# No 1 (2014)

**Year:**2014**Articles:**17**URL:**https://journals.rudn.ru/miph/issue/view/504

## Articles

### On the Cauchy Problem for a Semilinear Functional Differential Inclusion of the Fractional Order with Impulse Response and Infinite Delayina Banach Space

#### Abstract

In this paper, applying the theory of topological degree of condensing multi-valued mappings, we prove the existence of solution and the compactness of the set of solutions of the Cauchy problem for a semilinear functional diﬀerential inclusion of fractional order with inﬁnite delay and impulse responses in a Banach space. The article consists of an introduction and three sections. In the introduction the urgency of this problem, outlines the background and provides links to articles and monographs in which the reader can ﬁnd the applications of the theory of functional diﬀerential equations and inclusions of fractional order. In the second section we describe the formulation of the problem, we introduce the space, which addresses this problem and give a criterion for the relative compactness of the set in the input space. The third section consists of four sub-items, which provide preliminary information. In the ﬁrst subparagraph the concept of fractional derivative and fractional primitive is given. Second paragraph provides the necessary information from the theory of multi-valued mappings. The third sub-paragraph is devoted to information from the theory of measurable multifunctions. In the fourth paragraph we formulate a modiﬁed phase space entered by Hale and Kato. In the last section we formulate conditions that we impose on the elements included in the original inclusion and on the basis of auxiliary statements prove our main result.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(1):5-22

5-22

### Evaluation of IPTV Service Session Setup Time

#### Abstract

This paper describes an algorithm for evaluation of SIP session set up time for IPTV services based on IMS as open mixed BCMP network. We suggest a model of IPTV network composed of seven functions. To evaluate session set up time we design a call ﬂow diagram between STB and MS. First of all we determine call ﬂow rate for a BCMP network and after that use decomposition and aggregation methods to estimate the service time for each function. We calculate session set up time as sum of all service intervals. A mathematical model of the open exponential queuing network is designed to estimate mean session set up time. We also suggest a numerical experiment for the algorithm with the initial data close to the real one. According to the initial data expected session set up time amounts by 2 seconds which is corresponded to the international standards.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(1):23-29

23-29

### Research of the Reliability of a Homogeneous Redundant Warm Standby System in a Random Environment

#### Abstract

This paper investigates the inﬂuence of randomness of the environment on the reliability of work of technical systems and extends research of the previous paper to the case of the warm redundant standby. A short review of the papers devoted to the queueing systems operating in random environments is cited. A general Markov model of the reliability of a system operating in Markov random environment is proposed. Diﬀerential equations for the time dependent state probabilities of such a system and appropriate formulas for the stationary and non-stationary its reliability characteristics is given. An expression for the moment generation function and appropriate moments of the system life time are given. For the purpose of inﬂuence of environmental variability on the system reliability characteristics some parameter c is introdeced, which indicates the inﬂuence of variability on the intensity of failures and recoveries of elements in diﬀerent states of the environment. With using a specially developed software module in the environment MATLAB the numerical study and comparison of the reliability characteristics of the warm redundancy two-units system, operating in a stable and random environments with two states are conducted. Results of the numerical investigation, presented in the form of tables and graphs show both similarities and diﬀerences in the systems in a random and stable environments.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(1):30-42

30-42

### Application of Functional Polynomials to Approximation of Matrix-Valued Functional Integrals

#### Abstract

The matrix-valued functional integrals, generated by solutions of the Dirac equation are considered. These integrals are deﬁned on the one-dimensional continuous path x : |s,t|→ ℝ and take values in the space of complex d × d matrices. Matrix-valued integrals are widely used in relativistic quantum mechanics for investigation of particle in electromagnetic ﬁeld. Namely integrals are applied to represent the fundamental solution of the Cauchy problem for the Dirac equation. The method of approximate evaluation of matrix-valued integrals is proposed. This method is based on the expansion of functional in a series. Terms of a series have the form of a product of linear functionals with increasing total power. Taking a ﬁnite number of terms in the series and evaluating functional integrals of a product of linear functionals we obtain approximate value of the matrix-valued functional integral. Proposed method can be used for a wide class of integrals because the series converges for a large class of functionals. Application of the suggested method in the case of small and large parameters included in the integral is considered.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(1):43-46

