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Prof. Alexander Kazakov
Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences

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0 Partial Differential Equations
0 numerical methods
0 Markov processes
0 transportation and vehicle routing
0 Mathematical modelling

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Partial Differential Equations
numerical methods

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Journal article
Published: 13 May 2021 in Symmetry
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The article deals with nonlinear second-order evolutionary partial differential equations (PDEs) of the parabolic type with a reasonably general form. We consider the case of PDE degeneration when the unknown function vanishes. Similar equations in various forms arise in continuum mechanics to describe some diffusion and filtration processes as well as to model heat propagation in the case when the properties of the process depend significantly on the unknown function (concentration, temperature, etc.). One of the exciting and meaningful classes of solutions to these equations is diffusion (heat) waves, which describe the propagation of perturbations over a stationary (zero) background with a finite velocity. It is known that such effects are atypical for parabolic equations; they arise as a consequence of the degeneration mentioned above. We prove the existence theorem of piecewise analytical solutions of the considered type and construct exact solutions (ansatz). Their search reduces to the integration of Cauchy problems for second-order ODEs with a singularity in the term multiplying the highest derivative. In some special cases, the construction is brought to explicit formulas that allow us to study the properties of solutions. The case of the generalized porous medium equation turns out to be especially interesting as the constructed solution has the form of a soliton moving at a constant velocity.

ACS Style

Alexander Kazakov. Solutions to Nonlinear Evolutionary Parabolic Equations of the Diffusion Wave Type. Symmetry 2021, 13, 871 .

AMA Style

Alexander Kazakov. Solutions to Nonlinear Evolutionary Parabolic Equations of the Diffusion Wave Type. Symmetry. 2021; 13 (5):871.

Chicago/Turabian Style

Alexander Kazakov. 2021. "Solutions to Nonlinear Evolutionary Parabolic Equations of the Diffusion Wave Type." Symmetry 13, no. 5: 871.

Journal article
Published: 09 March 2021 in Applied Sciences
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Among the micro-logistic transport systems, railway stations should be highlighted, such as one of the most important transport infrastructure elements. The efficiency of the transport industry as a whole depends on the quality of their operation. Such systems have a complex multi-level structure, and the incoming traffic flow often has a stochastic character. It is known that the most effective approach to study the operation of such systems is mathematical modeling. Earlier, we proposed an approach to transport hub modeling using multiphase queuing systems with a batch Markovian arrival process (BMAP) as an incoming flow. In this paper, we develop the method by applying more complex models based on queuing networks that allow us to describe in detail the route of requests within an object with a non-linear hierarchical structure. This allows us to increase the adequacy of modeling and explore a new class of objects—freight railway stations and marshalling yards. Here we present mathematical models of two railway stations, one of which is a freight railway station located in Russia, and the other is a marshalling yard in the USA. The models have the form of queuing networks with BMAP flow. They are implemented as simulation software, and a numerical experiment is carried out. Based on the numerical results, some “bottlenecks” in the structure of the studied stations are determined. Moreover, the risk of switching to an irregular mode of operation is assessed. The proposed method is suitable for describing a wide range of cargo and passenger transport systems, including river ports, seaports, airports, and multimodal transport hubs. It allows a primary analysis of the hub operation and does not need large statistical information for parametric identification.

ACS Style

Igor Bychkov; Alexander Kazakov; Anna Lempert; Maxim Zharkov. Modeling of Railway Stations Based on Queuing Networks. Applied Sciences 2021, 11, 2425 .

AMA Style

Igor Bychkov, Alexander Kazakov, Anna Lempert, Maxim Zharkov. Modeling of Railway Stations Based on Queuing Networks. Applied Sciences. 2021; 11 (5):2425.

Chicago/Turabian Style

Igor Bychkov; Alexander Kazakov; Anna Lempert; Maxim Zharkov. 2021. "Modeling of Railway Stations Based on Queuing Networks." Applied Sciences 11, no. 5: 2425.

