Volume 25, Nº 2 (2025)
Mathematics
On estimates of the best M-term approximations of functions of many variables in a space with a uniform metric
Resumo
The paper considers the space of continuous functions with a uniform metric and the anisotropic Lorentz – Zygmund space of periodic functions of many variables and the Nikol’skii – Besov class in this space. We have established estimates of the best M-term trigonometric approximations of functions from the Nikol’skii – Besov class in a uniform metric.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(2):154-166
154-166
Estimation of the difference of partial sums of expansions by the root functions of the differential operator and into trigonometric Fourier series
Resumo
We consider a linear ordinary differential operator defined by an $n$-th order differential expression with a nonzero coefficient for $(n-1)$th derivative and Birkhoff regular two-point boundary conditions. The question of the uniform convergence of expansions of a function into a series of root functions of the operator $L$ and the usual trigonometric Fourier series, as well as the estimation of the difference of the corresponding partial sums, is investigated. Estimates of the difference of the partial sums of these expansions are obtained in terms of the general (integral) modules of continuity of the expandable function and the coefficient at the $(n-1)$th derivative. The proof essentially uses the estimate (previously obtained by the author) of the difference between the partial sums of the expansions of a function in a series with respect to the root functions of the operator $L$ and in the modified trigonometric Fourier series, as well as the author's analogue of the Steinhaus theorem in terms of general modules of continuity.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(2):167-172
167-172
On the projection method for solving the heat equation with lumped heat capacity
Resumo
This paper presents some results of the possibility of using the least squares projection method for solving heat equations with concentrated heat capacity on a half-line. An order estimate of the error is given of the considered projection scheme corresponding to an approximate solution of the heat equation using a basis of Laguerre – Jacobi polynomials. The results of calculations for a two-dimensional model problem are presented.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(2):173-183
173-183
Discontinuous Steklov operator and approximate polynomial splines
Resumo
For a continuous function specified on a uniform grid of the segment [0,1], a method for constructing approximation polynomial splines is presented. This method does not require any additional information about the function and ensures uniform convergence. This convergence also occurs for an approximately given grid function. In the case of a parabolic spline, the article provides formulas ready for direct use.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(2):184-188
184-188
Mechanics
Tabulation of the Riemann problem solution in Godunov method for Soave – Redlich – Kwong equation of state
Resumo
The work is devoted to the using of the exact solutions of the Riemann problem on the decay of an arbitrary discontinuity to describe the real gases flows with the Soave – Redlich – Kwong equation of state. The governing mathematical expressions are formulated for constructing an exact solution to the Riemann problem. The features of the functions included in the solution are investigated. It is demonstrated that the form of the Soave – Redlich – Kwong equation of state does not allow to define explicitly the relationship between pressure and internal energy of the gas. The connection between these is determined through gas temperature that leads to significant complication of the task solution technique on the discontinuities. The arising difficulties are determined, firstly, by the features of the mathematical formulation of the problem. It includes a number of nonlinear equations and definite integrals that require the using of the iterative methods to find an exact Riemann solution. This leads to a significant increase in numerical algorithm complexity. Secondly, the specific behavior of some functions in the mathematical model does not guarantee the correct construction of an exact solution to the Riemann problem in using the iterative methods. All these reasons make the classical approach inappropriate for solving complex problems of nonstationary gas dynamics for a real Soave – Redlich – Kwong gas. The approach proposed in this work uses interpolation of solutions constructed on the preliminary accurate calculations of the Riemann problem, performed without additional assumptions over the entire range of changes in gas-dynamic parameters. The use of tabulated values provides the accuracy of constructing an approximate solution and reduces the complexity of the computational algorithm. In the present work this approach is used for numerical simulation of the hydrogen flow in shock tube in a wide range of gas parameters in the fields of classical and non-classical gas dynamics and for numerical simulation of the gas dynamics of a hydrogen safety valve. The obtained results confirm that the use of tabulated parameters is justified in a wide range of gas parameters variations, and the proposed approach can be used to solve complex problems of non-stationary gas dynamics, including those with areas of mixed nonlinearity.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(2):189-202
189-202
Modeling fluid injection into a porous spherical composite with anisotropy
Resumo
