## Vol 25, No 4 (2017)

**Year:**2017**Articles:**10**URL:**http://journals.rudn.ru/miph/issue/view/1029**DOI:**https://doi.org/10.22363/2312-9735-2017-25-4

###### Abstract

Original results of a research of an eﬃcient computing method - group method of data han-dling are presented. Key shortcomings on each signiﬁcant procedure of a classical algorithm arerevealed and systematized, and also ways of their elimination, including author’s modiﬁcationsare presented. In particular, the use of dispersion and an assessment of dispersion (Fischer’scriterion) is proposed as an assessment of accuracy of the received result, additional “internal”criterion for evaluation of adequacy of model in various tests during the ﬁxing of input dataand changing of characteristics of an algorithm, and determining the optimal complexity of themodel. To solve the convergence problem of the classical algorithm, it was proposed to usethe methods of dispersion, factor and correlation analysis to eliminate non-informative features,modify the criterion for stopping the algorithm. The use of regularizing functionals is suggestedto solve the problem of multicollinearity of input characteristics and increase the stability of theobtained model, etc. A complex of computer modeling programs was developed, realizing an ef-ﬁcient modiﬁed algorithm of GMDH with the considered modiﬁcations and also methods of adispersion analysis, correlation analysis, component analysis, elements of the regression analy-sis and others. The conducted researches and the received practical results can become a basisfor development with use of Machine Learning and Data Science technologies of the automaticsystem of computer modeling, the intellectual analysis and the data processing.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(4):323-330

###### Abstract

This work researches the problem of controllability with phase space change. Nowadays theinterest to the controllability problems with variable structure is on the rise due to the continuouswidening of their practical application space. The tasks of this sort are appearing in physics,biology as well as in economics. The problem of transfer of object from the constrain set of onespace to the constrain set of diﬀerent space through the null point at the given lengths of timeis examined. The spaces may be of the diﬀerent dimensionality. The transfer is possible bothfrom the space of higher dimensionality to the space of lower dimensionality and vice versa. Themovement of the object is described by two nonlinear systems of diﬀerential equations, whilethe control action of the ﬁrst system has a special form, due to some physical applications. Thetransfer of the object from one space to another is given by certain mapping. For the problemin which the nonlinear system in the initial space is locally null-controlled and the right partof diﬀerential inclusion in the second space is the concave mapping the suﬃcient controllabilityconditions were achieved. The problem is researched using the controllability theory apparatus,convex analysis and multiple-valued mapping theory. Taking into the account the practical valueof the given problem the results achieved are of both theoretical and practical signiﬁcance.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(4):331-339

###### Abstract

In the paper we study integral properties of convolutions of functions with kernels generalizingthe classical Bessel-Macdonald kernels (), ∈ , 0 < < . The local behavior of Bessel-Macdonald kernels in the neighborhood of the origin are characterized by the singularity ofpower type ||-. The kernels of generalized Bessel-Riesz potentials may have non-powersingularities in the neighborhood of the origin. Their behavior at the inﬁnity is restricted onlyby the integrability condition, so that the kernels with compact support are included too. In thepaper the general criteria for the embedding of potentials into rearrangement invariant spacesare concretized in the case when the basic space coincides with the weighted Lorentz space.We obtain the explicit descriptions for the optimal rearrangement invariant space for such anembedding.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(4):340-349

###### Abstract

This paper is the ﬁrst in a series of two articles devoted to the review of “fork-join” (inthe western classiﬁcation) queuing systems or systems with the splitting of incoming queries.This system is a natural model for many other real systems. The article describes the fork-joinqueueing model construction and main characteristics of this model. Special attention is paid tomethods of analysis of the response time of the system. Since the exact expression for the meanresponse time is known only for the case of two servers, the article gives a detailed descriptionof the approach to obtaining an accurate expression of this characteristic. For the case whenthe number of servers is more than two, approximations of the mean response time are obtainedby diﬀerent methods, which is explained by the complexity of the studies due to the existingdependence between the queues of subqueries due to common arrival moments. The paperpresents several methods of approximate analysis: various variants of empirical approximation,i.e. methods that reﬁne the obtained characteristics by using the results of simulation modeling;interpolation methods using system load limit values in cases when the incoming ﬂow and servicetime distributions are not exponential.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(4):350-362