Journal article
Published: 01 March 2021 in Trudy Instituta Matematiki i Mekhaniki UrO RAN
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ACS Style

П. Д. Лебедев; А. Л. Казаков. Итерационные алгоритмы построения наилучших покрытий выпуклых многогранников наборами различных шаров. Trudy Instituta Matematiki i Mekhaniki UrO RAN 2021, 27, 1 .

AMA Style

П. Д. Лебедев, А. Л. Казаков. Итерационные алгоритмы построения наилучших покрытий выпуклых многогранников наборами различных шаров. Trudy Instituta Matematiki i Mekhaniki UrO RAN. 2021; 27 (1):1.

Chicago/Turabian Style

П. Д. Лебедев; А. Л. Казаков. 2021. "Итерационные алгоритмы построения наилучших покрытий выпуклых многогранников наборами различных шаров." Trudy Instituta Matematiki i Mekhaniki UrO RAN 27, no. 1: 1.

Originalpaper
Published: 01 January 2021 in Journal of Applied Mechanics and Technical Physics
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Solutions to a nonlinear parabolic convection–diffusion equation are constructed in the form of a diffusion wave that propagates over a zero background at a finite velocity. The theorem of existence and uniqueness of the solution is proven. The solution is constructed in the form of a characteristic series whose coefficients are determined using a recurrent procedure. Exact solutions of the considered type and their characteristics, including the domain of existence, are determined, and the behavior of these solutions on the boundaries of this domain of existence is studied. The boundary element method and the dual reciprocity method are used to develop, implement, and test an algorithm for constructing approximate solutions.

ACS Style

A. L. Kazakov; L. F. Spevak. EXACT AND APPROXIMATE SOLUTIONS OF A PROBLEM WITH A SINGULARITY FOR A CONVECTION–DIFFUSION EQUATION. Journal of Applied Mechanics and Technical Physics 2021, 62, 18 -26.

AMA Style

A. L. Kazakov, L. F. Spevak. EXACT AND APPROXIMATE SOLUTIONS OF A PROBLEM WITH A SINGULARITY FOR A CONVECTION–DIFFUSION EQUATION. Journal of Applied Mechanics and Technical Physics. 2021; 62 (1):18-26.

Chicago/Turabian Style

A. L. Kazakov; L. F. Spevak. 2021. "EXACT AND APPROXIMATE SOLUTIONS OF A PROBLEM WITH A SINGULARITY FOR A CONVECTION–DIFFUSION EQUATION." Journal of Applied Mechanics and Technical Physics 62, no. 1: 18-26.

Conference paper
Published: 23 November 2020 in E3S Web of Conferences
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The paper considers the main ideas of proposed computational technology for scenario modeling and forecasting the development of the national fuel and energy complexes of Russia and Mongolia, regarding to the intercountry trade in fuel and energy resources. The proposing technology exploits the ideas of multi-agent systems (MASs) and agent-based simulation models (ABSMs) as they can act as unifying means for different types of decision-making methods by distributed objects. Methodological principles and architecture for ABSM of the national fuel and energy complexes of Russia and Mongolia were proposed. The implementation issues of the model in the Adskit software tool is discussed. The problem of laying routes of extended energy objects is also considered. Based on the principles of geometric optics the author’s solution algorithm in the form of a special case of the variational problem was proposed to deal with this problem. The application of MAS and ABSM paradigms for forecasting and evaluating the state of the fuel and energy sector in Russia and Mongolia allows one to organize a step-by-step research of the energy system with the gradual development of the model: increasing the types of objects and agents; improving decision-making algorithms, including those based on mathematical models; creating complex scenarios. The technology forms methodological basis for supporting decision-making process of evaluation the prospective variants of bilateral energy cooperation of Russia and Mongolia and related project effectiveness.

ACS Style

Alexander Kazakov; Anna Lempert; Alexander Stolbov. On technology for modeling and forecasting the interrelated development of regional fuel and energy complexes of Russia and Mongolia. E3S Web of Conferences 2020, 209, 07019 .

AMA Style

Alexander Kazakov, Anna Lempert, Alexander Stolbov. On technology for modeling and forecasting the interrelated development of regional fuel and energy complexes of Russia and Mongolia. E3S Web of Conferences. 2020; 209 ():07019.