43-46

### Simulation of Interaction of Colliding Nanoclusters Beam with Solid Surface

#### Abstract

One of the eﬀective methods changing surface physical and chemical properties of a material is high energy impact of nanoclusters with solid surface. Molecular dynamic simulation is one of the most popular approach to study this process. It is very important for material science and nanotechnology to know as much as possible about conditions for control of getting given properties of the deposited layer. This work is devoted to the simulation of an angular impact for Cun (n = 13,55,147) nanoclusters with the substrate, consisting of 54000 atoms of copper. As contrast to our previous paper [Batgerel B., Nikonov E.G., Puzynin I.V. Simulation of Impact Intereaction of Uncharged Metallic Nanoclusters with Metallic Surface // Bulletin of Peoples’ Friendship University of Russia. Series “Mathematics. Information Sciences. Physics”. — 2013. — No 4. — Pp. 42–56.] we have studied properties of deposited layer on the surface particularly a penetration depth of the cluster atoms and a thickness in angular impact conditions. It is found that these parameters depend on the energy and size of nanoclusters, a number of clusters in the beam, a frequency of irradiation and a value of impact angle.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(1):47-51

47-51

### The Model of Tunneling of Clusters Through Repulsive Barriers in Symmetrized Coordinates Representation

#### Abstract

Formulation of a mathematical model for the system A identical particles with pair interaction of oscillator type in the repulsive barrier potentials in the form of a boundary-value problem for elliptic equations in new symmetrized coordinates, eﬀective methods, algorithms and program complexes for the analysis of its solutions are presented. Reduction of the problem for a cluster of A identical particles to subsystems “(one particle) + (cluster of (A − 1) particles)” and “(a cluster of Ab1 particles) + (cluster of Ab2 particles)” is considered. The solution of the boundary-value problem for a cluster of A identical particles sought the form of an expansion over the cluster (A − 1)-dimensional oscillator basis functions, symmetric and antisymmetric with respect to permutations of A identical particles, i.e. in the symmetrized coordinates representation [Gusev A.A. // Bulletin of PFUR. Series “Mathematics, Information Sciences. Physics”. — 2013. — No 3. — P. 52–67]. The problem is reduced to the boundary value problem for a set of coupled second-order ordinary diﬀerential equations with the R-matrix third type boundary conditions in the close coupling channel method. The amplitude matrix of transmission and reﬂection and the eigenfunctions of the continuous spectrum of the scattering problem with respect to the center of mass are calculated with help of the program complex KANTBP 3.0. The eﬀectiveness of the approach is demonstrated by the analysis of the solutions of the quantum tunneling of clusters consisting of a number of identical particles with oscillator-type pair interaction through a repulsive barrier in the s-wave approximation. The analysis of the eﬀect of quantum transparency, i.e. resonant tunneling of a cluster of several identical particles through the repulsive barriers, which is due to the presence of quasi-stationary states embedded in a continuum is given. To calculate the positions of the energy of quasistationary states and their classiﬁcation, the algorithm of solving the boundary value problems for elliptic equations in A-dimensional domain of a special type based on the decomposition of solutions A-dimensional oscillator basis is developed. The developed approach and set of programs focused on the analysis of the quantum diﬀusion of molecules, channeling and tunneling clusters and ions in crystals, as well as tetrahedral and octahedral symmetry of the nuclei.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(1):52-70

