Статьи журнала - Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование
Все статьи: 739
Статья научная
Constant structure closed semantic systems are the systems each element of which receives its definition through the correspondent unchangeable set of other elements (neighbors) of the system. The definitions of the elements change iteratively and simultaneously based on the neighbor portraits from the previous iteration. In this paper, I study the behavior of such model systems, starting from the zero state, where all the system's elements are equal. The development of constant-structure discrete time closed semantic systems may be modelled as a discrete time coloring process on a connected graph. Basically, I consider the iterative redefinition process on the vertices only, assuming that the edges are plain connectors, which do not have their own colors and do not participate in the definition of the incident vertices. However, the iterative coloring process for both vertices and edges may be converted to the vertices-only coloring case by the addition of virtual vertices corresponding to the edges assuming the colors for the vertices and for the edges are taken from the same palette and assigned in accordance with the same laws. I prove that the iterative coloring (redefinition) process in the described model will quickly degenerate into a series of pairwise isomorphic states and discuss some directions of further research.
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L-stability of nonlinear systems represented by state models
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Stability theory plays a key role in systems theory and engineering. The stability of equilibrium points is usually considered within the framework of the stability theory developed by the Russian mathematician and mechanic A.M. Lyapunov (1857-1918), who laid its foundations and gave it its name. Nowadays, the point of view on stability has become very widespread, as stability in relation to disturbance of the input signal. The research is based on the space-state approach for modelling nonlinear dynamic systems and an alternative "input-output'' approach. The input-output model is implemented without explicit knowledge of the internal structure determined by the equation of state. The system is considered as a "black box'', which is accessed only through the input and output terminals ports. The concept of stability in terms of "input-output'' is based on the definition of L-stability of a nonlinear system, the method of Lyapunov functions and its generalization to the case of nonlinear dynamical systems. The interpretation of the problem on accumulation of perturbations is reduced to the problem on finding the norm of an operator, which makes it possible to expand the range of models under study, depending on the space in which the input and output signals act.
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Статья научная
Modelling of various natural and technical processes often results in systems that comprise ordinary differential equations and algebraic equations This paper studies systems of quasi-linear integral-differential equations with a singular matrix multiplying the higher derivative of the desired vector-function. Such systems can be treated as differential algebraic equations perturbed by the Volterra operators. We obtained solvability conditions for such systems and their initial problems and consider possible ways of linearization for them on the basis of the Newton method. Applications that arise in the area of thermal engineering are discussed and as an example we consider a hydraulic circuit presented as a system comprising an interconnected set of discrete components that transport liquid. Numerical experiments that employed the implicit Euler scheme showed that the mathematical model of the straight-through boiler with a turbine and a regeneration system has a solution and this solution tends to the stationary mode preset by regulators.
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Lord Kelvin and Andrey Andreyevich Markov in a queue with single server
Статья научная
We use Lord Kelvin's method of images to show that a certain infinite system of equations with interesting boundary conditions leads to a Markovian dynamics in an L1-type space. This system originates from the queuing theory.
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Статья научная
The problem of three-dimensional mathematical modelling of the effect of air flow on an optical-mechanical unit (OMU) located in the unpressurised compartment of the aircraft, is considered. To solve this problem, a mathematical model of gas dynamics based on the solution of a complete system of Navier - Stokes equations that describe the dynamics of a turbulent, spatially unsteady flow of a viscous gas is developed. The software for simulating the process of flow past a WMU model in the aircraft compartment was created. The effect of the air flow on the OMU is described by the torque acting on the OMU from the airflow side. A numerical method for solving the three-dimensional gasdynamic problem is presented. The numerical method is based on the numerical high order Godunov scheme, realized on an irregular grid with arbitrary cells (tetrahedral, prismatic shape). Flows of conservative variables are calculated by solving the Riemann problem with an approximate AUSM method. The system of equations is supplemented by a two-parameter k-model of turbulence, modified for the calculation of high-speed compressible flows. To significantly reduce the cost of computing resources, it is suggested to use stochastic models of the effect of air flow on WMU. A general simulation algorithm is described.
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Mathematical model of a successful stock market game
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All available predictive models of stock market trade (like regression or statistical analysis, for instance) are based on studying of price fluctuation. This article proposes a new model of a successful stock market strategy based on studying of the behavior of the largest successful players. The main point of this model is that a relatively weak player repeats the actions of stronger players in the same fashion as in a race after leader a cyclist following a motorbike reaches greater velocity. We represent the leader as a vector in the nonnegative orthant Rn+ depending on the most successful traders (hedge funds). When buying and selling stocks, we should always keep the vector of own resources collinear to the leader's. This strategy will not yield significant profit, but it prevents considerable loss.
