Статьи журнала - Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование
Все статьи: 729
The Barenblatt-Zheltov-Kochina model on the segment with Wentzell boundary conditions
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In terms of the theory of relative p-bounded operators, we study the Barenblatt-Zheltov-Kochina model, which describes dynamics of pressure of a filtered fluid in a fractured-porous medium with general Wentzell boundary conditions. In particular, we consider spectrum of one-dimensional Laplace operator on the segment [0, 1] with general Wentzell boundary conditions. We examine the relative spectrum in one-dimensional Barenblatt-Zheltov-Kochina equation, and construct the resolving group in the Cauchy-Wentzell problem with general Wentzell boundary conditions. In the paper, these problems are solved under the assumption that the initial space is a contraction of the space L2(0, 1).
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The Cauchy problem for the Sobolev type equation of higher order
Статья научная
Of concern is the semilinear mathematical model of ion-acoustic waves in plasma. It is studied via the solvability of the Cauchy problem for an abstract complete semilinear Sobolev type equation of higher order. The theory of relatively polynomially bounded operator pencils, the theory of differentiable Banach manifolds, and the phase space method are used. Projectors splitting spaces into direct sums and an equation into a system of two equivalent equations are constructed. One of the equations determines the phase space of the initial equation, and its solution is a function with values from the eigenspace of the operator at the highest time derivative. The solution of the second equation is the function with values from the image of the projector. Thus, the sufficient conditions were obtained for the solvability of the problem under study. As an application, we consider the fourth-order equation with a singular operator at the highest time derivative, which is in the base of mathematical model of ion-acoustic waves in plasma. Reducing the model problem to an abstract one, we obtain sufficient conditions for the existence of a unique solution.
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The Lyapunov stability of the Cauchy-Dirichlet problem for the generalized Hoff equation
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We consider the initial boundary value problem with homogeneous Dirichlet boundary conditions for the generalized Hoff equation in a bounded domain. This equation models the dynamics of buckling of a double-tee girder under constant load and belongs to a large class of Sobolev type semilinear equations (We can isolate the linear and non-linear parts of the operator acting on the original function). The paper addresses the stability of zero solution of this problem. There are two methods in the theory of stability: the first one is the study of stability by linear approximation and the second one is the study of stability by Lyapunov function. We use the second Lyapunov''s method adapted to the case of incomplete normed spaces. The main result of this paper is a theorem on the stability and asymptotic stability of zero solution to this problem.
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The Pyt'ev-Chulichkov method for constructing a measurement in the Shestakov-Sviridyuk model
Статья научная
One of the approaches to solution of the problem on restoring a distorted input signal by the recorded output data of the sensor is the problem on optimal dynamic measurement, i.e. the Shestakov-Sviridyuk model. This model is the basis of the theory of optimal dynamic measurements and consists of the problem on minimizing the difference between the values of a virtual observation obtained using a computational model and experimental data, which are usually distorted by some noise. We consider the Shestakov-Sviridyuk model of optimal dynamic measurement in the presence of various types of noises. In the article, the main attention is paid to the preliminary stage of the study of the problem on optimal dynamic measurement. Namely, we consider the Pyt'ev-Chulichkov method of constructing observation data, i.e. transformation of the experimental data to make them free from noise in the form of ``white noise" understood as the Nelson-Gliklikh derivative of the multidimensional Wiener process. In order to use this method, a priori information about the properties of the functions describing the observation is used.
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The blow-up solutions to nonlinear fractional differential Caputo-system
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In this paper, we establish the finite time blow-up of solutions to nonlinear differential systems governed by Caputo fractional differential equation. Then, we derive sufficient conditions on parameters with positive given data. Moreover, for this purpose under some assumptions, we prove the non existence of global solutions to the considered class of nonlinear fractional differential Caputo-system subject to the initial condition. To prove our main result, we apply the test function method, Riemann-Liouville integral, Caputo derivative operator and some general analysis tools. Our result is new and generalizes the existing one.
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The contribution of Angelo Favini in twenty years of joint research (1996 - 2016)
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We study conditions for the loss of stability in a plastic deformation of a layer of weaker material in a sheet specimen. The layer is not collinear with the exterior forces acting in the sheet plane, which are orthogonal to each other and have opposite signs. The parameters of the problem are: the angle between the layer and the direction of exterior forces; the ratio of stresses due to exterior forces; the ratio of strengths of the layer material and the main material of the sheet specimen; the strengthening law of the layer material; the ratio of thicknesses of the layer and the specimen. Basing on Swift's plastic instability criterion for a deformation of the layer material, we obtain an algorithm for calculating critical stress in the layer and critical exterior loading in dependence on the indicated parameters. When contact strengthening of the layer is absent, our results have explicit analytic expressions. We find conditions under which the layer does not lower the strength of the specimen. We find conditions for the stressed state of the layer to be a pure shear and study this case.
