Vol 23, No 2 (2023)
Articles
On estimates of the order of the best M–term approximations of functions of several variables in the anisotropic Lorentz – Zygmund space
Abstract
The article considers the anisotropic Lorentz – Karamata space of periodic functions of several variables and the Nikol'skii – Besov class in this space. The order-sharp estimates are established for the best $M$-term trigonometric approximations of functions from the Nikol'skii-Besov class in the norm of another Lorentz – Zygmund space.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2023;23(2):142-156
142-156
Rate of interpolation of analytic functions with regularly decreasing coefficients by simple partial fractions
Abstract
We consider the problems of multiple interpolation of analytic functions $f(z)=f_0+f_1z+\dots$ in the unit disk with node $z=0$ by means of simple partial fractions (logarithmic derivatives of algebraic polynomials) with free poles and with all poles on the circle $|z|=1$. We obtain estimates of the interpolation errors under a condition of the form $|f_{m-1}|< C/\sqrt{m}$, $m=1,2,\dots$. More precisely, we assume that the moduli of the Maclaurin coefficients $f_m$ of a function $f$ do not exceed the corresponding coefficients $\alpha_m$ in the expansion of $a/\sqrt{1-x}$ ($-1< x< 1$, $0< a\le a^*\approx 0.34$) in powers of $x$. To prove the estimates, the constructions of Pad\'{e} simple partial fractions with free poles developed by V. I. and D. Ya. Danchenko (2001), O. N. Kosukhin (2005), V. I. Danchenko and P. V. Chunaev (2011) and the construction of interpolating simple partial fractions with poles on the circle developed by the author (2020) are used. Our theorems complement and improve a number of results of the listed works. Using properties of the sequence $\{\alpha_m\}$ it is possible to prove, in particular, that under the condition $|f_m|\le \alpha_m$ all the poles of the Pad\'{e} simple partial fraction of a function $f$ lie in the exterior of the unit circle.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2023;23(2):157-168
157-168
On the approximation of bounded functions by trigonometric polynomials in Hausdorff metric
Abstract
The article discusses a method for constructing a spline function to obtain estimates that are exact in order to approximate bounded functions by trigonometric polynomials in the Hausdorff metric. The introduction provides a brief history of approximation of continuous and bounded functions in the uniform metric and the Hausdorff metric. Section 1 contains the main definitions, necessary facts, and formulates the main result. An estimate for the indicated approximations is obtained from Jackson's inequality for uniform approximations. In section 2 auxiliary statements are proved. So, for an arbitrary $2\pi$-periodic bounded function, a spline function is constructed. Then, estimates are obtained for the best approximation, variation, and modulus of continuity of a given spline function. Section 3 contains evidence of the main results and final comments.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2023;23(2):169-182
169-182
The uniqueness of the solution of an initial boundary value problem for a hyperbolic equation with a mixed derivative and a formula for the solution
Abstract
An initial boundary value problem for an inhomogeneous second-order hyperbolic equation on a finite segment with constant coefficients and a mixed derivative is investigated. The case of fixed ends is considered. It is assumed that the roots of the characteristic equation are simple and lie on the real axis on different sides of the origin. The classical solution of the initial boundary value problem is determined. The uniqueness theorem of the classical solution is formulated and proved. A formula is given for the solution in the form of a series whose members are contour integrals containing the initial data of the problem. The corresponding spectral problem for a quadratic beam is constructed and a theorem is formulated on the expansion of the first component of a vector-function with respect to the derivative chains corresponding to the eigenfunctions of the beam. This theorem is essentially used in proving the uniqueness theorem for the classical solution of the initial boundary value problem.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2023;23(2):183-194
183-194
Cylindrical shell with a circular hole under various loads: Comparison of analytical and numerical solutions
Abstract
In this paper, the authors present the results of calculations of the stress field of a cylindrical shell weakened by a circular hole and under the influence of various loads: uniaxial tension, internal pressure and torsion. Six simplified equations of the theory of cylindrical shells with a high variability index (coinciding with the equations of the theory of shallow shells) are reduced to an equation of mathematical physics with respect to the stress potential, which is solved by the Fourier method. The main obstacle to obtaining an answer is the need to search for coefficients in the decomposition of the solution into the sum of the basis functions for which this solution satisfies the boundary conditions. Also, this equation depends on the parameter $\beta$, which is responsible for the relationship between the geometric characteristics of the shell and the hole. From a mechanical point of view, for small and medium holes, this parameter has limitations of $\beta\leq 4$, because for large values, the hole is considered large, and the general equations of the theory of cylindrical shells are used to describe the stress-strain state. At the same time, a detailed study of classical works has led to the understanding that none of the previously