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Mathematical model of a wide class memory oscillators
Статья научная
A mathematical model is proposed for describing a wide class of radiating or memory oscillators. As a basic equation in this model is an integro-differential equation of Voltaire type with difference kernels - memory functions, which were chosen by power functions. This choice is due, on the one hand, to broad applications of power law and fractal properties of processes in nature, and on the other hand it makes it possible to apply the mathematical apparatus of fractional calculus. Next, the model integro-differential equation was written in terms of derivatives of fractional Gerasimov - Caputo orders. Using approximations of operators of fractional orders, a non-local explicit finite-difference scheme was compiled that gives a numerical solution to the proposed model. With the help of lemmas and theorems, the conditions for stability and convergence of the resulting scheme are formulated. Examples of the work of a numerical algorithm for some hereditary oscillators such as Duffing, Airy and others are given, their oscillograms and phase trajectories are constructed.
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Статья научная
The mathematical model of liquid hydrogen sulfide injection into the semi-infinite porous layer saturated with the oil and water accompanied by H2S gas hydrate formation is presented here. We considered the case when the hydrate formation occurs at the frontal border and the oil displacement's front by hydrogen sulfide is ahead of this boundary. Solutions for pressure and temperature in every layer's area are built by help of the self-similar variable formation method. The values of the parameters of the moving interphase boundaries are found as the result of the iteration procedure. The coordinate dependence of phase boundaries on the injection pressure was studied on the basis of the obtained solutions. We have established that for the existence of solution with two different interphase boundaries, the injection pressure must be above a certain limiting value. The dependence of the limiting value of pressure on the initial temperature of the layer at different temperatures of the injected hydrogen sulphide is constructed. The results of the calculations showed that the constructed mathematical model with three areas in the reservoir gives an adequate description of the process at high injection pressures, the temperature of the injected hydrogen sulfide and the initial temperature of the layer.
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Статья научная
Based on the conjugate Darcy-Brinkman-Forchheymer hydrodynamic model and Schumann thermal model with boundary conditions of the second kind, a model with lumped parameters was proposed by means of geometric 2D averaging to identify the integral kinetics of the temperature fields of a porous matrix and a Newtonian coolant without phase transitions. The model was adapted for a heat-stressed surface by means of a porous compact heat exchanger with uniform porosity and permeability, obeying the modified Kozeny-Carman relation, in the form of a Cauchy problem, the solution of which was obtained in the final analytical representation for the average volume temperatures of the coolant and the porous matrix. The possibility of harmonic damped oscillations of the temperature fields and the absence of coolant overheating in the starting condition of the cooling system were shown. For the dimensionless time of establishing the stationary functioning of the porous heat exchanger, an approximate estimate was obtained correlating with the known data of computational and full-scale experiments.
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Статья научная
In this article we construct a mathematical model and examine the characteristics of the injection of carbon dioxide into the natural layer, rich of the methane and its hydrate in its initial state. We consider the case in which due to the injection of carbon dioxide in the layer two areas are formed: in the first (proximal) region the pores of the layer are saturated with carbon dioxide and its hydrate but in the second (distant) region the pores of the layer are saturated with methane and its hydrate. In the numerical experiments it was established when the pressure increasing on the right border of the layer and decreasing the permeability of the layer or of the pressure of the of carbon dioxide injected, the temperature of the layer can rise at the front of the replacement above the equilibrium temperature of gas hydrate decomposition of methane that corresponds to the appearance of the dissociation border of gas hydrate to methane and water.
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Mathematical model of the downward two-phase flow of a heat-transfer agent in an injection well
Статья научная
At the present time, the main method of developing highly viscous and bituminous oil reservoirs is the injection of hot water or steam into such reservoirs. When injecting heat-transfer agent into a porous reservoir, its characteristics at the wellhead are known. It is important to know the parameters of a heat-transfer agent (pressure, temperature, mass content of steam in a two-phase mixture "water-steam", etc.) directly at the reservoir entrance. In order to calculate various parameters of a heat-transfer agent along the injection well depth (including the bottomhole), we propose a mathematical model of the downward flow of a hot "water-steam" mixture in a vertical channel. The model takes into account phase transitions occurring in a two-phase "water-steam" mixture, and external heat exchange of the well product with surrounding rocks (including permafrost). Based on the proposed mathematical model, we develop an algorithm to solve a quasistationary problem. In this case, we use the Runge-Kutta method in order to solve the system of differential equations describing the stationary flow of a heat-transfer agent in a well. Also, in order to solve the non-stationary problem of temperature distribution in the rocks that surround the well (including permafrost), we use the author enthalpy method with implicit scheme. For each time moment, the developed software allows to find the distributions along the well depth of various parameters of the downward two-phase flow, taking into account external heat exchange, as well as the temperature distribution in the rocks that surround the well and the permafrost thawing radius.
