mathematical study and modelling in this paper is an inbound call center that receives calls initiated by customers. A closed exponential queueing network with customer retrials and impatient customers is used as a stochastic model of call processing. A brief review of published results on the application of queueing models in the mathematical modeling of customer service processes in call centers is discussed. The network model is described. The possible customer states, customer routing, parameters, and customer service features are given. The allocation of customers by network nodes at a fixed time fully describes the situation in the call center at that time. The state of the network model under study is represented by a continuous-time Markov chain on finite state space. The model is studied in the asymptotic case under the critical assumption of a large number of customers in the queueing network. The mathematical approach used makes it possible to use the passage to the limit from a Markov chain to a continuous-state Markov process. It is proved that the probability density function of the model state process satisfies the Fokker – Planck – Kolmogorov equation. Using the drift coefficients of the Fokker – Planck – Kolmogorov equation, a system of ordinary differential equations for calculating the expected number of customers in each network node over time can be written. The solution of this system allows for predicting the dynamics of the expected number of customers at the model nodes and regulating the parameters of the call center operation. The asymptotic technique used is applicable both in transient and steady states. The areas of implementation of research results are the design of call centers and the analysis of their workload.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(2):287-297
287-297
Using parallel computing to evaluate the transport of pollutants in shallow waters
摘要
In many countries of the world, there is a deterioration in the geoecological state of water bodies, associated with a significantly increased anthropogenic impact on natural waters. At the same time, the guaranteed quality of water resources and its reliability are ensured by comparing the realized quality and the guaranteed one. This article discusses models for the entry and movement of pollutants contained in the aquatic environment. Based on the currently used approaches and water pollution criteria, a set of parallel programs for high-performance computing systems has been developed, which allows modeling of the processes under consideration, as well as assessing risks and vulnerabilities in relation to anthropogenic impacts, zoning the water area of a shallow water body in accordance with the levels of anthropogenic loads, environmental design from the standpoint of sustainable development.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(2):298-315
298-315