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Статья научная
Krasnoyarsk Subsidiary of Kutateladze Institute of Thermophysics SB RAS and the Department of Thermophysics of the Siberian Federal University are developing a freely distributable "SigmaFW" software package for numerical simulation of the hydrodynamics, heat and mass transfer problems. It is assumed that the software package will be used in scientific and educational institutions as well as industrial enterprises in Russian Federation. Mathematical models realized in the software package describe steady and unsteady laminar and turbulent single - and multicomponent flow taking into account the dispersed phase, the conjugate and radiative heat transfer, and homogeneous gas-phase chemical reactions. The "SigmaFW" contains the necessary tools for building numerical domains, carrying out multi-threaded calculation, and visual analysis of the results. The article describes the three main blocks of software package: the grid generator, calculation module and analysis of the results. In additition, a number of test and application tasks are presented to demonstrate the capabilities of the software.
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The elements of the operator convexity in the construction of the programmed iteration method
Статья научная
The problem of retention studied here can be regarded (in the case of bounded control interval) as a variant of the approach problem within the given constraints in the phase space and the target set given by the hyperplane of the space positions corresponding to the terminal moment of the process (the retention problem on the infinite horizon also fits the problem stated in the work). The main difference of the problem from the previously considered formulation is the possibility of variation of the spaces of system trajectories and disturbance realizations depending on the initial moment of control. It is shown that the unsolvability set of the retention problem is the operator convex hull of the empty set constructed on the base of programmed absorption operator. Under some additional coherence conditions (on the spaces of system trajectories and disturbance realizations corresponding to different initial moments) the set of successful solvability is constructed as the limit of the iterative procedure in the space of sets, elements of which are positions of the game; in this case the structure of resolving quasistrategy is also given.
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The existence of a unique solution to a mixed control problem for Sobolev-type equations
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This article studies a mixed control problem for Sobolev-type equations in the case of a relatively radial operator. We use the Showalter-Sidorov initial condition. The difference in the statement of our problem from those studied previously by other researchers amounts to the form of the quality functional, which, in the authors' opinion, is more adequate to model applications in economics and technology. We prove an existence and uniqueness theorem for the solution to this problem.
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Статья научная
This paper describes a method of use of equations in M.F. Shul'gin's form in Lagrangian variables for steady motion stability and stabilization problems of systems with geometric constraints. These equations of motion are free from Lagrange multipliers; we substantiate their advantage for solving stability and stabilization problems. Depended coordinates corresponding to zero solutions of characteristic equation are allocated in the disturbed equations of motion. These variables are necessarily present in systems with geometric constraints for any control method. It is suggested to present equations of motion in Routh variables for finding stabilizing control coefficients; Lagrangian variables are more useful for constructing an estimation system of object state. In addition to previous results, we evaluate the ability to reduce the dimension of measured output signal obtained in conformity with the chosen modelling method. Suppose the state of system is under observations and the dimension of measurement vector is as little as possible. Stabilizing linear control law is fulfilled as feedback by the estimation of state. We can determine uniquely the coefficients of linear control law and estimation system can be determined uniquely by solving of the corresponding linear-quadratic problems for the separated controllable subsystems using the method of N.N. Krasovsky. The valid conclusion about asymptotical stability of the original equations is deduced using the previously proved theorem. This theorem is based on the nonlinear stability theory methods and analysis of limitations imposed by the geometric constraints on the initial disturbances.
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The mathematical modelling of the production of construction mixtures with prescribed properties
Статья научная
We propose a method for the mathematical modelling of the preparation of construction mixes with prescribed properties. The method rests on the optimal control theory for Leontieff-type systems. Leontieff-type equations originally arose as generalizations of the well-known input-output model of economics taking supplies into account. Then they were used with success in dynamical measurements, therefore giving rise to the theory of optimal measurements. In the introduction we describe the ideology of the proposed model. As an illustration, we use an example of preparing of simple concrete mixes. In the first section we model the production process of similar construction mixtures (for instance, concrete mixtures) depending on investments. As a result, we determine the price of a unit of the product. In the second section we lay the foundation for the forthcoming construction of numerical algorithms and software, as well as conduction of simulations. Apart from that, we explain the prescribed properties of construction mixes being optimal with respect to expenses.
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The multipoint initial-final value condition for the Navier - Stokes linear model
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The Navier - Stokes system models the dynamics of a viscous incompressible fluid. The problem of existence of solutions of the Cauchy - Dirichlet problem for this system is included in the list of the most serious problems of this century. In this paper it is proposed to consider the multipoint initial-final conditions instead of the Cauchy conditions. It should be noted that nowadays the study of solvabilityof initial-final value problems is a new and actively developing direction of the Sobolev type equations theory. The main result of the paper is the proof of unique solvability of the stated problem for the system of Navier - Stokes equations.