Chicago/Turabian Style

Alexander Kazakov; Anna Lempert; Alexander Stolbov. 2020. "On technology for modeling and forecasting the interrelated development of regional fuel and energy complexes of Russia and Mongolia." E3S Web of Conferences 209, no. : 07019.

Journal article
Published: 01 October 2020 in Diagnostics, Resource and Mechanics of materials and structures
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The paper deals with the construction of exact solutions to a nonlinear heat equation with degeneration in the case of the zero value of the required function. Generically self-similar solutions and traveling wave solutions are considered, the construction of which reduces to solving Cauchy problems for a nonlinear second-order ordinary differential equation with a singularity before the higher derivative. Two approaches are proposed to solve the Cauchy problems: the analytical solution by the power series method and the numerical solution by the boundary element method on a specified segment. A complex computational experiment is carried out to compare the above two methods with each other and with the finite difference methods, namely the Euler method and the fourth-order Runge-Kutta method. Power series segments are used on the first step of the finite difference solutions in order to resolve the singularity. The comparison of the application domains, the accuracy of the solutions and their dependence on the parameters of a certain problem shows that the boundary element method is the most universal, although not the most accurate for some particular examples. The conclusions drawn allow us to construct benchmark solutions to verify the approximate solutions of the nonlinear heat equation by various methods in a wide range of parameter values.

ACS Style

A. L. Kazakov; L. F. Spevak; E. L. Spevak. On numerical methods for constructing benchmark solutions to a nonlinear heat equation with a singularity. Diagnostics, Resource and Mechanics of materials and structures 2020, 26 -44.

AMA Style

A. L. Kazakov, L. F. Spevak, E. L. Spevak. On numerical methods for constructing benchmark solutions to a nonlinear heat equation with a singularity. Diagnostics, Resource and Mechanics of materials and structures. 2020; (5):26-44.

Chicago/Turabian Style

A. L. Kazakov; L. F. Spevak; E. L. Spevak. 2020. "On numerical methods for constructing benchmark solutions to a nonlinear heat equation with a singularity." Diagnostics, Resource and Mechanics of materials and structures , no. 5: 26-44.

Conference paper
Published: 14 September 2020 in Communications in Computer and Information Science
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The paper is devoted to the multiple covering problem by circles of two types. The number of circles of each class is given as well as a ratio radii. The circle covering problem is usually studied in the case when the distance between points is Euclidean. We assume that the distance is determined using some particular metric arising in logistics, which, generally speaking, is not Euclidean. The numerical algorithm is suggested and implemented. It based on an optical-geometric approach, which is developed by the authors in recent years and previously used only for circles of an equal radius. The results of a computational experiment are presented and discussed.

ACS Style

Alexander Kazakov; Anna Lempert; Quang Mung Le. On Multiple Coverings of Fixed Size Containers with Non-Euclidean Metric by Circles of Two Types. Communications in Computer and Information Science 2020, 120 -132.

AMA Style

Alexander Kazakov, Anna Lempert, Quang Mung Le. On Multiple Coverings of Fixed Size Containers with Non-Euclidean Metric by Circles of Two Types. Communications in Computer and Information Science. 2020; ():120-132.

Chicago/Turabian Style

Alexander Kazakov; Anna Lempert; Quang Mung Le. 2020. "On Multiple Coverings of Fixed Size Containers with Non-Euclidean Metric by Circles of Two Types." Communications in Computer and Information Science , no. : 120-132.

Journal article
Published: 11 June 2020 in Symmetry
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The paper deals with a system of two nonlinear second-order parabolic equations. Similar systems, also known as reaction-diffusion systems, describe different chemical processes. In particular, two unknown functions can represent concentrations of effectors (the activator and the inhibitor respectively), which participate in the reaction. Diffusion waves propagating over zero background with finite velocity form an essential class of solutions of these systems. The existence of such solutions is possible because the parabolic type of equations degenerates if unknown functions are equal to zero. We study the analytic solvability of a boundary value problem with the degeneration for the reaction-diffusion system. The diffusion wave front is known. We prove the theorem of existence of the analytic solution in the general case. We construct a solution in the form of power series and suggest recurrent formulas for coefficients. Since, generally speaking, the solution is not unique, we consider some cases not covered by the proved theorem and present the example similar to the classic example of S.V. Kovalevskaya.

