Математическое моделирование. Рубрика в журнале - Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование
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
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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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Model of conveyer with the regulable speed
Статья научная
This article is devoted to mathematical modelling of the production flow lines of the conveyor-type. Here is an analytical method for calculating the parameters of a production line with a regulated speed of the movement of the subjects of the labour along the conveyor developed. The description of the parameters of the state of the production line is made in the one-moment approximation using partial differential equations. There has been derived a solution that determines the state of the parameters of the production line for a given technological position as a function of the time. The transitional period during which the initial condition of the distributing of the subjects of the labour along a conveyor has the influence on the state parameters of a production line is certain. The developed method of the calculation of the flow parameters of the production line allows designing control systems of the production line of the conveyor-type with a regulated rate of the movement of the subjects of the labour.
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Статья научная
The paper considers a boundary-value problem for a hybrid system of differential equations, which represents a generalized mathematical model for a system of interconnected rigid bodies attached to the rod by elastic-damping links. A hybrid system of differential equations is understood as a system of differential equations composed of ordinary differential equations and partial differential equations. For the theoretical foundations of our approach to investigation of the boundary value problem for the hybrid system of differential equations we propose a method of finding eigenvalues for the boundary-value problem. The comparative analysis of the results of numerical computations conducted with the use of the proposed method and the results obtained by other techniques known from the literature have proved the validity and the universal character of the proposed approach.
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Modelling of the axisymmetric precision electrochemical shaping
Статья научная
The problem on modelling of a precision shaping and boundary conditions are formulated according to Faraday's law and with applying of stepwise dependence current efficiency on current density. The problem is reduced to the solution of a boundary problem for definition of two analytical functions of the complex variable. The first function is a conformal mapping of region of parametrical variable on the physical plane. In order to determine this function we use the Schwartz's integral and a spline interpolation. Unlike a plane problem for determination of potential and stream function of an axisymmetric field, the integration transformations of the second analytical function are used. The analytical function is defined in the form of a sum of two addends. The first addend takes into account the singularities of the function so that the second addend has no singularities. The second function is defined by the Schwartz's integral. Interpolation by spline functions is carried out, where the spline coefficients are derivatives of these functions by means of which the intensity vector components are calculated. We propose the method to solve the axisymmetric stationary problems, which differs from the known methods by the accuracy. By means of the method, we obtain the numerical results, describing the workpiece form. The error estimation of the obtained results is carried out. Also, we show qualitative coincidence with results of plane problem solution.
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Modelling the flow of character recognition results in video stream
Статья научная
The paper considers problems of developing stochastic models consistent with results of character image recognition in video stream. A set of assumptions that define the models structure and properties is stated. A class of distributions, namely the Dirichlet distribution and its generalizations, that set a description of the model components is pointed out; and methods for statistical estimation of the distribution parameters are given. To rank the models, the Akaike information criterion is used. The proposed theoretical distributions are verified vs sample data.
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Modelling the human papilloma virus transmission in a bisexually active host community
Статья научная
In this article, we construct a mathematical model describing the transmission dynamics of Human Papilloma Virus (HPV) in a bisexually active host community. Comprehensive mathematical techniques are used to qualitatively and quantitatively analyze the model. We analyze the local and global stabilities of the model's equilibria and show that if the basic reproduction number is less than unity, then the model is locally and globally asymptotically stable at the HPV-free static states. Also, if the basic reproduction number is less than unity, then the HPV-endemic static state is globally asymptotically stable. Numerical simulations are carried out and graphical illustrations are presented to validate the theoretical results.
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Modelling the process of non-equilibrium hydrate formation in a porous reservoir
Статья научная
This paper presents a solution to the flat-dimensional problem of gas hydrate formation in a porous medium. Highly permeable reservoirs are considered, as a result of which it is assumed that the process accompanied by the transition of gas into the hydrate composition is nonequilibrium. Based on the numerical solution, the influence of injection pressure and formation permeability on the peculiarities of phase transitions process has been studied. It is shown that with an increase in the injection pressure, both the maximum possible temperature of the system and the length of the hydrate-containing region increase. It has been found that the maximum temperature realized in the system, depending on the permeability of the reservoir, has a non-monotonic character. The influence of the initial temperature of the porous reservoir on the dynamics of phase transitions has been studied.