proposed methods for finding coefficients can be considered definitively successful, and the results obtained by these methods vary. Among the variety of works by Soviet and Western scientists of the 1960-70s years of the twentieth century, the numerical results of engineer Van Dyke, which he obtained by collocation, stand out. Unlike his contemporaries, who lay out the solution in a row for a small parameter $\beta$ and therefore get results only close to the flat case, Van Dyke first published results for the entire working range of the parameter $\beta$ in the framework of considering small and medium holes. The authors proposed a new approach based on the decomposition of basic functions into a Fourier series. For the first time, it was possible to compose an infinite system of linearly independent equations for finding unknown coefficients. It is essential that the proposed method, unlike the well-known small parameter method, has no mathematical limitations and can be used not only for the values of the parameter $\beta$ close to zero, but for any values. Restrictions up to $\beta=4$ are imposed by the mechanical model. In this paper, systems for finding unknown coefficients for basic functions for three types of loads are compiled, and the results obtained by the authors are compared with the results obtained by the numerical method. At the same time, if in most sources only the results of calculating the circumferential stresses at the boundary of the hole are given, in the proposed work the stress field for the entire cylindrical shell is found, arising due to the presence of the hole, depending on the polar coordinates $(r,\theta)$.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2023;23(2):195-206
195-206
Passive damping of vibrations of a cylindrical shell interacting with a flowing fluid
Abstract
The possibility of passive damping of vibrations of a thin-walled cylindrical shell interacting with a flowing fluid is evaluated. The mechanism is based on connecting the open piezoelectric ring fixed on the surface of the structure to an external shunt electric circuit consisting of series-connected resistance and inductance coil. Their optimal values were selected numerically using the developed finite-element algorithm. The proposed approach is based on solving a series of modal problems. It allows us to obtain higher damping ratios compared to those evaluated by the commonly used analytical expressions and leads to the smallest difference in the natural frequencies of the structure and the electric circuit. In modeling a spatial shell, its curvilinear surface is approximated by a set of flat segments. Each of them is supposed to comply with the relations of the theory of multilayer plates and the equations of linear theory of piezoelasticity written for the case of plane stress state. With this approach, in the vectors of electric field and electric induction it is possible to keep nonzero only such components that are normal to the electroded surface of the piezoelectric ring. The constitutive relations, describing the vortex-free dynamics of an ideal compressible fluid in the case of small perturbations, are formulated in terms of the perturbation velocity potential. The corresponding wave equation is written in the coordinate system associated with the structure and is transformed together with the impermeability condition and boundary conditions to a weak form using the Bubnov – Galerkin method. The paper analyzes the variation of the complex eigenvalues of an electromechanical system depending on the values of resistance and inductance of a series electric circuit. Different methods for calculating the optimal parameters of the system are compared. The frequency response curves demonstrating a decrease in the amplitude of forced harmonic vibrations at a given fluid flow velocity are obtained.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2023;23(2):207-226
207-226
Kinetics of residual stresses in thin-walled cylindrical specimens after bilateral surface hardening under creep conditions with rigid constraints on angular and axial linear displacements
Abstract
A method for solving the problem of relaxing residual stresses after bilateral surface hardening of a hollow cylinder under creep conditions with rigid constraints on the initially specified axial deformation and twist angle is presented. The solution is developed for complex loading regimes including pure thermal exposure, axial loading, torque, internal pressure, and their combinations. A numerical simulation was conducted on a thin-walled cylindrical specimen comprised of X18N10T steel, subjected to a temperature of $T\!=\!600\,^\circ$C, with the inner and outer surfaces subjected to ultrasonic peening. The reconstruction of the initial fields of residual stresses and plastic deformations was carried out based on the known experimental information on the distribution of axial and circumferential stress components in thin surface-hardened areas on the inner and outer surfaces. A phenomenological model of creep of steel alloy X18N10T at $T\!=\!600\,^\circ$C is constructed. The rheological deformation problem within the first two stages of creep was numerically solved using time and radius discretization. The calculations established that the presence of rigid constraints on angular and linear axial displacements resulted in a decrease in the rate of relaxation of residual stresses compared to the case where these constraints are absent. Graphs illustrating the kinetics of residual stresses with respect to the sequence of temperature and loading forces at different timestamps are presented.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2023;23(2):227-240
227-240