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Mathematical modelling and analysis of activities of PJSC "Fortum"
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The article discusses a mathematical model of the production activity of PJSC "Fortum", which is the main supplier of electrical and thermal energy in the Southern Urals. The model is constructed on the basis of the modified production function of Cobb - Douglas for 2010-2020. The author's algorithm for constructing the model is given, according to which a computer program for modelling the operating activities of an enterprise was written and registered in the State Register of computer programs in the Java language. The adequacy of the constructed model was checked by the coefficient of determination, the value of which showed its high reliability. The analysis of the constructed mathematical model made it possible to draw the following conclusions: the strategic goal of reducing tariffs declared in the course of the restructuring of OJSC "RAO UES of Russia" was not fulfilled, on the contrary, their steady growth is observed; the efficiency of operating activities does not meet modern requirements - during the entire analyzed period, there is a diminishing economies of scale; the sustainable development strategy implemented by PJSC "Fortum" does not have a proper economic power supply unit due to a decrease in the return on resources involved in production; relatively high financial results of PJSC "Fortum" are associated with an unreasonable increase in tariffs; consumers of electric and thermal energy of the Urals and Western Siberia, in the conditions of the monopoly on the energy market, finance the richest country in the world - the Netherlands - through payments from Fortum Holding B.V., which directly and indirectly owns 98% of the shares of PJSC "Fortum"; with the existing management mechanism and state control by the antimonopoly service, PJSC "Fortum" has no incentives to replace the outdated cogeneration option based on hydrocarbon fuels with modern, progressive green technologies. The results of the study are recommended to the Federal Antimonopoly Service to strengthen control over the activities of PJSC "Fortum", reduce tariff pressure on consumers of electric and thermal energy and unjustified enrichment of foreign subjects of the electric power industry at the expense of Russian consumers.
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Mathematical modelling economy
Статья обзорная
The article presents an overview of the main methods of economic modelling used in scientific research over the past twenty years. This overview does not claim to cover all areas, methods and models used in scientific research in the field of economics, since it is impossible to do within a single article. We consider mathematical modelling of only two branches of economic theory: macroeconomics and microeconomics. At the same time, we present no literature review of methods and models of research in the section of microeconomics, which take place in the tools of scientific research, but were described in the section of macroeconomics. We believe that this review is useful to scientists engaged in the indirect study of economic phenomena and processes.
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Mathematical modelling of a transport system with minimal maintenance costs
Статья научная
We suggest a mathematical model of a transport system. The model describes the delivery of products from several suppliers to different points of consumption. It is assumed that consumer demands are random. The model is a two-stage stochastic programming problem. At the first stage suppliers make the commodity stocks. At the second stage we consider the product distribution to the points of consumption while minimizing compensation expenses for the goods shortage caused by the random demand. The model takes into account a random loss that occurs during the transportation of goods and the detection of defective products. The total cost of the transport system operation is minimized. The algorithm for solving the problem is proposed. This algorithm is based on reduction of the original problem to an equivalent mixed-integer linear programming problem after discretization. An example is considered.
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Mathematical modelling of industrial equipment operation based on Markov processes
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The existence of an almost unlimited number of methods for evaluating the productivity of industrial equipment contributes to the uncertainty of choosing the most effective approach. At the same time, the presence of many possible states of equipment (from working to repair and other downtime) complicates the problem of modelling the operation of such a system. The problem of modelling does not lose its relevance, first of all, for large industrial companies. The article presents the methodological features of modelling the operation of industrial equipment based on the Markov method. This approach is used as a base for estimating the probabilities of equipment transitions between states, as well as for predicting the final state of operation of such a system. In terms of practical application, we consider an example of the functioning of the same type of industrial equipment in the framework of three possible states (functional, broken, and also in the mode of forced repair). Based on the results of calculations, we carry out assessment of the reality of the state transitions of equipment, designed rate of these transitions, as well as the predicted level of productivity equipment system after the period "''. The reliability of the research results is confirmed by their practical implementation. The obtained results are recommended to be used by the management and analysts of industrial companies in the process of making operational decisions and in the development of equipment repair strategies.