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Статья научная
At the application of bifurcation theory methods to nonlinear boundary value problems for ordinary differential equations of the fourth and higher order there usually arise technical difficulties, connected with determination of bifurcation manifolds, spectral investigation of the direct and conjugate linearized problems and the proof of their Fredholm property. For overcoming of this difficulty here the roots separation method is applied to the relevant characteristic equations with subsequent presentation of critical manifolds, that allows to investigate nonlinear problems in the precise statement. Such approach is applied here to two-point boundary value problem for the nonlinear ODE of the fourth order describing the buckling (divergence) of an elongated plate in a supersonic flow of gas, subjected to compressed or extended boundary stresses at the various boundary fastenings.
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Статья научная
Sobolev type equations theory has been an object of interest in recent years, with much attention being devoted to deterministic equations and systems. Still, there are also mathematical models containing random perturbation, such as white noise. A new concept of "white noise", originally constructed for finite dimensional spaces, is extended here to the case of infinite dimensional spaces. The main purpose is to develop stochastic higher-order Sobolev type equations theory and provide some practical applications. The main idea is to construct "noise" spaces using the Nelson-Gliklikh derivative. Abstract results concerning initial-final problems for higher order Sobolev type equations are applied to the Boussinesq-Love model with additive "white noise". We also use well-known methods in the investigation of Sobolev type equations, such as the phase space method, which reduces a singular equation to a regular one, as defined on some subspace of the initial space.
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Статья научная
In this paper, we present two new P-type and D-type iterative learning control (ILC) update laws for linear stochastic systems with random data dropout modeled with a Bernoulli random variable. We prove that the P-type and D-type ILC update laws converge to the desired input in the almost sure sense. We show that the convergence conditions of the inputs corresponding to the P-type and D-type ILC update laws for networked control systems are the same. We present the performance comparison of the P-type and D-type ILC update laws. In this comparison, we conclude that the P-type ILC update law is more effective than the D-type ILC update law for networked control systems.
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Статья научная
In this paper, we prove some new results on operational second order differential equations of elliptic type with general Robin boundary conditions in a non-commutative framework. The study is performed when the second member belongs to a Sobolev space. Existence, uniqueness and optimal regularity of the classical solution are proved using interpolation theory and results on the class of operators with bounded imaginary powers. We also give an example to which our theory applies. This paper improves naturally the ones studied in the commutative case by M. Cheggag, A. Favini, R. Labbas, S. Maingot and A. Medeghri: in fact, introducing some operational commutator, we generalize the representation formula of the solution given in the commutative case and prove that this representation has the desired regularity.
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Статья научная
Degenerate differential equations, as part of the differential-algebraic equations, the last few decades cause increasing interest among researchers, both because of the attractiveness of the considered theoretical questions, and by virtue of their applications. Currently, advanced methods developed in this area are used for system modelling and analysis of electrical and electronic circuits, chemical reaction simulations, optimization theory and automatic control, and many other areas. In this paper, the theory of normal forms of differential equations, originated in the works of Poincare and recently developed in the works of Arnold and his school, adapted to the simplest case of a degenerate differential equations. For this purpose we are using technique of Jordan chains, which was widely used in various problems of bifurcation theory. We study the normal forms of degenerate differential equations in the case of the existence of the maximal Jordan chain. Two and three dimensional spaces are studied in detail. Normal forms are the simplest representatives of the degenerate differential equations, which are equivalent to more complex ones. Therefore, normal forms should be considered as a model type of degenerate differential equations.
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Numerical analysis of fractional order integral dynamical models with piecewise continuous kernels
Статья научная
Volterra integral equations find their application in many areas, including mathematical physics, control theory, mechanics, electrical engineering, and in various industries. In particular, dynamic Volterra models with discontinuous kernels are effectively used in power engineering to determine the operating modes of energy storage devices, as well as to solve the problem of load balancing. This article proposes the numerical scheme for solution of the fractional order linear Volterra integral equations of the first kind with piecewise continuous kernels. The developed approach is based on a polynomial collocation method and effectively approximate such a weakly singular integrals. The efficiency of proposed numerical scheme is illustrated by two examples.
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Numerical study of the SUSUPLUME air pollution model
Статья научная
In this paper, we propose a SUSUPLUME air pollution as a modern application of the classical Gaussian plume model. The presented model takes into account meteorological conditions and parameters of the pollution sources. The classical model is supplemented by the equations of motion of the center of mass of a single emission. A numerical study has shown that in stationary weather conditions the presented model qualitatively coincides with other known models. The results of calculating the concentrations of pollutants do not contradict the obtained values based on the official methodology for calculating the maximum concentrations of pollutants approved for usege in the territory of the Russian Federation. The SUSUPLUME model contains a number of identifiable parameters and it can be adapted to real conditions. The computational model consists of two blocks: a block for recording measurement information and a block for calculating the concentrations of pollutants. The measurement information registration unit has a low labor intensity (over a million registrations per second). The pollutant concentrations calculation block is laborious (400 points of calculations per second). Concentrations are calculated independently, it allows to use parallelization of the computational process in the future.