ACS Style

Alexander Kazakov; Pavel Kuznetsov; Anna Lempert. Analytical Solutions to the Singular Problem for a System of Nonlinear Parabolic Equations of the Reaction-Diffusion Type. Symmetry 2020, 12, 999 .

AMA Style

Alexander Kazakov, Pavel Kuznetsov, Anna Lempert. Analytical Solutions to the Singular Problem for a System of Nonlinear Parabolic Equations of the Reaction-Diffusion Type. Symmetry. 2020; 12 (6):999.

Chicago/Turabian Style

Alexander Kazakov; Pavel Kuznetsov; Anna Lempert. 2020. "Analytical Solutions to the Singular Problem for a System of Nonlinear Parabolic Equations of the Reaction-Diffusion Type." Symmetry 12, no. 6: 999.

Journal article
Published: 02 June 2020 in Symmetry
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The paper deals with two-dimensional boundary-value problems for the degenerate nonlinear parabolic equation with a source term, which describes the process of heat conduction in the case of the power-law temperature dependence of the heat conductivity coefficient. We consider a heat wave propagation problem with a specified zero front in the case of two spatial variables. The solution existence and uniqueness theorem is proved in the class of analytic functions. The solution is constructed as a power series with coefficients to be calculated by a proposed constructive recurrent procedure. An algorithm based on the boundary element method using the dual reciprocity method is developed to solve the problem numerically. The efficiency of the application of the dual reciprocity method for various systems of radial basis functions is analyzed. An approach to constructing invariant solutions of the problem in the case of central symmetry is proposed. The constructed solutions are used to verify the developed numerical algorithm. The test calculations have shown the high efficiency of the algorithm.

ACS Style

Alexander Kazakov; Lev Spevak; Olga Nefedova; Anna Lempert. On the Analytical and Numerical Study of a Two-Dimensional Nonlinear Heat Equation with a Source Term. Symmetry 2020, 12, 921 .

AMA Style

Alexander Kazakov, Lev Spevak, Olga Nefedova, Anna Lempert. On the Analytical and Numerical Study of a Two-Dimensional Nonlinear Heat Equation with a Source Term. Symmetry. 2020; 12 (6):921.

Chicago/Turabian Style

Alexander Kazakov; Lev Spevak; Olga Nefedova; Anna Lempert. 2020. "On the Analytical and Numerical Study of a Two-Dimensional Nonlinear Heat Equation with a Source Term." Symmetry 12, no. 6: 921.

Journal article
Published: 01 June 2020 in Trudy Instituta Matematiki i Mekhaniki UrO RAN
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УДК 514.174.2 MSC: 52C17, 05B40, 51M16, 52A27 DOI: 10.21538/0134-4889-2020-26-2-173-187 Исследование П.Д. Лебедева поддержано грантом РНФ (проект № 19-11-00105), исследование А.Л. Казакова выполнено при поддержке РФФИ (проект № 18-07-00604), исследование А.А. Лемперт — при поддержке РФФИ (проект № 20-010-00724) и Правительства Иркутской области.

ACS Style

П. Д. Лебедев; А. Л. Казаков; А. А. Лемперт. Численные методы построения упаковок из различных шаров в выпуклые компакты. Trudy Instituta Matematiki i Mekhaniki UrO RAN 2020, 26, 1 .

AMA Style

П. Д. Лебедев, А. Л. Казаков, А. А. Лемперт. Численные методы построения упаковок из различных шаров в выпуклые компакты. Trudy Instituta Matematiki i Mekhaniki UrO RAN. 2020; 26 (2):1.

Chicago/Turabian Style

П. Д. Лебедев; А. Л. Казаков; А. А. Лемперт. 2020. "Численные методы построения упаковок из различных шаров в выпуклые компакты." Trudy Instituta Matematiki i Mekhaniki UrO RAN 26, no. 2: 1.