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Quantitative approaches based on mathematical modelling are used to justify a set of measures aimed at justifying a set of preventive measures to protect the life and health of older people in the context of COVID-19 pandemic. Analysis of the state of development of actuarial mathematical models of mortality in the COVID-19 epidemic shows the need to construct models that reflect the dynamics of the studied ratios of infection rates, morbidity, recovery and mortality in the dynamics of the pandemic, taking into account the influence of external factors on this process. Most of the known mathematical models for predicting the spread and consequences of COVID-19 are compartmental models that implement sequential transitions between states with the allocation of groups of individuals with different affiliation to the progression/decline of the spread of infection. To compensate for the shortcomings of the compartmental models due to the assumption of population homogeneity and the lack of adequate approaches to the scalability of the simulation results, models based on the Monte Carlo method and the concept of multi-agent systems are used. The development of modelling methods is associated with the need to expand information support for healthcare professionals and health care organizers with the possibility of online configuration of parameters of mathematical models and the use of data from «cloud services» with visualization of the results of modelling.
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Статья научная
The article is devoted to mathematical modelling of processes, occurring in the flowing part of the vortex flowmeter, by the finite element method. The urgency of the current study is due to the lack of research in this area. The analysis of research literature devoted to the study of non-stationary vortex shedding processes and other hydrogasdynamics effects occurring in the flowing part of the vortex flowmeter and similar devices has been performed. A brief description of the vortex generation process behind the bluff body placed in a circular cross-section pipe as well as the basic criteria for functional products are presented. Various mathematical models for describing turbulent flows in pipes with an object or obstruction were investigated. The available software packages suitable for modelling unsteady turbulent flows were analyzed. The ANSYS software package, in particular CFX module for fluid and gas, as well as various approaches to mathematical modelling were used to simulate the flowing part of the vortex flowmeter. The article provides a brief description of the basic computational domain settings, mesh formation and initial and boundary conditions setting. To verify the numerical calculations, physical experiments on fluid and gas test benches were performed. For this purpose the samples corresponding to the numerical model have been manufactured and tested. The research findings led us to conclude that in terms of accuracy and calculation time the optimal approach to numerical simulation of vortex generation processes (Karman vortex street) in the vortex flowmeter is the use of the Reynolds-averaged Navier - Stokes equations (or RANS equations) closed by means of a two-equation model of turbulence, known as the k-e model, which is confirmed by comparison with the experimental data.
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Статья научная
Development of periodic disturbances on free surface of water film falling down vertical plane for Reynolds' number Reє[5;10] is investigated. The investigation is implemented in a scope of the nonlinear differential equation for evolution of free surface of falling down liquid film. The equation is solved by a finite differencies method at rectangular uniformly spaced grid. By researching the growth of unit inaccuracy, the conditions on parameters of computation grid for inaccuracies to be not increasing are obtained. As a result, waveforms of water film, time spent to form the regular wave mode and amplitudes of periodic disturbances are calculated. Calculated amplitudes and experimental ones are compared.
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Mathematical terrain modelling with the help of modified Gaussian functions
Статья научная
Based on a fundamentally new approach, we present a complete mathematical model for estimating the mass of water in the flooded coastal relief, taking into account the water in the basin of the reservoir in a given region. Taking into account stochastic studies, we construct an approximate model of the relief of the reservoir basin bottom, as well as the relief of a possible section of the flooding of this basin coastline. The modelling is based on the empirical data of measurements of the reservoir depths, as well as on the study on the architecture of the lines of the coastal maps of the possible flooding zone. Based on the measurements of the depths and bumps of the bottom surface, we verify the hypothesis that the use of the two-dimensional Gauss distribution is adequate. Numerous confirmation of this hypothesis on the basis of empirical measurements allows to use localized elliptic Gauss surfaces as a model function in order to construct an approximate model of hillocks and valleys. At the same time, the coordinates of local extremes of the depths, as well as the values of these extremes are constant. In order to simulate the surfaces of the underwater slopes, we construct planes according to depth measurements. This simulation is not a real copy, but is stochastic in nature and allows to take into account the main goal of the model, i.e. a full adequate estimation of the water mass of the flooded coastal relief included the water in the basin of the reservoir in the region. The equation of the model of the entire flooded region includes all local functions constructed for the mounds and troughs of the reservoir, as well as the functions of the planes of the slope models. For an approximate construction of the surface equations of the coastal zone, we use maps with detailed level lines as empirical data.
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