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Observability of square membranes by Fourier series methods
Статья научная
Fourier series methods have been successfully applied in control theory for a long time. Some theorems, however, resisted this approach. Some years ago, Mehrenberger succeeded in establishing the boundary observability of vibrating rectangular membranes (and of analogous higher dimensional problems) by developing an ingenious generalization of Ingham's classical theorem on nonharmonic Fourier series. His method turn out to be useful for other applications as well. We improve Mehrenberger's approach by a shorter proof, and we improve and generalize some earlier applications.
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Статья научная
We analyze, by means of singular potentials defined in terms of Dirac functions and their derivatives, a one dimensional symmetry breaking in quantum mechanics. From a mathematical point of view we use a technique of selfadjoint extensions applied to a symmetric differential operator with a domain containing smooth functions which vanish at two inner points of the real line. As is well known, the latter leads to a two-point boundary problem. We compute the resolvent of the corresponding extension and investigate its behavior in the case in which the inner points change their positions. The domain of these extensions can contain some functions with non differentiability or discontinuity at the points mentioned before. This fact can be interpreted as a presence of singular potentials like shifted Dirac delta functions and/or their first derivative centered at the same points. Then, we study the existence of broken-symmetry bound states. For some given entanglement boundary conditions we can show the existence of a ground state, which leads to a spontaneous symmetry breaking. We also prove that within a frame of Pontryagin spaces this type of symmetry breaking is saved if the distance between the mentioned above interior points tends to zero and then we can reformulate this result in terms of a larger Hilbert space.
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On a model of spontaneous symmetry breaking in quantum mechanics
Статья научная
Our goal is to find a model for the phenomenon of spontaneous symmetry breaking arising in one dimensional quantum mechanical problems. For this purpose we consider boundary value problems related with two interior points of the real line, symmetric with respect to the origin. This approach can be treated as a presence of singular potentials containing shifted Dirac delta functions and their derivatives. From mathematical point of view we use a technique of selfadjoint extensions applied to a symmetric differential operator which has a domain containing smooth functions vanishing in two mentioned above points. We calculate the resolvent of corresponding extension and investigate its behavior if the interior points change their positions. The domain of these extensions can contain some functions that have non differentiability or discontinuity at the points mentioned above, the latter can be interpreted as an appearance of singular potentials centered at the same points. Next, broken-symmetry bound states are discovered. More precisely, for a particular entanglement of boundary conditions, there is a ground state, generating a spontaneous symmetry breaking, stable under the phenomenon of decoherence provoked from external fluctuations. We discuss the model in the context of the "chiral" broken-symmetry states of molecules like NH3. We show that within a Hilbert space approach a spontaneous symmetry breaking disappears if the distance between the mentioned above interior points tends to zero.
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On a nonlinear problem of the breaking water waves
Статья научная
The paper is devoted to the initial boundary value problem for the Korteweg-de Vries-Benjamin-Bona-Mahony equation in a finite domain. This particular problem arises from the phenomenon of long wave with small amplitude in fluid. For certain initial-boundary problems for the Korteweg-de Vries-Benjamin-Bona-Mahony equation, we obtain the conditions of blowing-up of global and travelling wave solutions in finite time. The proof of the results is based on the nonlinear capacity method. In closing, we provide the exact and numerical examples.
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Статья научная
This paper presents the results of applying the methods of the hydraulic circuit theory in the interactive modeling of hydrogasdynamic and thermal processes that occur in the equipment of thermal power plants. The problem statement of flow distribution in the energy plants with different pressure drop laws in the branches of complex gas air and steam-water ducts is formulated. The research shows that the application of traditional methods of hydraulic circuit theory is challenging for such problems. Some aspects of solvability of the related systems of nonlinear equations are studied. The numerical methods for solving these systems as applied to the problems that require calculations in real time are considered. A computation scheme is proposed. The scheme makes it possible to reduce the initial statement of the problem to the classical scheme of the nodal pressure method. The method of decomposition of the hydraulic circuit configuration into interconnected circuits of smaller dimension is considered to reduce the computational effort. The results of tests that demonstrate high reliability of the method developed for the flow distribution calculation are presented.
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