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D. S. Kulyabov

Publications and source records attributed to D. S. Kulyabov.

At least 19 recordsLinked to original sources
arXiv2019

Numerical analysis of eikonal equation

The Maxwell equations have a fairly simple form. However, finding solutions of Maxwell's equations is an extremely difficult task. Therefore, various simplifying approaches are often used in optics. One such simplifying approach is to use the approximation of geometric optics. The approximation of geometric optics is constructed with the assumption that the wavelengths are small (short-wavelength approximation). The basis of geometric optics is the eikonal equation. The eikonal equation can be obtained from the wave equation (Helmholtz equation). Thus, the eikonal equation relates the wave and geometric optics. In fact, the eikonal equation is a quasi-classical approximation (the Wentzel-Kramers-Brillouin method) of wave optics. This paper shows the application of geometric methods of electrodynamics to the calculation of optical devices, such as Maxwell and Luneburg lenses. The eikonal equation, which was transformed to the ODE system by the method of characteristics, is considered. The resulting system is written for the case of Maxwell and Luneburg lenses.

physics.comp-ph↗
arXiv2019

New Features in the Second Version of the Cadabra Computer Algebra System

In certain scientific domains, there is a need for tensor operations. To facilitate tensor computations,computer algebra systems are employed. In our research, we have been using Cadabra as the main computer algebra system for several years. Recently, an operable second version of this software was released. In this version, a number of improvements were made that can be regarded as revolutionary ones. The most significant improvements are the implementation of component computations and the change in the ideology of the Cadabra's software mechanism as compared to the first version. This paper provides a brief overview of the key improvements in the Cadabra system.

cs.SC↗
arXiv2018

Algorithm for Lens Calculations in the Geometrized Maxwell Theory

Nowadays the geometric approach in optics is often used to find out media parameters based on propagation paths of the rays because in this case it is a direct problem. However inverse problem in the framework of geometrical optics is usually not given attention. The aim of this work is to demonstrate the work of the proposed the algorithm in the framework of geometrical approach to optics for solving the problem of finding the propagation path of the electromagnetic radiation depending on environmental parameters. The methods of differential geometry are used for effective metrics construction for isotropic and anisotropic media. For effective metric space ray trajectories are obtained in the form of geodesic curves. The introduced algorithm is applied to well-known objects - Maxwell and Luneburg lenses. The similarity of results obtained by classical and geometric approach is demonstrated.

physics.optics↗
arXiv2018

The Riemannian geometry is not sufficient for the geometrization of the Maxwell's equations

The transformation optics uses geometrized Maxwell's constitutive equations to solve the inverse problem of optics, namely to solve the problem of finding the parameters of the medium along the paths of the electromagnetic field propagation. The quadratic Riemannian geometry is usually used for the geometrization of Maxwell's constitutive equations, because of the usage of the general relativity approaches. However, the problem of the insufficiency of the Riemannian structure for describing the constitutive tensor of the Maxwell's equations arises. The authors analyze the structure of the constitutive tensor and correlate it with the structure of the metric tensor of Riemannian geometry. It was concluded that the use of the quadratic metric for the geometrization of Maxwell's equations is insufficient, since the number of components of the metric tensor is less than the number of components of the constitutive tensor. The possible solution to this problem may be a transition to Finslerian geometry, in particular, the use of the Berwald-Moor metric to establish the structural correspondence between the field tensors of the electromagnetic field.

physics.gen-ph↗
arXiv2018

Implementing a Method for Stochastization of One-Step Processes in a Computer Algebra System

When modeling such phenomena as population dynamics, controllable ows, etc., a problem arises of adapting the existing models to a phenomenon under study. For this purpose, we propose to derive new models from the rst principles by stochastization of one-step processes. Research can be represented as an iterative process that consists in obtaining a model and its further re nement. The number of such iterations can be extremely large. This work is aimed at software implementation (by means of computer algebra) of a method for stochastization of one-step processes. As a basis of the software implementation, we use the SymPy computer algebra system. Based on a developed algorithm, we derive stochastic di erential equations and their interaction schemes. The operation of the program is demonstrated on the Verhulst and Lotka-Volterra models.

cs.SC↗
arXiv2017

Approaches to Stochastic Modeling of Wind Turbines

Background. This paper study statistical data gathered from wind turbines located on the territory of the Republic of Poland. The research is aimed to construct the stochastic model that predicts the change of wind speed with time. Purpose. The purpose of this work is to find the optimal distribution for the approximation of available statistical data on wind speed. Methods. We consider four distributions of a random variable: Log-Normal, Weibull, Gamma and Beta. In order to evaluate the parameters of distributions we use method of maximum likelihood. To assess the the results of approximation we use a quantile-quantile plot. Results. All the considered distributions properly approximate the available data. The Weibull distribution shows the best results for the extreme values of the wind speed. Conclusions. The results of the analysis are consistent with the common practice of using the Weibull distribution for wind speed modeling. In the future we plan to compare the results obtained with a much larger data set as well as to build a stochastic model of the evolution of the wind speed depending on time.