Porous spherical composites are widely found both in natural objects and in technical applications. To describe the mechanical behavior of such structures, it is necessary to take into account the interaction of solid and fluid phases, since the fluid inside the porous body supports and redistributes part of the external load. This paper proposes a model of a spherical composite including a porous core and an elastic transversely isotropic shell. The developed model is applied for stress and strain analysis in different loading regimes, including the effect of external normal pressure and the injection of an additional volume of fluid, for example, when modeling intravitreal injections in biomedical research. The analysis has shown that when modeling intravitreal injections and describing the eyeball as a poroelastic composite, the drop in intraocular pressure with increasing degree of anisotropy is less significant than in the case of a model with an elastic shell under the influence of internal pressure alone. The increase in the degree of anisotropy is more significant for the pressure reduction in the porous core in the mode of external normal pressure application than the injection of additional fluid volume. The study also investigates the influence of the geometry and mechanical properties of the porous core on the variation of the shell thickness. The results obtained provide a comprehensive understanding of the stress distribution and fluid pressure, which allows to consider the influence of shell anisotropy, core porosity and their mechanical characteristics on the behavior of spherical composites.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(2):203-213
203-213
Harmonic wave propagation in viscoelastic media modelled via fractional derivative models
Resumo
In the present paper, harmonic waves propagating in 3D isotropic viscoelastic media are analyzed using the fractional derivative Kelvin – Voigt model, Maxwell model and standard linear solid model. Asymptotic values of the wave velocities, their coefficients of attenuation and logarithmic decrements have been found.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(2):214-230
214-230
On the issue of studying the structural and mechanical characteristics of bovine cancellous bone
Resumo
Mechanical experiments with human bones are difficult, so many authors study mechanical parameters of bovine cancellous bone, which is close to human bone in its properties. Studies on estimation of the effective modulus of elasticity of cancellous bone of vertebrae and other bovine bones are known. However, its mechanical properties depending on the direction of loading and structural properties have not been studied yet. The aim of this work was to comprehensively study the mechanical properties of bovine cancellous bone depending on the loading direction, volumetric mineral density and porosity. The objectives of this work were: to develop requirements for the size of samples in uniaxial compression to estimate the effective modulus of elasticity within the framework of the rod theory; to conduct uniaxial mechanical experiments on compression of bone specimens in three directions; to measure the volumetric bone mineral density and porosity; to construct regression relationships linking mechanical and structural properties of cancellous bone. As a result of the study, the dependences linking the effective modulus of elasticity with mineral density, as well as porosity of cancellous bone were revealed. The author's method of determining the porosity of cancellous bone was developed and presented. As a result of the study, the dependences linking the effective modulus of elasticity with mineral density and porosity of cancellous bone were revealed. The author's method of determining the porosity of cancellous bone was developed and presented. The paper also presents the requirements for the relative height (ratio of height to the largest cross-sectional dimension) of cancellous bone specimens. It was revealed that when conducting uniaxial compression experiments and further calculation of the effective modulus of elasticity according to the rod theory, the relative height of the specimens should be at least 5 units.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(2):231-245
231-245
The influence of thermal load on convection in a system of two binary mixtures with a phase transition
Resumo
A mathematical model of convective flow in a horizontal channel with evaporation at the interface between a binary liquid and a gas-vapor mixture, derived on the basis of the Navier – Stokes equations in the Oberbeck – Boussinesq approximation taking into account thermal diffusion effect is under consideration. Using the constructed exact solution of the governing equations, various types of boundary conditions for the temperature on the walls of the channel, inside which the two-layer system of media with a phase transition is located, are tested. The boundary conditions can be the following: heating of the both walls, thermal insulation of the both walls, or a combination of these conditions. The different nature of the dependence (linear and quadratic) of surface tension on the concentration of one components is taken into account. It has been studied which of the problem statements under consideration make sense to discuss from the point of view of analyzing the evaporation process. The application of the constructed solution for description of the flow in the system «70% aqueous solution of ethanol — a mixture of ethanol vapor and nitrogen» shows that the solution adequately reflects the main characteristics of the process: the relationship of the evaporation mass flow rate with the thermal load of the channel wall and flow rate of co-current gas. A qualitative comparison with available experimental data is provided.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(2):246-258
246-258
Plastic strain of the bimetal tube subjected to inner pressure
Resumo
