Vol 26, No 3 (2018)

Mathematical Modeling

Wave Processes Modeling in Two Coaxial Shells Filled with a Viscous Liquid and Surrounded by Elastic Medium

Blinkov Y.A., Evdokimova E.V., Mogilevich L.I., Rebrina A.Y.


The investigation of deformation waves behavior in elastic shells is one of the important trends in the contemporary wave dynamics. There exist mathematical models of wave motions in infinitely long geometrically non-linear shells, containing viscous incompressible liquid based on the related hydroelasticity problems, which are derived by the shells dynamics and viscous incompressible liquid equations in the form of centralized Korteweg-de Vries (KdV) equations. In addition, mathematical models or the wave process in infinitely long geometrically non-linear coaxial cylindrical elastic shells are obtained by the perturbation method. These models differ from the known ones by the consideration of incompressible liquid between the shells, based on the related hydroelasticity problems. These problems are described by shell dynamics and viscous incompressible liquid equations with corresponding edge conditions in the form of generalized KdV equations system. The paper presents the investigation of wave occurrences in two geometrically non-linear elastic coaxial cylindrical shells of Kirchhoff-Love type, containing viscous incompressible liquid both in between and inside, and surrounded by an elastic medium, acting in both normal and longitudinal directions. The difference schemes of Crank-Nicholson type are obtained for the considered equations system by taking into account liquid impact and with the help of Grobner basis construction. To generate these difference schemes, the basic integral difference correlations, approximating initial equations system, were used.

Discrete and Continuous Models and Applied Computational Science. 2018;26(3):203-215
pages 203-215 views

The Increase of Negative Hydrogen Ions Production in ECR Source by use of the Additional Low-Temperature Emitters of Electrons

Dugar-Jabon V.D., Karyaka V.I., Terletsky A.Y.


Analysis of the mechanisms of formation of negative hydrogen ions in the plasma source, operating at the electron cyclotron resonance, leads to the conclusion about the fundamentally important role played in this process by low-temperature electrons. In the source the negative ion production is realized in two stages. First, hydrogen molecules colliding in a plasma with energetic electrons, are exited to high-laying Rydberg electron states and to high vibration levels in the plasma volume. Further, pulling the low-energy electrons, excited molecules acquire a negative charge. Negative atomic ions result from dissociation of excited negatively charged hydrogen molecules. Necessary for this process, the electrons of low energies are the result of collisions of fast plasma electrons with plasma electrodes. In the presented experiments to further increase the number of low-energy electrons were used electrons, emitted from the heated tungsten filaments and ceramic electrodes LaB6 placed in the chamber of the source. The experiments found that emission of electrons from tungsten heaters have improved stability of the discharge and expanded the range of pressure under which there was a discharge, without changing substantially the magnitude of current of negative ions. The emission of electrons from the LaB6 electrodes increased the current of negative ions from a source more than 3 times.

Discrete and Continuous Models and Applied Computational Science. 2018;26(3):216-225
pages 216-225 views

Finite Element Method of High-Order Accuracy for Solving Two Dimensional Elliptic Boundary-Value Problems of Two and Three Identical Atoms in a Line

Gusev A.A.


We considered models of three identical atoms in a line with molecular pair interactions and diatomic molecule scattered by an atom or tunneling through potential barriers. The models are formulated as 2D elliptic boundary-value problems (BVPs) in the Jacobi and polar coordinates. The BVP in Jacobi coordinates solved by finite element method of high-order accuracy for discrete spectrums of models under consideration. To solve the scattering problems the BVP in polar coordinates are reduced by means of Kantorovich method to a system of second-order ordinary differential equations with respect to the radial variable using the expansion of the desired solutions in the set of angular basis functions that depend on the radial variable as a parameter. The efficiency of the elaborated method, algorithms and programs is demonstrated by benchmark calculations of the resonance scattering, metastable and bound states of the considered models and also by a comparison of results for bound states of the three atomic system in the framework of direct solving the BVP by FEM and Kantorovich reduction.
Discrete and Continuous Models and Applied Computational Science. 2018;26(3):226-243
pages 226-243 views

Simple Model of Nonlinear Spin Waves in Graphene Structures

Kulyabov D.S., Lovetskiy K.P., Le A.N.


A series of theoretical and experimental works is known which investigated the magnetic properties of graphene structures. This is due, among other things, to the prospects of using graphene as a material for the needs of the future nanoelectronics and spintronics. In particular, it is known about the presence of ferromagnetic properties at temperatures up to 200 C and above in a single-layer graphene films that are free from impurities. Previously there was proposed a quantum field theoretical model describing the possible mechanism of ferromagnetism in graphene as a result of spontaneous breaking of spin symmetry of the surface density of valence electrons. The possible spatial configurations of the localized spin density were described. In this paper we investigate such spatially localized nonlinear spin configurations of the valence electron density on the graphene surface such as kinks, and their interactions, as well as quasibound metastable states of the interacting kinks and antikinks, that are breathers. The spectrum of such breathers is investigated. It is shown that under certain conditions, this spectrum has a discrete sector, which, in turn, allows us to speak about the possibility of coherent quantum generation of spin waves in graphene structures, which is important in terms of practical applications in nanoelectronics and spintronics.
Discrete and Continuous Models and Applied Computational Science. 2018;26(3):244-251
pages 244-251 views

Optical Characterization of a Thin-Film Material Based on Light Intensity Measurements

Lovetskiy K.P., Nikolaev N.E., Sevastianov A.L.