52-70

### Influence of Stochastization on One-Step Models

#### Abstract

It is assumed that the introduction of probability in mathematical model makes it more adequate. There are practically no methods of the agreed (depending on structure of the system) introduction of probability in deterministic models. Authors have improved the method of constructing stochastic models for the class of one-step processes and illustrated it by models of population dynamics. Population dynamics was chosen for study because its deterministic models are suﬃciently well explored that allows to compare the obtained results with the results already known. We have examined the impact of the introduction of stochastics in the deterministic model, on the example of population dynamics system of type “predator–prey”. Previously obtained stochastic diﬀerential equations are studied by the methods of the qualitative theory of diﬀerential equations. Stationary state and ﬁrst integral of the system are obtained. To demonstrate the results the numerical simulations on the basis of Runge–Kutta method for stochastic diﬀerential equations are performed. The ﬁrst integral of deterministic system (phase volume) in the stochastic case does not remain unchanged, but increases, which ultimately leads to the death of one or both populations. One of the disadvantages of the classical system of type “predator–prey” is preservation of the amplitude of populations oscillations. In the stochastic model the process terminates with the death of one or both populations, which from the authors’ point of view makes the model more adequate.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(1):71-85

71-85

### A Brief Description of Higher-Order Accurate Numerical Solution of Burgers’ Equation

#### Abstract

Two new higher-order accurate ﬁnite-diﬀerence schemes for the numerical solution of boundary-value problem of the Burgers’ equation are suggested. Burgers equation is a one-dimensional analogue of the Navier-Stokes equations describing the dynamics of ﬂuids and it possesses all of its mathematical properties. Besides the Burgers’ equation, one of the few nonlinear partial diﬀerential equations which has the exact solution, and it can be used as a test model to compare the properties of diﬀerent numerical methods. A ﬁrst scheme is purposed for the numerical solution of the heat equation. It has a sixth-order approximation in the space variable, and a third-order one in the time variable. A second scheme is used for ﬁnding a numerical solution for the Burgers’s equation using the relationship between the heat and Burgers’ equations. This scheme also has a sixth-order approximation in the space variable. The numerical results of test examples are found in good agreement with exact solutions and conﬁrm the approximation orders of the schemes proposed.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(1):86-91

86-91

### Modification of the Numerical Code for Gas-Dynamical Flowsin Cylindrical Coordinates

#### Abstract

The goal of this article is to develop a robust and accurate numerical method for solving hyperbolic conservation laws in three dimensions. The basic equations are the three-dimensional Euler equations describing the motion of an inviscid gas. The mathematical description of the model is represented by the system of equations of continuity, motion and energy (three dimensional nonstationary partial diﬀerential equations). We used the equation for adiabatic motion in this article. The numerical method for solution of the gas-dynamical equations in strict divergent form has been used in this work. The three-dimensional numerical code for perfect non-stationary gas-dynamical ﬂows simulation in cylindrical coordinates is constructed. This code is based on the explicit quasimonotonic, ﬁrst-order TVD scheme. This scheme admit introduction of the limits on the anti-diﬀusion ﬂows, which enhances the approximation order (to third order in the spatial coordinates) with minimal numerical dissipation and preservation of the monotonicity of the scheme. In order to ensure numerical stability, the time step is restricted by a well-known Courant-Friedrich-Lewy stability condition. The proposed scheme is comparable to the high order over the classical TVD schemes. Our scheme has the added advantage of simplicity and computational eﬃciency. The numerical tests which were fulﬁled by the author in additional researches, validated the robustness and eﬀectiveness of the proposed scheme.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(1):92-98

92-98

### Distributed System for Detection and Prevention Network Attacks to Cloud Computing

#### Abstract

The paper describes the problem of detection of intrusion for distributed systems and cloud computing. The goal is to detect and prevent both classical and distributed network attacks such as Denial of Service (DoS, DDoS). The paper identiﬁes a number of problems of various popular cloud computing systems that represent a danger not only obtaining access to the user data, but also it could compromise the integrity and eﬃciency of the computer system. As solution it is proposed to develop a system for detecting and preventing network attacks. The system consists of several modules designed to perform diﬀerent functions: detection and prevention of attacks, interaction of system modules, data management and storage. Recognition algorithms are based on the methods of artiﬁcial intelligence and the theory of probability. The new solution uses some intellectual methods to recognize attacks as counter to signature-based approach. This paper describes the architecture and functioning of the proposed solutions. It presents the advantages and disadvantages of the described approach. The solution results were presented in the conclusion of the paper. Solution was tested with diﬀerent types of network attacks on a specially prepared experimental stand.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(1):99-105