Hierarchical risk analysis of unmanned aerial vehicle threat models
Abstract
The relevance of the issues considered in the work is determined by the widespread use of unmanned aerial vehicles in human activities, the high accident rate and the need for scientific justification of the cost of their safety when used in various operating conditions, in particular, in systems for monitoring objects distributed in space. The purpose of the study is to develop recommendations for optimizing the costs of protection systems for unmanned aerial vehicles based on a hierarchical risk analysis of threat models. Results include the developed concept of research which is based on the principle of determining adequate ratios between the risks from threats to aircraft and the costs of their protection. In the process of research, a method and a software tool were developed that allow the construction and analysis of a three-level risk model from threats. At the lower level of this model, risk assessments are determined for private threat models of individual sections of the route, at the middle level the same is done with risk assessments for clusters of private threat models, and at the upper level, maximum risk assessments for the entire route are performed. In the process of computational experiments using the developed method, it was determined, in particular, that the cost of protection for the general threat model exceeded the private and cluster models by 47% and 20%, respectively. The application of the hierarchical analysis method makes it possible to calculate the total risks and evaluate the corresponding cost costs for various modifications of protection systems. The results of the calculation can be used as recommendations when estimating the costs of building a protection system for unmanned aerial vehicles.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2023;23(2):241-252
241-252
Investigation of the need to use the variable value of the ballistic coefficient when modeling the trajectory of the bullet in the shooter simulator
Abstract
When developing electronic shooting simulators for manual automatic weapons that do not use ammunition, it is necessary to achieve the maximum realistic modeling of the bullet flight path for each shot taking into account a set of factors. Traditionally, a system of differential equations of outer ballistics is used in modeling the trajectory. The use of a constant value of the ballistic coefficient in such a mathematical model does not allow to achieve high accuracy of modeling the trajectory for such important for solving the “task of the meeting” parameters as complete flight time and excess of the trajectory for all targeted range of small arms. The initial values in the mathematical model based on the system of differential equations of the outer ballistic are the casting angle (depends on the settings of the sight), the initial speed and the ballistic coefficient of the bullet, and such parameters as the current excess, range, time, speed and direction are calculated. Estimates of the errors of the calculation of the coordinates of the ballistic trajectory at various approaches to the use of the value of the ballistic coefficient are given. It is concluded that at the moment when modeling the flight trajectory of the bullet, simplification based on the use of a constant value of the ballistic coefficient is quite justified but with the relevant requirements of the tactical and technical task the study of ways to increase the accuracy of the trajectory modeling will become relevant. One of these paths is using the value of the ballistic coefficient, depending on the casting angle proposed in this article.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2023;23(2):253-263
253-263
Optimal solution in the model of control over an economic system in the condition of a mass disease
Abstract
In the conditions of а mass disease, governing bodies of an economic system face a number of tasks related to the need to minimize its negative effects. This requires a tool that allows timely predicting the dynamics of the situation and determining what measures need to be taken. In this paper, a specialized mathematical model is proposed as such a tool, taking into account socio-biological and economic factors. The model is a dynamic optimal control problem with a delay in phase variables. The values of the model parameters were estimated using statistical data on the COVID-19 pandemic in the Russian Federation and the Ulyanovsk region. As target functionals, the following are considered: “social criterion” — a decrease in the number of cases; “economic criterion” — an increase in the relative profit of an economic system. To solve the problem, the authors apply a modification of the numerical parameterization method developed in their early studies. The article presents and analyzes the results of the numerical experiment aimed at studying the obtained optimal solutions. It is shown that: the optimal solution for social and economic criteria when changing budgets is stable; most of the parameters of the optimal solution are weakly elastic relative to the values of variables considered; the parameters of the optimal solution when using the economic criterion are more susceptible to change than when using the social criterion; the nature of the change in the parameters of the optimal solution for the Ulyanovsk region and for the Russian Federation is similar. Thus, the paper offers a tool for analyzing an economic problem in conditions of mass disease and confirms the applicability of the tool for finding optimal management strategies in various economic systems.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2023;23(2):264-273
264-273