Краткие сообщения. Рубрика в журнале - Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование
Computational experiment for a class of mathematical models of magnetohydrodynamics
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The first initial-boundary value problem for the system modelling the motion of the incompressible viscoelastic Kelvin - Voigt fluid in the magnetic field of the Earth is investigated considering that the fluid is under external influence. The problem is studied under the assumption that the fluid is under different external influences depending not only on the coordinates of the point in space but on time too. In the framework of the theory of semi-linear Sobolev type equations the theorem of existence and uniqueness of the solution of the stated problem is proved. The solution itself is a quasi-stationary semi-trajectory. The description of the problem's extended phase space is obtained. The results of the computainal experiment are presented.
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Computational experiment for one class of evolution mathematical models in quasi-Sobolev spaces
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In the article the mathematical model representing one class of evolution equations in quasi-Banach spaces is studied. A theorem on the unique solvability of the Cauchy problem is stated. The conditions for the phase space existence are presented. We also give the conditions for exponential dichotomies of solutions. Based on the theoretical results there was developed an algorithm for the numerical solution of the problem. The algorithm is implemented in Maple. The article includes description of the algorithm which is illustrated by variety of model examples showing the work of the developed program and represent the main properties of solutions.
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Computational experiment for one mathematical model of ion-acoustic waves
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In the article the mathematical model of ion-acoustic waves in a plasma in an external magnetic field is studied. This model can be reduced to a Cauchy problem for a Sobolev type equation of the fourth order with polynomially (A,p)-bounded operator pencil. Therefore abstract results on solvability of the Cauchy problem for such equation can be used. In the article a theorem on the unique solvability of the Cauchy - Dirichlet problem is mentioned. Based on the theoretical results there was developed an algorithm for the numerical solution of the problem, using a modified Galerkin method. The algorithm is implemented in Maple. The article includes description of this algorithm. It is illustrated by model examples showing the work of the developed program.
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We investigate of a special dam optimal location at the Volga river in the area of the Akhtuba left sleeve beginning (7 km to the south of the Volga Hydroelectric Power Station dam). We claim that a new water-retaining dam can resolve the key problem of the Volga-Akhtuba floodplain related to insufficient water amount during spring floodings due to the overregulation of the Lower Volga. Using a numerical integration of Saint-Vanant equations we study the water dynamics across the northern part of the Volga-Akhtuba floodplain taking into account its actual topography. As the result we found an amount of water VA passing to the Akhtuba during spring period for a given water flow through the Volga Hydroelectric Power Station (so-called hydrograph which characterises the water flow per unit of time). By varying the location of the water-retaining dam xd,yd we obtained various values of VA(xd,yd) as well as various flow spatial structure on the territory during the flood period. Gradient descent method provides the dam coordinated with the maximum value of VA. Such approach to the dam location choice let us find the best solution, that the value VA increases by a factor of 2. Our analysis demonstrates a good potential of the numerical simulations in the field of hydraulic works.
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This paper considers the parameter estimation problem for models of one-dimensional chaotic systems. The guaranteed algorithm is proposed in the context of set-membership approach, which assumes that only intervals of possible values are known for the uncertain variables in the model (initial condition, parameter and measurement errors). The algorithm recursively computes the interval estimates of the parameter at every time step. If the prior information is correct, found interval estimates always contain the true value of the parameter. For certain models of measurement errors the result of the algorithm is the exact value of the parameter (the final interval estimate contains a single point). The goal of this study is to derive conditions under which the guaranteed algorithm improves the interval estimate of the parameter.
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Convergence modelling in international integration associations
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The article considers mathematical tools for modelling economic policy as a whole, as well as convergence in the field of labor, foreign economic activity, monetary and debt policy. Convergence was estimated using the σ-convergence model, which characterizes the decrease in time spread in the levels of development of countries and regions, reflecting the negative relationship between economic growth rates and the initial level of development of countries and regions. The σ-convergence was estimated by the coefficient of variation and by the dispersion-based model. To assess β-convergence, we used the Barro and Sala-i-Martin models, as well as the Baumol, Solow-Svan, and Quadrado-Rour models. The use of this mathematical toolkit allows to explore the presence and speed of convergence before and after joining international integration associations. The proposed mathematical modelling tools are recommended to be used in order to analyze convergence processes, study the dynamics of convergence or divergence, and also to adjust the directions and methods of state and regional economic policies of countries included in the integration association.