cs.CE↗
arXiv2017

The Stochastic Processes Generation in OpenModelica

Background: Component-based modeling language Modelica (OpenModelica is open source implementation) is used for the numerical simulation of complex processes of different nature represented by ODE system. However, in OpenModelica standard library there is no routines for pseudo-random numbers generation, which makes it impossible to use for stochastic modeling processes. Purpose: The goal of this article is a brief overview of a number of algorithms for generation a sequence of uniformly distributed pseudo random numbers and quality assessment of the sequence given by them, as well as the ways to implement some of these algorithms in OpenModelica system. Methods: All the algorithms are implemented in C language, and the results of their work tested using open source package DieHarder. For those algorithms that do not use bit operations, we describe there realisation using OpwnModelica. The other algorithms can be called in OpenModelica as C functions Results: We have implemented and tested about nine algorithms. DieHarder testing revealed the highest quality pseudo-random number generators. Also we have reviewed libraries Noise and AdvancedNoise, who claim to be adding to the Modelica Standard Library. Conclusions: In OpenModelica system can be implemented generators of uniformly distributed pseudo-random numbers, which is the first step towards to make OpenModelica suitable for simulation of stochastic processes.

cs.MS↗
arXiv2016

Stochastic Runge-Kutta Software Package for Stochastic Differential Equations

As a result of the application of a technique of multistep processes stochastic models construction the range of models, implemented as a self-consistent differential equations, was obtained. These are partial differential equations (master equation, the Fokker--Planck equation) and stochastic differential equations (Langevin equation). However, analytical methods do not always allow to research these equations adequately. It is proposed to use the combined analytical and numerical approach studying these equations. For this purpose the numerical part is realized within the framework of symbolic computation. It is recommended to apply stochastic Runge--Kutta methods for numerical study of stochastic differential equations in the form of the Langevin. Under this approach, a program complex on the basis of analytical calculations metasystem Sage is developed. For model verification logarithmic walks and Black--Scholes two-dimensional model are used. To illustrate the stochastic "predator--prey" type model is used. The utility of the combined numerical-analytical approach is demonstrated.

physics.comp-ph↗
arXiv2016

Using Two Types of Computer Algebra Systems to Solve Maxwell Optics Problems

To synthesize Maxwell optics systems, the mathematical apparatus of tensor and vector analysis is generally employed. This mathematical apparatus implies executing a great number of simple stereotyped operations, which are adequately supported by computer algebra systems. In this paper, we distinguish between two stages of working with a mathematical model: model development and model usage. Each of these stages implies its own computer algebra system. As a model problem, we consider the problem of geometrization of Maxwell's equations. Two computer algebra systems---Cadabra and FORM---are selected for use at different stages of investigation.

cs.SC↗
arXiv2016

Operator Approach to the Master Equation for the One-Step Process

Presentation of the probability as an intrinsic property of the nature leads researchers to switch from deterministic to stochastic description of the phenomena. The procedure of stochastization of one-step process was formulated. It allows to write down the master equation based on the type of of the kinetic equations and assumptions about the nature of the process. The kinetics of the interaction has recently attracted attention because it often occurs in the physical, chemical, technical, biological, environmental, economic, and sociological systems. However, there are no general methods for the direct study of this equation. Leaving in the expansion terms up to the second order we can get the Fokker-Planck equation, and thus the Langevin equation. It should be clearly understood that these equations are approximate recording of the master equation. However, this does not eliminate the need for the study of the master equation. Moreover, the power series produced during the master equation decomposition may be divergent (for example, in spatial models). This makes it impossible to apply the classical perturbation theory. It is proposed to use quantum field perturbation theory for the statistical systems (the so-called Doi method). This work is a methodological material that describes the principles of master equation solution based on quantum field perturbation theory methods. The characteristic property of the work is that it is intelligible for non-specialists in quantum field theory. As an example the Verhulst model is used because of its simplicity and clarity (the first order equation is independent of the spatial variables, however, contains non-linearity). We show the full equivalence of the operator and combinatorial methods of obtaining and study of the one-step process master equation.