Journal article
Published: 19 March 2020 in Numerical Methods and Programming (Vychislitel'nye Metody i Programmirovanie)
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Рассматривается задача упаковки шаров двух типов в замкнутое ограниченное множество в трехмерном пространстве как с евклидовой, так и со специальной неевклидовой метрикой. Требуется максимизировать радиус шаров при известном количестве шаров каждого типа и заданном отношении между радиусами. Предложен вычислительный алгоритм, основанный на комбинации метода бильярдного моделирования и оптико-геометрического подхода, базирующегося на фундаментальных физических принципах Ферма и Гюйгенса. Приведены результаты вычислительного эксперимента. The problem of packing balls of two types into a closed bounded set in three-dimensional space with the Euclidean metric and a special non-Euclidean metric. It is required to maximize the radius of the balls for a given number of balls of each type and a known ratio of radii. We propose a computational algorithm based on a combination of the billiard modeling method and the optical-geometric approach employing the fundamental physical principles of Fermat and Huygens. The results of numerical experiments are discussed.

ACS Style

А.л. Казаков; А.а. Лемперт; Ч.т. Та. An algorithm for packing balls of two types in a three-dimensional set with a non-Euclidean metric. Numerical Methods and Programming (Vychislitel'nye Metody i Programmirovanie) 2020, 21, 152 -163.

AMA Style

А.л. Казаков, А.а. Лемперт, Ч.т. Та. An algorithm for packing balls of two types in a three-dimensional set with a non-Euclidean metric. Numerical Methods and Programming (Vychislitel'nye Metody i Programmirovanie). 2020; 21 (2):152-163.

Chicago/Turabian Style

А.л. Казаков; А.а. Лемперт; Ч.т. Та. 2020. "An algorithm for packing balls of two types in a three-dimensional set with a non-Euclidean metric." Numerical Methods and Programming (Vychislitel'nye Metody i Programmirovanie) 21, no. 2: 152-163.

Journal article
Published: 01 February 2020 in Siberian Advances in Mathematics
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The article is devoted to the construction and investigation of exact solutions with free boundary to a second-order nonlinear parabolic equation. The solutions belong to the classes of generalized self-similar and generalized traveling waves. Their construction is reduced to Cauchy problems for second-order ordinary differential equations (ODE), for which we prove existence and uniqueness theorems for their solutions. A qualitative analysis of the ODE is carried out by passing to a dynamical system and constructing and studying its phase portrait. In addition, we present geometric illustrations.

ACS Style

A. L. Kazakov. Construction and Investigation of Exact Solutions with Free Boundary to a Nonlinear Heat Equation with Source. Siberian Advances in Mathematics 2020, 30, 91 -105.

AMA Style

A. L. Kazakov. Construction and Investigation of Exact Solutions with Free Boundary to a Nonlinear Heat Equation with Source. Siberian Advances in Mathematics. 2020; 30 (2):91-105.

Chicago/Turabian Style

A. L. Kazakov. 2020. "Construction and Investigation of Exact Solutions with Free Boundary to a Nonlinear Heat Equation with Source." Siberian Advances in Mathematics 30, no. 2: 91-105.

Journal article
Published: 01 January 2020 in The Bulletin of Irkutsk State University. Series Mathematics
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ACS Style

A. L. Kazakov; Matrosov Institute for System Dynamics and Control Theory SB RAS; P. D. Lebedev; A. A. Lempert; Irkutsk National Research Technical University; Krasovskii Institute of Mathematics and Mechanics of UB RAS. On Covering Bounded Sets by Collections of Circles of Various Radii. The Bulletin of Irkutsk State University. Series Mathematics 2020, 31, 18 -33.

AMA Style

A. L. Kazakov, Matrosov Institute for System Dynamics and Control Theory SB RAS, P. D. Lebedev, A. A. Lempert, Irkutsk National Research Technical University, Krasovskii Institute of Mathematics and Mechanics of UB RAS. On Covering Bounded Sets by Collections of Circles of Various Radii. The Bulletin of Irkutsk State University. Series Mathematics. 2020; 31 (31):18-33.