99-105

### On Transport Task with Due Regard to Real Requirements

#### Abstract

The cargo transportation volumes increase annually all over the world. Transportation companies face a very diﬃcult task concerning the deﬁnition of the optimal routing and vehicle loads. Such task is known as Vehicle Routing Problem (VPR). The application of the classical approach to the task description is quite complicated due to the fact that it does not take into account a lot of parameters which deﬁne the crucial criteria of the successful operation of the company such as: consideration of the vehicle characteristics and characteristics of the cargo to be transported, variety of depots and open route, the possibility of partial loading/unloading of the vehicle at the itinerary points, transportation of cargo which consists of various goods, consideration of the service priority of the point. So the article deals with the complex transportation task. Actual local features for transport enterprises were found out. The article also contains the formulation of the problem for wide-spread practical applications, the mathematical model of the complex transportation task.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(1):106-112

106-112

### Remarks to the Problem of Neutrino Passing through Matter

#### Abstract

A critical analysis of the mechanism of resonance ampliﬁcation of neutrino oscillations in matter in two diﬀerent approaches is presented. The ﬁrst approach is based on the fact that weak interactions are chiral invariant and therefore these interactions not can generate neutrino mass at the exchange of W boson. Then only the neutrino momentum changes and not the neutrino mass, and the gain of neutrino oscillations in matter should not arise. The second approach is based on the fact that in Wolfenstein’s equation, which gives the resonant ampliﬁcation of neutrino oscillations in matter, it is assumed that with the change of the neutrino energy only its mass changes, and its momentum remains unchanged. In fact, if the energy neutrinos in matter changes, then its momentum must also change. In this case, in the solution of the equation there is no appreciable enhancement of neutrino oscillations in the solar matter. Experimental status of the mechanism of resonant ampliﬁcation of neutrino oscillations in matter at the enhancement of neutrino oscillations in the solar matter and at the so-called Day-Night eﬀect. Experimental data on the detection of the gain of neutrino oscillations in solar matter have no indication on the presence of ampliﬁcation . Observation of Day-Night eﬀect is important, since it is a direct checking of the resonance mechanism . But the available experimental data also have no indication on the realization of this eﬀect.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(1):113-123

113-123

### A Sequential Growth Dynamics for a Directed Acyclic Dyadic Graph

#### Abstract

A model of discrete spacetime on a microscopic level is considered. It is a directed acyclic dyadic graph (an x-graph). The dyadic graph means that each vertex possesses no more than two incident incoming edges and two incident outgoing edges. This model is the particular case of a causal set because the set of vertices of x-graph is a causal set. The sequential growth dynamics is considered. This dynamics is a stochastic sequential additions of new vertices one by one. A new vertex can be connected with existed vertex by an edge only if the existed vertex possesses less than four incident edges. There are four types of such additions. The probabilities of diﬀerent variants of addition of a new vertex depend on the structure of existed x-graph. These probabilities are the functions of the probabilities of random choice of directed paths in the x-graph. The random choice of directed paths is based on the binary alternatives. In each vertex of the directed path we choose one of two possible edges to continue this path. It is proved that such algorithm of the growth is a consequence of a causality principle and some conditions of symmetry and normalization. The probabilities are represented in a matrix form. The iterative procedure to calculate probabilities is considered. Elementary evolution operators is introduced. The second variant to calculate probabilities is based on these elementary evolution operators.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(1):124-138

