# Vol 26, No 2 (2018)

## Mathematical Modeling

### Eigen Waves of a Plane Symmetric Anisotropic Waveguide

Bikeev O.N., Lovetskiy K.P., Sevastianov A.L.

#### Abstract

Precise dispersion equations for a plane symmetric dielectric anisotropic waveguide are obtained, in which the waveguide layer is isotropic, and the framing media are assumed to be anisotropic uniaxial media. The tensors of the dielectric permittivity of the framing media are not assumed to be diagonal, namely, in one of them this tensor is formed by rotating the diagonal tensor by some angle between the optical axis of the anisotropic medium and the direction of propagation of the electromagnetic wave. The tensor of dielectric permittivity of another anisotropic medium is rotated by the same angle, but in the opposite direction, with the optical axes of both framing media lying in a plane parallel to the boundaries of the waveguiding structure. Thus, in framing media, the existence of six-component electromagnetic waves is maintained. In the dispersion properties of such a waveguide, certain features are observed in comparison with the case when the framing media are assumed to be isotropic. It is found that the first symmetric mode of such a waveguide has a finite deceleration with zero thickness of the isotropic layer, which indicates the possibility of the appearance of surface electromagnetic waves (the so-called Dyakonov waves) at the boundaries of this isotropic layer. It is noted that the transition of the antisymmetric mode to the Dyakonov wave occurs with a finite thickness of the waveguiding layer. Dependencies of the deceleration of the elementary (symmetric) mode on the angle of rotation of the optical axis of anisotropic media relative to the direction of propagation of the guided wave of the waveguide structure are given.

Discrete and Continuous Models and Applied Computational Science. 2018;26(2):119-128
119-128

### On the Calculation of Electromagnetic Fields in Closed Waveguides with Inhomogeneous Filling

Tyutyunnik A.A.

#### Abstract

The paper investigates waveguides of constant cross-section with ideally conducting walls and arbitrary filling. The problem of finding the normal modes of a waveguide in a full vector formulation has been set and discretized. In the framework of numerical experiments, the guiding and evanescent modes of the waveguide are calculated for several variants of the fillings. The problem of diffraction of the normal waveguide mode incident on the joint of two waveguides, the cross-sections of which coincide, and the filling at the junction varies abruptly, is set and discretized. The results of numerical experiments for specific configurations of waveguide joints are presented, and the transmission and reflection coefficients of the guided modes are calculated. The solution of the Maxwell equations system is based on the decomposition of fields with the help of four potentials, and in the present work a symbolic-numerical method is realized that uses this approach. The numerical experiments presented in this paper show that the proposed approach and the method on its basis allow the effective calculation of various characteristics of waveguide systems. The adequacy of the approach used is also evidenced by comparing the results obtained with the results of V.V. Shevchenko for the diffraction problem at the junction of two open waveguides The symbolic-numerical method used in the work is implemented in the computer algebra system Maple, in particular, the calculations of matrix elements in the framework of the incomplete Galerkin method are carried out in symbolic form to accelerate further calculations using numerical methods.

Discrete and Continuous Models and Applied Computational Science. 2018;26(2):129-139
129-139

### An Inviscid Analogue of the Poiseuille Problem

Koptev A.V.

#### Abstract

We consider a plane problem of steady-state motion of an ideal incompressible fluid flow in a channel between two parallel planes under the action of a given pressure drop. The problem is considered in Cartesian coordinates. The formulation is analogous to the well-known Poiseuille problem with the difference that an ideal fluid is considered instead of a viscous one. The non-flow condition is set as the boundary ones on the channel walls. So, that the velocity vector is parallel to the bounding surfaces over the channel walls. The pressure drop is set as a given positive quantity. An approach proposed based on the use of the first integral of the Euler equations while preserving nonlinear terms. We represent the derivation of main relations for the case of 2D steady-state motion of an incompressible fluid. The solution of equations for hydrodynamic characteristics in the form of expansions in powers of the Cartesian coordinates was found out by analytical way. The standard programs of Maple package are used to determine the coefficients of decomposition for some values of defining parameters. As a result expressions for hydrodynamic characteristics are obtained and their features investigated. In particular, zones of recurrent motions and zones of intense vortex motion of fluid were revealed.

Discrete and Continuous Models and Applied Computational Science. 2018;26(2):140-154
140-154

### Construction of the Mathematical Model of Pricing for Telecommunication Services with Allowance for Congestion in Networks

Vasilyev S.A., Haroun H.S.

#### Abstract

This paper considers a model of dynamic pricing in the telecommunications market incomplete competition and taking into account overloads in multiservice networks. The model consists in the use of mathematical modeling methods, game theory and queueing theory. It is assumed that telecommunication companies agree on the rules of incoming and outgoing traffic charging in pairs, and this charging is built as a function of the tariffs that companies offer their subscribers for service. Companies are limited the agreement on mutual rules of reciprocal proportional charging for access traffic at first, which subsequently determine the tariffs for the multiservice network users. The reciprocity of the rules means that companies are subject to the same rules for the entire time interval during which the agreement is in force. Taking into account imperfect competition in the telecommunications market and using profit optimization method for each company the equilibrium tariffs and the volume of services are found with subject to congestion in multi-service networks.

Discrete and Continuous Models and Applied Computational Science. 2018;26(2):155-166
155-166

### Analysis of Queueing Systems with an Infinite Number of Servers and a Small Parameter

Vasilyev S.A., Tsareva G.O.