Chicago/Turabian Style

A. L. Kazakov; Matrosov Institute for System Dynamics and Control Theory SB RAS; P. D. Lebedev; A. A. Lempert; Irkutsk National Research Technical University; Krasovskii Institute of Mathematics and Mechanics of UB RAS. 2020. "On Covering Bounded Sets by Collections of Circles of Various Radii." The Bulletin of Irkutsk State University. Series Mathematics 31, no. 31: 18-33.

Journal article
Published: 01 January 2020 in The Bulletin of Irkutsk State University. Series Mathematics
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The paper deals with the problem of the motion of a heat wave with a specified front for a general nonlinear parabolic heat equation. An unknown function depends on two variables. Along the heat wave front, the coefficient of thermal conductivity and the source function vanish, which leads to a degeneration of the parabolic type of the equation. This circumstance is the mathematical reason for the appearance of the considered solutions, which describe perturbations propagating along the zero background with a finite velocity. Such effects are generally atypical for parabolic equations. Previously, we proved the existence and uniqueness theorem for the problem considered in this paper. Still, it is local and does not allow us to study the properties of the solution beyond the small neighborhood of the heat wave front. To overcome this problem, the article proposes an iterative method for constructing an approximate solution for a given time interval, based on the boundary element approach. Since it is usually not possible to prove strict convergence theorems of approximate methods for nonlinear equations of mathematical physics with a singularity, verification of the calculation results is relevant. One of the traditional ways is to compare them with exact solutions. In this article, we obtain and study an exact solution of the required type, the construction of which is reduced to integrating the Cauchy problem for an ODE. We obtained some qualitative properties, including an interval estimation of the wave amplitude in one particular case. The performed calculations show the effectiveness of the developed computational algorithm, as well as the compliance of the results of calculations with qualitative analysis.

ACS Style

A. L. Kazakov; Matrosov Institute for System Dynamics and Control Theory SB RAS; L. F. Spevak; Institute of Engineering Science UB RAS. Approximate and Exact Solutions to the Singular Nonlinear Heat Equation with a Common Type of Nonlinearity. The Bulletin of Irkutsk State University. Series Mathematics 2020, 34, 18 -34.

AMA Style

A. L. Kazakov, Matrosov Institute for System Dynamics and Control Theory SB RAS, L. F. Spevak, Institute of Engineering Science UB RAS. Approximate and Exact Solutions to the Singular Nonlinear Heat Equation with a Common Type of Nonlinearity. The Bulletin of Irkutsk State University. Series Mathematics. 2020; 34 (34):18-34.

Chicago/Turabian Style

A. L. Kazakov; Matrosov Institute for System Dynamics and Control Theory SB RAS; L. F. Spevak; Institute of Engineering Science UB RAS. 2020. "Approximate and Exact Solutions to the Singular Nonlinear Heat Equation with a Common Type of Nonlinearity." The Bulletin of Irkutsk State University. Series Mathematics 34, no. 34: 18-34.

Journal article
Published: 20 November 2019 in Program Systems: Theory and Applications
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Статья посвящена разработке вычислительной технологии для сценарного моделирования и прогнозирования взаимосвязанного развития национальных топливно/энергетических комплексов России и Монголии с учетом межстрановой торговли топливно/энергетическими ресурсами. Целью исследования является создание методологической базы для определения наиболее перспективных вариантов двухстороннего взаимодействия, которая позволит давать обоснованные оценки эффективности проектам сотрудничества России и Монголии в области энергетики. Научной основой для создаваемой технологии послужили принципы агентного имитационного моделирования, в соответствии с которыми изучаемые объекты рассматриваются как элементы многоагентной системы. Для создания агентной имитационной модели (АИМ) топливно/энергетического комплекса России и Монголии выбрано инструментальное средство разработки агентных имитационных моделей Adskit. Проведено обоснование выбора программных средств,...