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The optimal control problem for output material flow on conveyor belt with input accumulating bunker
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The article is devoted to the synthesis of optimal control of conveyor belt with the accumulating input bunker. Much attention is given to the model of the conveyor belt with a constant speed of the belt. Simulation of the conveyor belt is carried out in the one-moment approximation using partial differential equations. The conveyor belt is represented as a distributed system. The used PDE-model of the conveyor belt allows to determine the state of the flow parameters for a given technological position as a function of time. We consider the optimal control problem for flow parameters of the conveyor belt. The problem consists in ensuring the minimum deviation of the output material flow from a given target amount. The control is carried out by the material flow amount, which comes from the accumulating bunker into the conveyor belt input. In the synthesis of optimal control, we take into account the limitations on the size of the accumulating bunker, as well as on both max and min amounts of control. We construct optimal control of the material flow amount coming from the accumulating bunker. Also, we determine the conditions to switch control modes, and estimate time period between the moments of the switching.
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The optimal design of pressure swing adsorption process of air oxygen enrichment under uncertainty
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The paper formulates and studies the problem of optimal (by the criterion of profits from oxygen production) design of a pressure swing adsorption (PSA) unit for air oxygen enrichment under partial uncertainty of the source data (the air composition, temperature, atmospheric pressure) with limitations on oxygen purity, unit capacity, and resource saving granular adsorbent. A heuristic iterative algorithm was developed for solving an optimal design problem under partial uncertainty of the source data. An auxiliary optimization problem related to the class of nonlinear programming problems (assuming the approximation of continuous control functions at the stages of the adsorption-desorption cycle by step-functions) was formulated and then solved by the sequential quadratic programming method. The problem of optimal design was solved for the range of PSA units with a capacity of 1 to 4 l/min allowing to obtain oxygen with a purity of 40 to 90% vol. According to the findings, we analyze the most promising operational and design parameters ensuring the maximum profit in the operation of the PSA unit, taking into account the saving of the granular adsorbent. It was established that the introduction of limitations on the gas flow rate in the frontal layer of the PSA unit adsorbent allows to increase the reliability of its operation and the adsorbent service life.
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We consider the problem of identifying the trajectory of a mobile point source described by the Delta function in a one-dimensional linear convective transport equation under a given additional boundary condition. To solve this problem, the Delta function is approximated by a continuous function and a discrete analog of the problem is constructed using finite-difference approximations in the form of an implicit difference scheme. To solve the resulting difference problem, we propose a special representation that allows to split the problem into two mutually independent linear first-order difference problems at each discrete value of a time variable. The result is an explicit formula for determining the position of a mobile point source for each discrete value of a time variable. Based on the proposed computational algorithm, numerical experiments were performed for model problems.
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The rate of convergence of hypersingular equations numerical computation
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Numerical methods for solving hypersingular equations based on Chebyshev polynomials of the second kind with a weight taking into account the Meixner physical conditions on the edge are developed. We obtained estimates of the rate of convergence using the analytical form of the matrix of an integral operator with a logarithmic singularity. Authors considered a delta function model, and its inapplicability in diffraction problems and vibrator antennas are shown. Previously, a numerical-analytical method for solving the excitation problems of vibrator antennas was proposed, but in the present work, the rationale for the numerical-analytical method is given for the first time. Unlike the reduction method, the numerical-analytical method demonstrates reliable convergence, not only in diffraction problems but also in antenna excitation problems. The specific feature of the excitation problems is that the right-hand side of the hypersingular equation is localized in a small region, in comparison with the characteristic dimensions of the antenna. Mathematically, this means that the right-hand side of the hypersingular equation decomposes into a slowly-convergent series. A similar property is also possessed by the solution of the equation. That is why the method of reduction is not effective enough. An example of a numerical solution is considered. Estimates of the rate of convergence are obtained. The applicability of developed methods for investigating a wide range of diffraction problems is shown.
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Статья научная
This article focuses on the application of wavelet theory to the problem of modelling the processes of manufacturing the shells of fibrous composite materials (CM). The basic methods for preparing such shells are two related ones: filament winding, when the strip made of CM is laid out on the outstretched surface, and laying out, when the tape is placed by dint of pressing rollers. In both cases, laying the tape is carried out in accordance with the program of moving spreader. To create such a program the mathematical model of the process of placing the tape is needed. The article describes semi orthogonal wavelet systems on the segment that are based on B-spline of arbitrary order. The matrices which compose the filter bank for such wavelet systems are represented. Some algorithms for geometric modelling are reviewed and summarized from the point of view of the wavelet theory. The results are applied to the mathematical modelling and software of manufacturing process of shells made of fibrous composite materials. As an example, consider the process of making the ventilator blade.
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Статья научная
In this paper, the problem of training the Viola-Jones detector for 3D objects is considered on the example of an inflatable life raft PSN-10. The detector is trained on a fully synthetic training dataset. The paper discusses in detail the methods of modelling an inflatable life raft, water surface, various weather conditions. As a feature space, we use edge Haar-like features, which allow training the detector that is resistant to various lighting conditions. To increase the computational efficiency, the L1 norm is used to calculate the magnitude of the image gradient. The performance of the trained detector is estimated on real data obtained during the rescue operation of the trawler "Dalniy Vostok". The proposed method for training the Viola-Jones detectors can be successfully used as a component of hardware and software "assistants" of the UAV.
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