The problem of deformation of a two-layer (bimetallic) thick tube under inner pressure is considered. A dependence of critical load from physical and geometrical properties of the product at which plastic flow is occurred at the beginning of either inner or outer layer is derived. The relations for the semi-analytical solution of the problem of irreversible deformation of a two-layer pipe under the influence of internal pressure are derived. A special case of deformation of a bimetallic pipe made of 09G2C steel with an internal plating layer made of corrosion-resistant 13XFA steel at various thicknesses is considered. The calculation of the autofrettage process was performed in order to improve the operational properties of the product. Conclusions about the choice of the optimal (in terms of strength properties) thickness of the cladding layer, based on the physical parameters of the materials used and the geometric dimensions of the product are drawn.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(2):259-270
259-270
Computer Sciences
Structural-parametric identification of boundary conditions in inverse heat conduction problems using an ensemble of correctness classes
Resumo
An approach to structural-parametric identification of boundary conditions of technological thermal physics processes based on the solution of inverse heat conduction problems is proposed. The stage of structural identification under conditions of a priori uncertainty is reduced to generation of alternatives of possible classes of solutions, which are given in the form of compact sets. Taking into account the restrictions on the membership of the solution to the corresponding classes, the initial incorrectly posed problem was decomposed into a set of conditionally correct problems. Parametrization of the identified characteristic and the resulting state function corresponding to it is carried out at the stage of parametric identification on the basis of the given model structure. Thus, the obtained problems are reduced to parametric optimization problems. Its solution is realized on the basis of methods of optimal control of systems with distributed parameters estimating of temperature discrepancy in a uniform metric. The analytical method of minimax optimisation, considering alternance properties of optimal distributions, allows solving mathematical programming problems concerning the values of the parameter vector for each of the formulated alternatives. The minimax criterion is used to select an appropriate mathematical model from all available variants. If necessary, the structure of the model can be refined by extending the classes of solutions. The presented approach demonstrates satisfactory quality of identification at typical modes of operation of thermal plants on sets of sufficiently smooth functions with the minimum possible number of parameters for the required accuracy of the solution. The aim of the approach is to provide information support for the decision making process on the structure of the model operator in inverse heat conduction problems. By generating hypotheses in the form of correctness classes parameterised by a vector of parameters of higher dimensionality, the quality of identification is improved in complex equipment operating modes.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(2):271-280
271-280
Statistical modeling of the depolarizing properties of optically dense dispersive systems in the small-angle scattering mode of probe light propagation
Resumo
Results of statistical modeling of the polarization degree decay in the case of forward propagation of a linearly polarized laser beam in multiple scattering dispersive systems are presented. Disordered ensembles of dielectric spherical particles with various values of the wave parameter are considered as these dispersive systems. The modeling algorithm is based on an iterative transformation of the Jones vectors for partial components of the multiple scattered light fields in the considered systems due to random sequences of scattering events; the transformation procedure is provided using the Monte-Carlo simulation. The average number of scattering events corresponding to the $1/e$ decay of the polarization degree, and the ratio of the depolarization length to the mean transport free path of probe light in the scattering systems are considered as the key parameters. It was found that the maximal depolarization length is achieved in the case when the wave parameter of scattering particles is close to the value corresponding to the first Mie resonance of the dependence of the scattering efficiency on the wave parameter. The modeling results are compared to the experimental and theoretical data obtained using a hybrid approach in the framework of the diffusion approximation of radiative transfer theory.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(2):281-294
281-294
Heuristic optimization methods for linear ordering of automata
Resumo
The rapid development of society is associated with two key areas of science and technology: methods of working with Big Data and Artificial Intelligence. There is a common belief that up to 80% of the data analysis process is the time spent on data preparation. One aspect of preparing data for analysis is structuring and organizing data sets (also known as data tidying). Order relations are ubiquitous, we meet them when we consider numbers, Boolean algebras, partitions, multisets, graphs, logical formulas, and many other mathematical entities. On the one hand, order relations are used for representing data and knowledge, on the other hand, they serve as important tools for describing models and methods of data analysis, such as decision trees, random forests, version spaces, association rules, and so on. Since a serious limitation of many methods of pattern mining is computational complexity, it is important to have an efficient algorithm for ordering data. In this paper, we consider deterministic automata without output signals and investigate the problem of linear ordering of such automata, which consists of building a linear order on the set of states of an automaton, that will be consistent with the action of each input signal of the automaton. To solve this problem, we consider heuristic methods of global optimization: simulated annealing method and artificial bee colony algorithm. For both methods, we made a software implementation and performed testing on a special kind of automata.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(2):295-302
295-302