ACS Style

Александр Леонидович Казаков; Анна Ананьевна Лемперт; Александр Борисович Столбов; Борис Григорьевич Санеев; Сергей Петрович Попов. Principles of creating technology for modeling and forecasting the development of regional fuel and energy complexes of Russia and Mongolia in respect the energy cooperation between the two countries. Program Systems: Theory and Applications 2019, 10, 1 .

AMA Style

Александр Леонидович Казаков, Анна Ананьевна Лемперт, Александр Борисович Столбов, Борис Григорьевич Санеев, Сергей Петрович Попов. Principles of creating technology for modeling and forecasting the development of regional fuel and energy complexes of Russia and Mongolia in respect the energy cooperation between the two countries. Program Systems: Theory and Applications. 2019; 10 (4):1.

Chicago/Turabian Style

Александр Леонидович Казаков; Анна Ананьевна Лемперт; Александр Борисович Столбов; Борис Григорьевич Санеев; Сергей Петрович Попов. 2019. "Principles of creating technology for modeling and forecasting the development of regional fuel and energy complexes of Russia and Mongolia in respect the energy cooperation between the two countries." Program Systems: Theory and Applications 10, no. 4: 1.

Conference paper
Published: 19 November 2019 in MECHANICS, RESOURCE AND DIAGNOSTICS OF MATERIALS AND STRUCTURES (MRDMS-2019): Proceedings of the 13th International Conference on Mechanics, Resource and Diagnostics of Materials and Structures
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The problem of constructing solutions to the nonlinear heat equation with power nonlinearity is considered. The solutions have the form of a traveling wave and simulate the propagation of disturbances over a cold background with a finite velocity. It is shown that the construction can be reduced to the Cauchy problem for an ordinary second- order differential equation with a singularity multiplying the highest derivative. Its solutions are constructed using the boundary element method based on the dual reciprocity method. A computational experiment is carried out. The results are compared with the solutions of the same problems by the power series method. The calculations have shown the correctness of the developed boundary element algorithm and its advantage compared to the power series segments and the step-by-step method previously proposed by the authors.

ACS Style

A. L. Kazakov; L. F. Spevak. Numerical study of travelling wave type solutions for the nonlinear heat equation. MECHANICS, RESOURCE AND DIAGNOSTICS OF MATERIALS AND STRUCTURES (MRDMS-2019): Proceedings of the 13th International Conference on Mechanics, Resource and Diagnostics of Materials and Structures 2019, 2176, 030006 .

AMA Style

A. L. Kazakov, L. F. Spevak. Numerical study of travelling wave type solutions for the nonlinear heat equation. MECHANICS, RESOURCE AND DIAGNOSTICS OF MATERIALS AND STRUCTURES (MRDMS-2019): Proceedings of the 13th International Conference on Mechanics, Resource and Diagnostics of Materials and Structures. 2019; 2176 (1):030006.

Chicago/Turabian Style

A. L. Kazakov; L. F. Spevak. 2019. "Numerical study of travelling wave type solutions for the nonlinear heat equation." MECHANICS, RESOURCE AND DIAGNOSTICS OF MATERIALS AND STRUCTURES (MRDMS-2019): Proceedings of the 13th International Conference on Mechanics, Resource and Diagnostics of Materials and Structures 2176, no. 1: 030006.

Conference paper
Published: 01 November 2019 in Journal of Physics: Conference Series
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The paper discusses solutions of the nonlinear heat equation, which have the form of a heat wave propagating on a zero background with a finite velocity. Such solutions are not typical for parabolic equations, and their existence is associated with the degeneration of the problem at the wave (zero) front. We propose a numerical algorithm for constructing a two-dimensional heat wave, symmetrical with respect to the origin, with a non-zero boundary condition defined on the moving boundary. The main difficulty of the new task is that at each time point a heat wave front (a domain boundary) is unknown. The solution is carried out in two stages. At first, we change the roles of unknown function and radial polar coordinate. For a new unknown function at each time point, we obtain a boundary value problem for the Poisson equation in a known region. The step-by-step solving of this problem by the method of boundary elements at a given time interval allows us to determine the law of the zero front moving. At second, we approximate the found zero front by an analytical function and construct a generalized self-similar solution. The developed algorithm is implemented and tested on a task set.