math-ph↗
arXiv2015

Maxwell's Optics Symplectic Hamiltonian

The Hamiltonian formalism is extremely elegant and convenient to mechanics problems. However, its application to the classical field theories is a difficult task. In fact, you can set one to one correspondence between the Lagrangian and Hamiltonian in the case of hyperregular Lagrangian. It is impossible to do the same in gauge-invariant field theories. In the case of irregular Lagrangian the Dirac Hamiltonian formalism with constraints is usually used, and this leads to a number of certain difficulties. The paper proposes a reformulation of the problem to the case of a field without sources. This allows to use a symplectic Hamiltonian formalism. The proposed formalism will be used by the authors in the future to justify the methods of vector bundles (Hamiltonian bundles) in transformation optics.

math-ph↗
arXiv2015

Designing Installations for Verification of the Model of Active Queue Management Discipline RED in the GNS3

The problem of RED-module mathematical model results verification, based on GNS3 experimental stand, is discussed in this article. The experimental stand consists of virtual Cisco router, traffic generator D-ITG and traffic receiver. The process of construction of such stand is presented. Also, the interaction between experimental stand and a computer of investigation in order to obtain and analyze data from stand is revised. A stochastic model of the traffic management RED type module was built. Verification of the model was carried out on the NS-2 basis. However, we would like to conduct verification on a real router. As a result was the task of designing an experimental stand. It was decided to verify the clean RED algorithm based on Cisco router. For the construction of the stand software package GNS3 (Graphical Network Simulator) was chosen. Thus, the purpose of the study is to build on the GNS3 basis a virtual stand consisting of a Cisco router, a traffic generator and a receiver. A traffic generator D-ITG (Distributed Internet Traffic Generator) is used as.

cs.NI↗
arXiv2015

The Method of Constructing Models of Peer to Peer Protocols

The models of peer to peer protocols are presented with the help of one-step processes. On the basis of this presentation and the method of randomization of one-step processes described method for constructing models of peer to peer protocols. As specific implementations of proposed method the models of FastTrack and Bittorrent protocols are studied.

cs.NI↗
arXiv2015

One-Step Stochastic Processes Simulation Software Package

Background. It is assumed that the introduction of stochastic in mathematical model makes it more adequate. But there is virtually no methods of coordinated (depended on structure of the system) stochastic introduction into deterministic models. Authors have improved the method of stochastic models construction for the class of one-step processes and illustrated by models of population dynamics. Population dynamics was chosen for study because its deterministic models were sufficiently well explored that allows to compare the results with already known ones. Purpose. To optimize the models creation as much as possible some routine operations should be automated. In this case, the process of drawing up the model equations can be algorithmized and implemented in the computer algebra system. Furthermore, on the basis of these results a set of programs for numerical experiment can be obtained. Method. The computer algebra system Axiom is used for analytical calculations implementation. To perform the numerical experiment FORTRAN and Julia languages are used. The method Runge--Kutta method for stochastic differential equations is used as numerical method. Results. The program compex for creating stochastic one-step processes models is constructed. Its application is illustrated by the predator-prey population dynamic system. Conclusions. Computer algebra systems are very convenient for the purposes of rapid prototyping in mathematical models design and analysis.

cs.SC↗
arXiv2014

The simplest geometrization of Maxwell's equations

For research in the field of transformation optics and for the calculation of optically inhomogeneous lenses the method of geometrization of the Maxwell equations seems to be perspective. The basic idea is to transform the coefficients of material equations, namely the dielectric permittivity and magnetic permeability in the effective geometry of space-time (besides the vacuum Maxwell equations). This allows us to solve the direct and inverse problems, that is, to find the permittivity and magnetic permeability for a given effective geometry (paths of rays), as well as finding an effective geometry on the dielectric permittivity and magnetic permeability. The most popular naive geometrization was proposed by J. Plebanski. Under certain limitations it is quite good for solving relevant problems. It should be noted that in his paper only the resulting formulas and exclusively for Cartesian coordinate systems are given. In our work we conducted a detailed derivation of formulas for the naive geometrization of Maxwell's equations, and these formulas are written for an arbitrary curvilinear coordinate system. This work is a step toward building a complete covariant geometrization of the macroscopic Maxwell's equations.

math-ph↗
arXiv2014

Tensor computations in computer algebra systems

This paper considers three types of tensor computations. On their basis, we attempt to formulate criteria that must be satisfied by a computer algebra system dealing with tensors. We briefly overview the current state of tensor computations in different computer algebra systems. The tensor computations are illustrated with appropriate examples implemented in specific systems: Cadabra and Maxima.

cs.SC↗
arXiv2013

Maxwell's Equations in Arbitrary Coordinate System

The article is devoted to application of tensorial formalism for derivation of different types of Maxwell's equations. The Maxwell's equations are written in the covariant coordinate-free and the covariant coordinate forms. Also the relation between vectorial and tensorial formalisms and differential operators for arbitrary holonomic coordinate system in coordinate form is given. The results obtained by tensorial and vectorial formalisms are verified in cylindrical and spherical coordinate systems.

math-ph↗