Вісник Київського національного університету імені Тараса Шевченка. Серія: фізико-математичні науки
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On a determination of the boundary function in the initial-boundary value problem for the second order hyperbolic equation
Get PDFIn the paper the problem of determination of the boundary function is studied in the initial boundary value problem described by the second order hyperbolic equation. With the help of the additional condition, the functional is constructed, and the problem under consideration is reduced to the optimal control problem. The differential of the function is calculated, a necessary and sufficient condition for optimality is proved.
Pages of the article in the issue: 56 - 60
Language of the article: Englis
Plane elastic wave interaction. Considering of quadratically and cubically nonlinearity
Get PDFThe interaction of elastic plane harmonic waves in the material, the nonlinear properties of which are described by the elastic potential of Murnaghan, is investigated theoretically. The displacement vector is depended of only one spatial variable and time, a record of the complete system of equations for plane waves moves along the abscissa axis is recorded and used. The interaction of longitudinal waves with a separate considering cubic nonlinearity is investigated. On the basis of the cubic equation of motion, the interaction of four harmonic waves is studied. The method of slowly variable amplitudes is used. Firstly the two-wave interaction is investigated, then the interaction of four waves is described. Shorten and evolutionary equations are obtained, the first integrals of these equations and the record of the law of conservation for a set of four interacting waves are obtained. An analogy is made between the triplets studied when taking into account the interaction of three waves and the triplets investigated in the case under consideration, taking into account the four-wave interaction, quadruplets.
Pages of the article in the issue: 50 - 53
Language of the article: Ukrainia
One approach to formalizing the process of information dissemination based on diffusion-limited aggregation
Get PDFThis article examines one of the approaches to the formalization of information dissemination processes based on the diffusion-limited aggregation model, using elements of cellular automata and their analogs. The model describes the dynamics of the information dissemination process without the influence of the mass media by taking into account the facts of information exchange that occurs during communication between participants of an arbitrary target audience. It is believed that the process is characterized by the property of self-similarity. An approach is proposed that makes it possible to study the dynamics of information dissemination processes, taking into account the attitude of the group members to each other and the attitude of the participants to the input information. As a result, an assessment of the effectiveness of the information dissemination process was obtained, which allows drawing conclusions regarding the success of information promotion measures. To demonstrate the processes of information dissemination modeled on the basis of the approach, the results of numerical experiments are presented, in which the implementation of the information exchange procedure for each person is limited to three members of the target group.
Pages of the article in the issue: 61 - 66
Language of the article: Ukrainia
Stability estimates in nonlinear differential equations of a special kind
Get PDFQuite a lot of works have been devoted to problems of stability theory and, in particular, to the use of the second Lyapunov method for this. The main ones are the following [1-7]. The main attention in these works is paid to obtaining stability conditions. At the same time, when solving practical problems, it is important to obtain quantitative characteristics of the convergence of solutions to an equilibrium position. In this paper, we consider nonlinear scalar differential equations with nonlinearity of a special form (weakly nonlinear equations). Differential equations of this type are encountered in the study of processes in neurodynamics [8,9]. In this paper, we obtain stability conditions for a stationary solution of scalar equations of this type. And also the characteristics of the convergence of the process are calculated. It is shown that the solution of stability problems is closely related to optimization problems [10-12].
Pages of the article in the issue: 67 - 71
Language of the article: Ukrainia
Stokes flows in 3D containers
Get PDFThis study consists of two parts. First we consider an analytical approach for solving the problem of steady Stokes flow in some 3D containers with arbitrary velocities prescribed over the surfaces. The approach is based on the superposition method. First we discuss the Stokes problem solution in a finite cylinder. This is the simplest problem because the flow domain is restricted with only two families of coordinate surfaces and the edge (rim) is a smooth line. Then we discuss the analytical solution of the Stokes problem in more complicated domains, such as a circular cone, a rectangular trihedral corner and a 3D rectangular cavity. The Moffatt eddies in such domains are described. In the second part of the study we consider the laminar mixing process in the Stokes flow in a 3D container. We show that in 3D flows a much richer variety of mixing regimes is observed than in 2D flow configurations. The mixing processes in a 3D flow, containing periodic lines, possess essentially two-dimensional characteristics. In the flows, where only isolated periodic points exist, the liquid elements stretch or compress in all three directions.