###### Abstract

The multidimensional models of the population dynamics are considered in the paper. Thesemodels are the generalizations of the Lotka-Volterra model in case of interaction of the ﬁnitenumber of populations. The deterministic description of the models is given by the systemsof the ordinary nonlinear diﬀerential equations presented in the paper in the form of themultidimensional vector diﬀerential equations. The qualitative properties of the speciﬁed modelsare suﬃciently well studied by means of Lyapunov methods. However, the probabilistic factorsinﬂuencing on the behavior of models are not taken into account at the deterministic descriptionof models. The new approaches to the modeling and stability analysis are of theoretical andapplied interest in the nondeterministic case.In this paper, the methods for design of multidimensional nondeterministic models ofinteraction of populations are considered. The ﬁrst method is connected with the transitionfrom the vector nonlinear ordinary diﬀerential equation to the corresponding vector diﬀerentialinclusions, fuzzy and stochastic diﬀerential equations. On the basis of the reduction principle,which makes it possible to reduce the problem of the stability of solutions of a diﬀerentialinclusion to the problem of stability of solutions of other types of equations, stability conditionsfor the constructed models are obtained. The second method is connected with the technique ofdesign of the self-consistent stochastic models. The scheme of interaction is received on the basisof this technique. This scheme includes a symbolical record of possible interactions between thesystem elements. The structure of the multidimensional stochastic Lotka-Volterra models isdescribed, and the transition to the corresponding Fokker-Planck vector equations is carriedout by means of the system state operators and the system state change operator. The rules forthe transition to the multidimensional stochastic diﬀerential equation in the Langevin form areformulated. The execution of the numerical experiment with the application of the developedprogram complex for solving the systems of the stochastic diﬀerential equations is possible forthe models which are the concretizations of the studied general models. The described approachto the modeling of the stochastic systems can be applied in the problems of comparing of thequalitative properties of the models in deterministic and stochastic cases. The obtained resultsare aimed at the development of the methods for the analysis of nondeterministic nonlinearmodels.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(4):363-372

###### Abstract

Solving the system of essentially nonlinear diﬀerential equations for diﬀerent we obtain thecurrent-voltage characteristics (IVC) for a system of Josephson junctions (JJ) as a hysteresisloop. When the current approaches on the back way the breakpoint the voltage ()falls sharply to zero. In addition, in numerical modelling (non-periodic boundary conditions(NPBC)) IVC multiple branching is observed near . It is interesting to study this phenomenonanalytically developing asymptotic methods. There had been developed simple “asymptotic”formulas suitable for calculation of all IVC points except near to . A numerical-analyticalmethod allowing to shorten IVC calculation time essentially was proposed. This method showedgood results in IVC multiple branching calculation in particular. All calculations were performedusing the REDUCE system. We succeeded ﬁrst to calculate analytically all points of IVC. Anapproximate solution at the breakpoint region (periodic boundary conditions (PBC)) has beendeveloped using the Bogolyubov-Krylov method.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(4):373-379

###### Abstract

Within the general theory of relativity the Bianchi type VIII cosmological models with rotationand expansion have been built. The ﬁrst case includes a ﬁeld of radiation, the second one -a perfect ﬂuid with dust-like equation of state. Perfect anisotropic ﬂuid imitates the rotatingdark energy. Static and dynamic cosmological modes have been observed, at the same timethe equations of state are partly postulated in the ﬁrst case for the anisotropic ﬂuid and inthe second case - for the perfect isotropic ﬂuid, that imitates baryon matter. The analysisof absence of closed time-like curves has been done, so the models have been proved to becasual when the metric parameters satisfy the found conditions. Also the conditions, when theanisotropic ﬂuid’s equation of state becomes vacuum-like, the energy of the ﬂuid dominatesand it becomes asymptotically isotropic, have been cleared out. Specialities of the oscillatingmode have been observed. The order of present angular velocity value, calculated within thecosmological models, has been found to be quite satisfactory when expanding from the Plank scale to the present size of observed part of the Universe. The found solutions may be used foreﬀects taking place nowadays and also during the inﬂationary stage.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(4):380-389