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Forecasting tariffs for the day-ahead market based on the additive model
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The problem of constructing an additive model for forecasting of the market tariff for the day ahead is solved. The trend component is constructed on the basis of the autoregressive model of already known values of the day-ahead market tariff and the external factor of electricity consumption according to the United Energy System (UES) of the Urals Wholesale Electricity and Power Market (OREM) of Russia for 2009-2018. Based on the construction of the autocorrelation function, three seasonal components are identified in the time series of hourly values of the market tariff for the day ahead: annual (8760 values), weekly (168 values), daily (24 values). A harmonic model of each component is constructed. The final additive model is constructed taking into account the specifics of the electricity market and the process of setting the market tariff for the day ahead and a balancing market. The practical significance of the developed additive model is adequate accuracy with the well-known models for forecasting of the market tariff for the day ahead of the UES of the Urals. The proposed model allows the subjects of the electric power industry to avoid penalties from the balancing market by ensuring high accuracy of forecasting.
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We study the possibility of influence on the saving of allocated funds for the elimination of consequences of natural disasters. At that, we take into account statistical data on the emergence of such phenomena and the degree of actual damage. The article describes the problem of determining the optimal share of funds that either replenish or spend the principal amount according to the distribution. We prove that, under certain conditions of the distribution and a positive mathematical expectation, it is possible to choose a share that ensures the maximum possible growth of the original deposit account. At the same time, the choice of the share allows not to lose the full provision of damage recovery. This process is presented as a serial multi-stage process based on a Markov chain that takes into account only the distribution based on the statistical data of this year to plan the size of the deposit share for the next year. For simplicity, we assume that the process is established and has a constant distribution for some time. The distribution table can be changed in the case of a major change in stochastic data. We consider a serial multi-stage process of changing the monetary amount that is purposefully deposited for the renewal, replacement and restoration of security and alarm systems at burned-out facilities. The optimal stochastic control of the change in the share of the money deposit providing this restoration is carried out based on the generalized Kelly formula. An example of model validity is shown. On the basis of statistical data, the analysis of the possibility to use this model is carried out.
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Income modelling of enterprises on the basic of vector prediction
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This paper proposes a new approach to income modelling of enterprises, based on the vector prediction methods. Existing approaches to the income modelling are based on the use of traditional methods of economic dynamics prediction such as the average absolute increase and average growth rate. Quantitative inaccuracy and highly approximate nature of predictions are inherent for traditional methods. The authors propose the economic-mathematical model for the enterprise income planning for a year ahead on a quarterly basis. For revenue prediction two methods of vector prediction are used (the method of orthogonal differences and multiplicative Holt - Winters' method). This model provides prediction for a few steps forward at the same time. Individually, each of these methods doesn't take into account the diversity of the process. Only in conjunction the methods allow to take into account the demolition of trends and seasonal nature of income, thus providing the necessary stability of the prediction. To summarize two predictions their linear combination is calculated, the choice of weighting coefficients being based on the accuracy of private predictions. The accuracy of the private prediction is defined as the average relative error of the forecast.
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Introducing a power of the operator in direct spectral problems
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The resolvent method, proposed by Sadovnichiy and Dubrovsky in the 1990s, is successfully applied in the direct spectral problem to calculate the asymptotics of eigenvalues of the perturbed operator, find formulas for the regularized trace, and recover perturbation. But the application of this method faces difficulties when the resolvent of the unperturbed operator is non-nuclear. Therefore, a number of physical problems could only be considered on the interval. This article describes a justification of the transition to the power of an operator in order to expand the area of possible applications of the resolvent method. Considering the problem of calculating the regularized trace of the Laplace operator on a parallelepiped of arbitrary dimension, we show that for every fixed dimension it is possible to choose the required power of the operator and to calculate the regularized traces. These studies are relevant due to the need to study important applied problems, particularly in hydrodynamics, electronics, elasticity theory, quantum mechanics, and other fields.