Pages of the article in the issue: 71 - 76
Language of the article: Englis
On the method of the damage accumulation analysis in front of the fatigue crack in thin plates
Get PDFThe problem on modeling the process of damage accumulation along the fracture front at fatigue in a thin isotropic plate is considered. The solution of the problem is based on joined of the concepts of fracture mechanics and mechanics of continuous damage. A numerical solution of the integral equation of crack front motion is proposed. Consideration of this equation as a superposition of a set of recurrent equations for each moment allows us to model the jumping nature of the fatigue crack growth and take into account the history of damage accumulation in the plate material during loading. A numerical solution of the test problem on the fatigue crack propagation in a thin plate made of aluminum alloy 7075-T6 with uniaxial asymmetric cyclic tension-compression is obtained. The constructed dependences of the fatigue crack length on the number of load cycles agree satisfactorily with the experimental data.
Pages of the article in the issue: 89 - 92
Language of the article: Ukrainia
Simulation of cylindrical rod destruction process under multi-cyclic symmetric torsion
Get PDFA fatigue model based on a decrease in the carrier mass of a substance in the first quarter of a cycle. Also a fatigue model based on an increase in its density in the second quarter of a counterclockwise rotation cycle. As well as this model based a decrease in a carrier mass in a third quarter cycle and an increase in its density in a fourth quarter of a clockwise rotation cycle. The tangential stress and shear angle are related by the Hooke linear relationship. Depending on the initial physical and mechanical properties of the rod, its structural changes are controlled, which quantitatively reflect the changes in mass, density, stresses, shear modulus, which are calculated on each cycle. It is accepted that the brittle fracture of the rod occurs in a cycle in which the inequality of the initial fracture energies and the potential elastic energy pumped on this cycle is not fulfilled. The criterion for achieving the limit of fatigue is not to fulfill the inequality outside the accepted test base. The model algorithm is implemented in the software environment of computer algebra.
Pages of the article in the issue: 46 - 49
Language of the article: Ukrainia
On a generalization of the concept of normal numbers
Get PDFThe paper considers the generalization of the concept of normal numbers in the context of the classical s-th representation of real numbers, in relation to the Q_s-representation, first considered by M. Pratsiovytyi. The result of I. Nivena and H. Zukerman is deepened in relation to the metric theory of normal E. Borel numbers. It is shown that the set of all Q_s-normal numbers has a Lebesgue measure 1. The connection between the property of normality and the uniform distribution of the sequence of numbers generated by the shift operator in relation to the corresponding number is established. It was found that the set of all numbers of the segment [0; 1] for which the corresponding sequence generated by the operator of left-hand shift Q_s-digits is uniformly distributed has a full Lebesgue measure. The corresponding theorems deepen the results of the metric theory Q_s-decompositions of real numbers of the segment [0; 1] obtained by M. Pratsiovytyi and G. Torbin.
Pages of the article in the issue: 58 - 62
Language of the article: Ukrainia
Duality theory for concavification of utility functions in incomplete market model
Get PDFThe main goal for this paper is to prove the existence of the optimal investment strategies for the standard and robust problems of maximization for the concavified utility function in an incomplete market model. We extend the existing results for strictly concave utility functions to concavification of non-concave utility functions. Moreover, we present an assumption under which the optimal strategies for concavified problems are also optimal strategies for non-concave problems.
Pages of the article in the issue: 10 - 17
Language of the article: Englis
Mathematical modeling of influence of strong winds on technical and plant structures on urban areas
Get PDFStormy events in recent years have shown that the destructive effects of wind on urban technical structures and plants pose a special threat. The paper provides an overview of mathematical models and approaches to experimental and theoretical studies of the problems associated with the effects of wind gusts and tornadoes on urban areas. Computer simulations of wind action on standard multistorey buildings in Ukraine are given. The coefficients of normal and shear components of forces and moments of forces acting on the surface of buildings, as well as vortex tracks over the residential complex at different wind speeds from moderate to severe have been computed. The calculations were performed by the finite element method using the model of turbulent air flow in the package AnSys2020. It is shown how with the help of a slight change in shape (roofs, additional passages, shields) the destructive effects of wind on the buildings and plants, as well as the threat to human life can be reduced.
Pages of the article in the issue: 39 - 45
Language of the article: Ukrainia