ACS Style

A L Kazakov; L F Spevak; A A Lempert; O A Nefedova. A computational algorithm for constructing a two-dimensional heat wave generated by a non-stationary boundary condition. Journal of Physics: Conference Series 2019, 1392, 012083 .

AMA Style

A L Kazakov, L F Spevak, A A Lempert, O A Nefedova. A computational algorithm for constructing a two-dimensional heat wave generated by a non-stationary boundary condition. Journal of Physics: Conference Series. 2019; 1392 (1):012083.

Chicago/Turabian Style

A L Kazakov; L F Spevak; A A Lempert; O A Nefedova. 2019. "A computational algorithm for constructing a two-dimensional heat wave generated by a non-stationary boundary condition." Journal of Physics: Conference Series 1392, no. 1: 012083.

Conference paper
Published: 27 October 2019 in Communications in Computer and Information Science
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The paper is devoted to the circle covering problem with unequal circles. The number of circles is given. Also, we know a function, which determines a relation between the radii of two neighboring circles. The circle covering problem is usually studied in the case when the distance between points is Euclidean. We assume that the distance is determined by means of some special metric, which, generally speaking, is not Euclidean. The special numerical algorithm is suggested and implemented. It based on optical-geometric approach, which is developed by the authors in recent years and previously used only for circles of the equal radius. The results of a computational experiment are presented and discussed.

ACS Style

Alexander Kazakov; Anna Lempert; Quang Mung Le. On the Thinnest Covering of Fixed Size Containers with Non-euclidean Metric by Incongruent Circles. Communications in Computer and Information Science 2019, 195 -206.

AMA Style

Alexander Kazakov, Anna Lempert, Quang Mung Le. On the Thinnest Covering of Fixed Size Containers with Non-euclidean Metric by Incongruent Circles. Communications in Computer and Information Science. 2019; ():195-206.

Chicago/Turabian Style

Alexander Kazakov; Anna Lempert; Quang Mung Le. 2019. "On the Thinnest Covering of Fixed Size Containers with Non-euclidean Metric by Incongruent Circles." Communications in Computer and Information Science , no. : 195-206.

Journal article
Published: 07 August 2019 in Sibirskie Elektronnye Matematicheskie Izvestiya
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ACS Style

Alexander Kazakov. On exact solutions to a heat wave propagation boundary-value problem for a nonlinear heat equation. Sibirskie Elektronnye Matematicheskie Izvestiya 2019, 16, 1057 -1068.

AMA Style

Alexander Kazakov. On exact solutions to a heat wave propagation boundary-value problem for a nonlinear heat equation. Sibirskie Elektronnye Matematicheskie Izvestiya. 2019; 16 ():1057-1068.

Chicago/Turabian Style

Alexander Kazakov. 2019. "On exact solutions to a heat wave propagation boundary-value problem for a nonlinear heat equation." Sibirskie Elektronnye Matematicheskie Izvestiya 16, no. : 1057-1068.

Journal article
Published: 01 August 2019 in Diagnostics, Resource and Mechanics of materials and structures
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ACS Style

A. L. Kazakov. ON THE ANALYTICAL CONSTRUCTION OF A HEAT WAVE FOR THE NONLINEAR HEAT EQUATION WITH A SOURCE IN POLAR COORDINATES. Diagnostics, Resource and Mechanics of materials and structures 2019, 16 -25.

AMA Style

A. L. Kazakov. ON THE ANALYTICAL CONSTRUCTION OF A HEAT WAVE FOR THE NONLINEAR HEAT EQUATION WITH A SOURCE IN POLAR COORDINATES. Diagnostics, Resource and Mechanics of materials and structures. 2019; (4):16-25.

Chicago/Turabian Style

A. L. Kazakov. 2019. "ON THE ANALYTICAL CONSTRUCTION OF A HEAT WAVE FOR THE NONLINEAR HEAT EQUATION WITH A SOURCE IN POLAR COORDINATES." Diagnostics, Resource and Mechanics of materials and structures , no. 4: 16-25.