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Inverse problem for a linearized quasi-stationary phase field model with degeneracy
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The inverse problem for a linearized quasi-stationary phase field model is considered. The inverse problem is reduced to a linear inverse problem for the first order differential equation in a Banach space with a degenerate operator at the derivative and an overdetermination condition on the degeneracy subspace. The unknown parameter in the problem dependens on the source time function. The theorem of existence and uniqueness of classical solutions is proved by methods of degenerate operator semigroup theory at some additional conditions on the operator. General results are applied to the original inverse problem.
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In this paper, we study the problem of boundary control and final observation for one degenerate mathematical model of motion speed potentials distribution of filtered liquid free surface with the Showalter-Sidorov initial condition. The mathematical model is based on the degenerate Boussinesq equation with an inhomogeneous Dirichlet condition. This model belongs to the class of semilinear Sobolev-type models in which the nonlinear operator is p-coercive and s-monotone. In the paper, the problem of boundary control and final observation for a semilinear Sobolev-type model is considered and conditions for the existence of a control-state pair of the problem are found. In applied studies of a research problem, it is allowed to find such a potentials distributionof filtered liquid free surface, at which the system transitions from the initial condition to a given final state within a certain period of time T.
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Investigation of the transient responses of a beam on an elastic polymeric foundation
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The negative impact of vibrations on various devices and mechanisms can be significant, so it is important to take this factor into account when designing, operating and maintaining various equipment and engineering systems. Various methods and technologies can be used to protect against the negative effects of vibrations. Special damping materials are often used. This research paper is devoted to the analysis of the effectiveness of vibration reduction taking into account the physical parameters of elastic polymeric materials. To conduct the study, a mathematical model describing motion of the beam resting on an elastic polymeric foundation is constructed. The model is based on a system of nonlinear differential equations. An algorithm was developed and applied for the numerical solution of this system of equations. Numerical experiments were carried out for the study of the system reaction to different cases of accelerations. As a result, the deflection structure for materials with different physical characteristics were obtained. These results can serve as a starting point for a deeper study of materials and creation of more complex structures.
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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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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 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 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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Modelling of "green" investments risks
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The practical lack of methods for quantitative assessment of capital-intensive renewable energy projects taking into account the present uncertainty leads to an increase in the riskiness and reduce the amount of "green" investment in the economy. It resolves the urgency of the developing approaches to the renewable energy projects assessment. The article presents the diagnostics of specific risks and assessment of limiting the investor market share in the renewable energy projects with the optimal risk level. The proposed authors' approach to risks formalization of renewable energy deals is based on the introduction of dummy-variable, evaluation of coefficients' significance, tendency and strength of relation of risks indicators. This approach yields an indicative assessment of the deals' riskiness taking into account sectoral specifics. The developed method to limiting the investor market share is based on the principles of the economic capital theory. It allows for consolidating the current states of investor, project and economy. Research veracity is confirmed by the practical implementation. The results of the study can be used by the management of energy companies, investors and analysts in making financial decisions.
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The aim of this work is to carry out numerical calculation of electric fields in axisymmetric systems for cathode protection of underground structures against corrosion. As is known, cathode protection is a complex multi-parameter system, which has a number of features depending on the type and form of the protected structure, properties of the surrounding environment, etc. As a result, the modelling of protection parameters is quite complex and requires new more advanced computer technologies. In this work, we construct a mathematical model of the problem on the electric field distribution in an axisymmetric system for cathode electrochemical protection of underground structures. In addition, we propose an algorithm to solve the problem. The algorithm is based on the use of integral Green transformations and, in particular, the second Green integral formula. Also, we give an example of a computer implementation of the proposed algorithm in the case of a homogeneous half-space. The Scilab application package is used to implement the example.
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