#### Abstract

In this paper we consider the dynamics of large-scale queueing systems with an infinite number of servers. We assume that there is a Poisson input flow of requests with intensity . We suppose that each incoming request selects two any servers randomly and at the next step of an algorithm is sending this request to the server with the shorter queue instantly. A share () of the servers that have the queues lengths with not less than can be described using a system of ordinary differential equations of infinite order. We investigate this system of ordinary differential equations of infinite order with a small real parameter. A small real parameter allows us to describe the processes of rapid changes in large-scale queueing systems. We use the simulation methods for this large-scale queueing systems analysis. The numerical simulation show that the solution of the singularly perturbed systems of differential equations have an area of rapid change of the solutions, which is usually located in the initial point of the problem. This area of rapid function change is called the area of the mathematical boundary layer. The thickness of the boundary layer depends on the value of a small parameter, and when the small parameter decreases, the thickness of the boundary layer decreases. The paper presents the numerical examples of the existence of steady state conditions for evolutions () and quasi-periodic conditions with boundary layers for evolutions ().
Discrete and Continuous Models and Applied Computational Science. 2018;26(2):167-175
167-175

### Device for Periodic Modulation of Laser Radiation

Komotskii V.A., Sokolov Y.M., Suetin N.V.

#### Abstract

In this paper we consider a new type of mechanical device for periodic modulation of laser radiation. The modulating unit consists of two phase diffraction gratings with a rectangular profile, one of which moves relative to the other. The output beam of radiation in this device can be either a zero-order beam of diffraction, or one of the first orders of diffraction. The results of numerical simulation of output waveforms are presented. In the first order, we obtain a sinusoidal form of output power modulation with an efficiency of up to 40 percent. Optimal parameters of the phase diffraction gratings are calculated. Modulation produced in the zero order of diffraction has an impulse form with an efficiency of about 80-90 percent. The specific shape of the pulses in the zero order of diffraction depends on the distance between the two gratings. The results of numerical calculations and experimental studies are in good agreement. A special advantage of this type of modulator is the possibility of increasing the frequency of mechanical modulation of the laser beam to hundreds of kHz. The results of experimental studies of the characteristics of the scheme under consideration are presented. The device makes it possible to obtain modulation frequencies up to hundreds of kHz with a harmonic waveform in the first orders of diffraction and periodic pulses in the zero order.
Discrete and Continuous Models and Applied Computational Science. 2018;26(2):176-182
176-182

## Mathematics

### The Solvability of the Inverse Problem for the Evolution Equation with a Superstable Semigroup

Tikhonov I.V., Vu Nguyen S.T.

#### Abstract

For the evolution equation in a Banach space, the linear inverse source problem is studied. It is required to recover an unknown nonhomogeneous term by means of an additional nonlocal condition written in the form of a Riemann-Stieltjes integral. The main assumption is related to the superstability (quasinilpotency) of the evolution semigroup. More precisely, it is assumed that the evolutionary semigroup associated with the abstract differential equation has an infinite negative exponential type. Without other restrictions, a theorem on the solvability of the inverse problem is obtained. It is shown that the solution can be represented by a convergent Neumann series. Exact conditions under which an infinite series becomes a finite sum are found. Here, the algorithm for calculating the solution becomes finite. Model examples are considered, including an important example of the inverse problem with final overdetermination. The above results can be applicated in special parts of mathematical physics related to the theory of elasticity and the linear transport theory. As is customary, our research takes place in the general case with a choice of the complex scalar field, but the main facts are also true in the real case. The created theory allows transfer to nonlocal problems for evolution equations, when instead of the traditional initial condition special time averaging is used to find the solution.

Discrete and Continuous Models and Applied Computational Science. 2018;26(2):103-118
103-118

## Interdisciplinary Research

### Corruption: Development Mechanisms, Ways of Prevention (Experience of Computer Modeling with Application of Numerical Methods)

Sukhodolov A.P., Kuznetsova I.A.

#### Abstract

Criminal activity of the corrupted employees of the Russian government institutions of various levels is one of the significant reasons of stay of the country in a condition of crisis. There is an association of considerable part of the operating bureaucracy in uniform antisocial system. A main goal of such a system is achievement of the highest level of the welfare and welfare of friends and relatives. On the other hand, the conglomerate of the interconnected participants of this criminal community, instead of fully performing fully their official duties, realizes plans of the destructive coordinated impact on the state social and economic basis, gaining and strengthening the power influence in political, economic and social spheres. The purpose of the authors of this work is a brief description of the idea of cognitive approach to the system analysis of corruption process. It is presented by simulation model of the avalanche type of special structure for demonstration of the dual mechanism of management of the social explosion caused by corruption. The idea of applying cognitive modeling of corruption process is based on the dialectic concept: control of an internal contradiction. It allows to reflect dynamics of a system condition and to show result of accumulation of a quantitative property with transition to the new quality. In the marked area development of corruption activity leads to explosion, breaking integrity of social and economic system. Cognitive research (in structural aspect) gives us the chance to reveal the mechanism of action of the factors influencing the course of “reproduction” of the corrupted elite. However, some factors promote braking, others, on the contrary, accelerate approach of national accident. Organizational and legal impacts on a source of degradation allow conducting fight against corruption. The Russian mentality, national, cultural features, and legislative basis do not allow adopting fully experience of fight against corruption of some countries, for example, of China, the USA, Singapore, etc. Therefore, it is necessary to develop the ways. In this study, the computer program of the avalanche process is used. Results of the computer experiments realizing the mechanism of double regulation of a condition of the studied phenomenon are given. The model is simple and evident, contains not many parameters and variables, and allows explaining inevitability of social and economic crisis in conditions of the current legislation and the weakened moral and ethical influence on the criminal antisocial union. Provisions and conclusions of this study prove inevitability of catastrophic consequences in social and economic space, but they can be delayed for considerable time due to disciplinary, legal, educational influence.

Discrete and Continuous Models and Applied Computational Science. 2018;26(2):183-193
183-193