Light interacts with materials in a variety of ways; this article focuses on determination of refraction and absorption characterized by a material’s refractive index. We discuss some of the useful models for the frequency dependence of the refractive index, and practical approaches to calculating refractive indices of thin films and thick substrates. The efficiency of manufacturing of existing and successful creation of new devices of solid-state micro- and nanoelectronics largely depends on the level of development of the technology for manufacturing layers of various materials with a thickness of several nanometers to tens of micrometers. A high degree of perfection of layered structures and particularly structures based on dielectric and/or metallic films with nanometer thickness is needed for their successful application in micro-, nano-, acousto-, microwave and optoelectronics. It is impossible to achieve high degree of perfection without the use of high-precision methods of measuring electrophysical parameters of dielectric and semiconductor materials and structures, metallic films. We have developed the program “Multilayer”, which serves both to simulate the propagation of light through multilayer thin-film layered media, and to determine the dielectric (permittivity tensor of anisotropic films) and geometric (physical and optical thicknesses of the film) parameters of various thin-film coatings. The base mathematical models applied for the description of the light wave propagation through a homogeneous optical medium and for the determination of the optical characteristics of thin layers of optical materials based on the results of light intensity measurements are described. The main mathematical formalism employed in the program is based on solving the Maxwell’s equations for propagation of light through anisotropic stratified media. The algorithm uses the Berreman matrices of order
Discrete and Continuous Models and Applied Computational Science. 2018;26(3):252-260
pages 252-260 views

Computer Science

RACH Session Initiation Procedure with Retranslations

Semenova O.V., Vlaskina A.S., Medvedeva E.G., Zaripova E.R., Gudkova I.A.


Due to the rapidly growing number of user’s equipments, operating in the LTE (Long-Term Evolution) mobile radio network, it became necessary to find methods to reduce the user load and signaling message’s traffic. New rules and procedures of devices communication are created to downsize network resources. This article explores the procedure for establishing a connection between mobile user’s equipment and the LTE base station. Such possible solution of transferring small data generated by machine-to-machine connections (M2M, Machine-to-Machine) was presented in the technical report TR 37.868 of the 3GPP consortium (3rd Generation Partnership Project). This approach implies the use of the random access channel (RACH) to interconnect between devices. Effective usage of the procedure for establishing a connection through a random access radio channel is based on the analysis of time-probability characteristics, such as the access success probability and failed access probability, average access delay. In this article we proposed analytical model in term of Markov chain to find these characteristics and present numerical analysis, considering collisions of preamble message, retransmissions of signalling messages and the boundary conditions on the number of possible retransmissions. The accuracy of the proposed analytical model was verified using programming environment R.

Discrete and Continuous Models and Applied Computational Science. 2018;26(3):261-271
pages 261-271 views

Parametric Study of the System with Active Queue Management Module

Velieva T.R.


Self-oscillating modes in control systems of computer networks quite negatively affect the characteristics of these networks. The problem of finding the areas of self-oscillations is actual and important as the study of parameters of self-oscillations. These studies are extremely labor-intensive because of the substantial non-linear nature of the mathematical model. It is of interest to obtain a so-called parametric portrait describing the zones of occurrence of self-oscillations depending on the value of the parameters: one parameter (two-dimensional graph), two parameters (three-dimensional graph), and so on. Such a parametric portrait allows us to purposefully manage the characteristics of the investigated control system. The investigation of the system under consideration on the basis of ordinary linearization by Taylor expansion is not possible because of the disappearance of the self-oscillatory regime. Therefore, the paper describes a parametric study technique based on the method of harmonic linearization. To verify the theoretical results obtained, simulation is used. In addition, it is proposed to use the computer algebra system for analytical calculations. For this, the criteria for choosing software were formulated. Based on these criteria, a set of software for analytical and numerical calculations was proposed.

Discrete and Continuous Models and Applied Computational Science. 2018;26(3):272-284
pages 272-284 views

Computer and Information Science

The Method of Finding Algebraic Integral for First-order Differential Equations

Malykh M.D., Yu Y.


Article is devoted to search of algebraic integrals of the ordinary differential equations in the systems of computer algebra. The main attention is paid to development of practical instructions for work with an original package for Sage called in honor of M. N. Lagutinski. At the beginning of article Beaune’s problem is formulated: for a given differential equation, we need to identify whether it is in the form of rational integral, and if the answer is true, we need to quadrature it. The difficulties of finding the upper bound of the integral order and its value for solving differential equations practically are discussed, bounded Beaune’s problem is formulated. Our work is based on the method of M. N. Lagutinski. The theory and its realization are tested on the problems from Text-Book on Differential Equations by A. F. Filippov. The numerical experiments, which were carried out, show that the method makes it possible to identify the existence of the rational integral without taking much resources and time. However, using the method to calculate integrals is very time-consuming. In conclusion, recommendations on the optimal use of the method of Lagutinski are given. All calculations are executed in the computer algebra system Sage.

Discrete and Continuous Models and Applied Computational Science. 2018;26(3):285-291
pages 285-291 views

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