###### Abstract

Supernovae of type Ia are used as standard candles in modern cosmology, they serve to testcosmological models. Interpretation of the Hubble diagram based on the standard cosmologicalmodel led cosmologists to conclusion that the Universe is ﬁlled mostly with cosmic dust andmysterious dark energy.In this paper we present exact solutions of the Friedmann equation in standard cosmologyand conformal cosmology. The theoretical curves interpolating the Hubble diagram for thelatest supernova data are expressed in an analytical form. The functions belong to the classof meromorphic Weierstrass functions. Both approaches describe the modern Hubble diagramwith the same accuracy. Physical interpretation from the standpoint of conformal cosmologyis preferable, since supernova data are described without using a Λ-term. In the standardcosmology, the Hubble diagram is described by some characteristics: a Hubble parameter (),a deceleration (), and a jerk (). As calculations show, the deceleration parameter changesits sign during the evolution of the Universe, the -parameter remains constant. In the modernera, the Universe expands with acceleration, and in the past its acceleration was negative. Thechange in the sign of acceleration, without a clear physical reason, puzzles cosmologists.It seems obvious to us that to study objects dislocated from us at distances of billions oflight years, we should not use the coordinate time customary for work in laboratories, butthe conformal time. In conformal coordinates, the behavior of photons is described as in theMinkowski space. The time intervals d and d are diﬀerent, they are related by the scale factor:d = d. The conformal luminosity distance is longer than the standard luminosity distance,which is manifested when observing distant stellar objects. As a result, the eﬀective magnitudevalue - the redshift relationship, on which the Hubble diagram is constructed, will be diﬀerent.Using the conformal Friedmann equation, we introduce the conformal parameters (), (),(). All parameters remain positive during the evolution of the Universe. The scale factorgrows with deceleration. The Universe does not experience a jerk.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(4):390-400

###### Abstract

In this paper of the nonlinear laser wave propagation in the hot plasma along the strong exter-nal magnetic ﬁeld under the electron cyclotron resonance conditions is investigated. The strongnonlinearity of such a process is caused by the relativistic electron movement and resonancewave ponderomotive force acting on the electrons. The system of equations for the enveloperight hand polarized laser pulse is derived using the hydrodynamics and Maxwell’s equations.The numerical integration of this system for the cold plasma case discovered the soliton solu-tions. This kind of solutions take a form of the envelope solitons containing inside them the plasma oscillations. The analytical expression for the energy density integral in a cold plasma isderived. It follows from the numerical results that for a hot plasma under cyclotron resonanceconditions the soliton solution becomes unstable. In this case the energy density conservationbreaks down, but electron momentum density conserves. It is concluded that the nonlinear sat-uration of the ﬁeld amplitude is due to the plasma charge separation under electromagneticradiation pressure. In this case the discrete set of the envelope soliton carrier frequency is de-termined by the ratio of the frequency of the nonlinear longitudinal electron oscillations to theLangmuir frequency of plasma. For the low density plasma the discrete frequency spectrumobtained by the numerical integration transforms to the continuous one.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(4):401-409

###### Abstract

The task of integrating information contained in databases of various companies that providecargo carriers is considered in the article. It is assumed that these companies are independentand each of them maintains its own database about its employees, customers, number and pa-rameters of carries, services and orders. The article simulates three such independent databases(information sources) and describes a scheme for constructing an integrated system based onthem, in particular, for the convenience of customers when searching for carriers with the neces-sary parameters (dimensions and load-carrying capacity). To solve this problem, it is proposedto use an approach combining the advantages of mediator and data repository technologies. Itis assumed that all sources have similar conceptual schemes, but they have their own speciﬁccharacteristics. In particular, there may be diﬀerent names of tables and attributes in diﬀer-ent sources and diﬀerent distribution of attributes throw tables. Certain attributes may alsobe absent at some sources. We construct correspondence tables and a mediator that translateuser queries to the sources. To identify the necessary sources, a small auxiliary repository ismaintained that contains information about the parameters of the carriers that each source has.The user of the integrated system enters data on the weight and dimensions of his cargo, then,using the information of the repository, the system determines the appropriate trucking compa-nies to which the user request is broadcast. After combining all the answers that came from thesources, the client receives full information with a list of suitable companies, prices and otherconditions for the execution of the order.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(4):410-419