124-138

### On Multidimensional Cosmology with Anisotropic Fluid: Asymptotical Acceleration and Zero Variation of G

#### Abstract

A multidimensional cosmological model describing the dynamics of n + 1 ﬂat factor-spaces Mi in the presence of a one-component anisotropic ﬂuid is oﬀered. The pressures in all spaces are proportional to the density: pi = wiρ, i = 0,…,n. Solutions with accelerated expansion of our 3-space M0 and zero variation of the gravitational constant G are studied. These solutions exist for two branches of the parameter w0: The ﬁrst branch describes the matter with w0 > 1, the second one may contain phantom matter with w0 < −1. It is shown that these solutions are special case of more general solutions with accelerated expansion of our 3-space M0 and asymptotically zero variation of the gravitational constant G. The model of an ideal many-dimensional substance with three isotropic dimensions of our space, additional dimensions and time is considered. Spacelike dimensions are presented by the power metric depending on parameters of an equation of state. It is shown, that association of dynamic parameters of our three-dimensional space on additional dimensions in the open view may be expressed through coeﬃcient of anisotropy of additional dimensions. Dependence from parameter of an equation of state of our isotropic 3-dimensional space to coeﬃcient of anisotropy of the additional dimensions, requiring the accelerated expansion of the Universe is received in an explicit aspect. The received association is presented pictorially.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(1):139-147

139-147

### Investigation of Potential Flow of Fluid in Porous Medium Taking Account of Darcy Law and Variable Diffusion Coefficient

#### Abstract

We have considered the potential ﬂow of the ﬂuid in the porous medium taking into account Darcy low and diﬀerent types of the diﬀusion coeﬃcient in a tube with radius a. The ﬂow is supposed to be stationary and cylindrically-symmetric and the Darcy force is a linear function of the velocity. We have established that a result of the potential ﬂow is identity ∂2P∕∂r∂z ≡ ∂2P∕∂z∂r, where ∂P∕∂r and vz = ∂Φ∕∂z are deﬁned from Euler equation for two components of the velocity: vr = ∂Φ∕∂r and vz = ∂Φ∕∂z, where Φ(r,z) is velocity potential. It means that Euler equation system is compatible and integrable, and the solution is reduced to the solution of the continuity equation. Continuity equation is linear diﬀerential equation for the potential Φ(r,z) and one assumes solution in divided variable: Φ(r,z) = U(r)W(z). For U(z) we have Bessel equation of zero order. This solution depends on the choice of the diﬀusion coeﬃcient in the continuity equation. In all the occasions we have exact solution and established that component of the velocity vz descreases like exponent with increase of z.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(1):148-152

148-152

### Singularities of the Green Function for the Schrödinger Operator with a Potential, Singular at the Origin

#### Abstract

We study the asymptote r → 0 of the Green function G+(r,0,k2) for the Schrödinger operator with a short-range potential of arbitrary form, singular at the origin as r−ρ with ρ > 0. A short-range potential by deﬁnition is a potential that decreases at inﬁnity more rapidly than the Coulomb one. This is done on the basis of integral Lippmann-Schwinger equation for the Green function in coordinate representation. It is shown that to describe the asymptote one has to distinguish three cases depending on the value of potential’s parameter ρ. If the singularity is weaker than that of the Coulomb potential, the Green function has a standard singularity, namely the singularity of the form r−1. In the case 1 ≤ ρ < 2 an additional singularity arises. If ρ = 1 the additional singularity has the same form as in the case of the Coulomb potential. In the case 1 < ρ < 2 it has the form of a polar singularity of the form r−ρ+1. In all cases described above the singular terms of asymptotic expansions are written in explicit forms via potential V ’s parameters that describe its behaviour at inﬁnity. The problem that we consider has interesting applications in physics, for example in a theory of zero range potentials.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(1):153-157

153-157

### Our authors

**Discrete and Continuous Models and Applied Computational Science**. 2014;(1):158-159

158-159