Математическое моделирование. Рубрика в журнале - Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование
Статья научная
We investigate the Cauchy-Dirichlet problem for a system of Oskolkov equations of nonzero order. The considered mathematical model describes the flow of an incompressible viscoelastic Kelvin-Voigt fluid in the magnetic field of the Earth. The model takes into account that the fluid is subject to various external influences, which depend on both the coordinate of the point in space and the time. The first part of the paper presents the known results obtained by the authors earlier and based on the theory of solvability of the Cauchy problem for semilinear nonautonomous Sobolev type equations. In the second part, we reduce the considered mathematical model to an abstract Cauchy problem. In the third part, we prove the main result that is the theorem on the existence and uniqueness of the solution. Also, we establish the conditions for the existence of quasi-stationary semitrajectories, and describe the extended phase space of the model under study. In this paper, we summarize our results for the Oskolkov system that simulates the motion of a viscoelastic incompressible Kelvin-Voigt fluid of zero order in the magnetic field of the Earth.
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A numerical method for inverse spectral problems
Статья научная
Basing on the Galerkin methods, we develop a new numerical method for solving the inverse spectral problems generated by discrete lower semibounded operators. The restrictions on the perturbing operator are relaxed in comparison with the method based on the theory of regular traces. A Fredholm integral equation of the first kind enables us to recover the values of the perturbing operator at the discretization nodes. We tested the method on spectral problems for the Sturm - Liouville operator, and the results of numerous simulations demonstrate its computational efficiency. We found simple formulas for the eigenvalues of a discrete lower semibounded operator avoiding the roots of the corresponding secular equations. The calculation of eigenvalues of these operators can start at an arbitrary index independently of the (un)availability of the eigenvalues with smaller indices. For perturbed selfadjoint operators we can calculate eigenvalues with large indices when the Galerkin method becomes difficult to apply.
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A parametric stochastic model of bone geometry
Статья научная
The aim of the present study is to develop a parametric bone modelling algorithm which takes into account bone microarchitecture. This approach allows to generate hematopoietic bone segment phantoms based on literature-derived micro - and macro dimensions. We propose a method for subdividing bones into small segments which can be described by simple geometric shapes filled with a stochastically generated rod-like model of the trabecular structure with appropriate voxel resolution. This approach avoids the disadvantages of non-parametric individual modelling based on computer tomography scans. The parametric method allows the simulation of individual variability in bone-specific dimensions. The model presented in this paper will be used to describe the geometry of hematopoietic sites, which in turn will serve as a basis for calculating the doses of irradiation of the hematopoietic cells of the bone marrow from the incorporated beta-emitters.
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A simulation of the thermal state of heavily loaded tribo-units and its evaluation
Статья научная
The thermal state of the elements of heavily loaded tribo-units is among the most important parameters affecting their performance. The temperature of the lubricating layer of bearings determines to a large extent their load-bearing capacity. The heat balance equation used to estimate the temperature of friction pairs fails to yield the temperature fields and the regions of their maximum values. This fact makes our problem important. We examine a mathematical model and a method for calculating the thermal state and thermohydrodynamic characteristics of heavily loaded sliding bearings, taking into account the non-Newtonian properties of the fluid as well as the heat exchange processes between the lubricating layer and the surrounding moving surfaces of tribo-units. To solve the energy equation, we propose to use finite difference approximation methods. To create the difference analogs of the energy equations for some structural elements and thin lubricant layers, we use the Pismen-Reckford scheme of implicit alternating directions. We present the calculated hydromechanical characteristics of the connecting rod bearing of a heat engine. We obtain three-dimensional distributions of temperature in the lubricant. The results show that, if we allow for convective heat transfer in the radial direction, the processes of heat exchange between the lubricating layer and the surrounding moving surfaces enable us to determine more accurately the mean lubricant temperature and the thermal stress of a tribo-unit as a whole. Our method can be used to assess the performance and efficiency of heavily loaded tribo-units of piston and rotary machines.
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About one approach to numerical solution of nonlinear optimal speed problems
Статья научная
Optimal speed problems are among the most important problems of the theory of controlled systems. In the qualitative theory of nonlinear speed problems one of the main results is the Pontryagin maximum principle. For the numerical solution of nonlinear speed problems, along with methods based on the maximum principle, methods of reducing to auxiliary problems of optimal control using linearization, parameterization, discretization, and other techniques are widely used. The complexity of numerical methods is determined by the number of iterations to find a solution to the speed problem with a given accuracy. A universal computational procedure that is effective for calculating a variety of speed problems does not currently exist. Therefore, it is actual to develop special approaches to reduce the amount of calculations and reduce the number of iterations. The paper proposes a new approach based on the reduction of a nonlinear speed problem to an auxiliary optimization problem with mixed control functions and parameters. To search for a solution to the emerging auxiliary problem, a specially developed form of conditions for nonlocal improvement of admissible control in the form of a fixed-point problem of the control operator, and a constructed iterative algorithm for successive improvement of admissible controls are used. Approbation and comparative analysis of the computational efficiency of the proposed fixed point approach is carried out on known models of optimal speed problems.
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Acoustic waves propagation in heated water with vapor bubbles
Статья научная
The problems of wave propagation in bubbly media are of great interest for researchers for nearly half a century due to the wide distribution of these systems in nature and their intense use in modern technology. It is known that the attenuation intensity of sound disturbances in the gas-liquid media is mainly determined by the thermophysical characteristics of the gas in bubbles. It turns out that these effects are significantly observable with increasing of steam concentration due to system temperature get higher. In this paper, we consider the propagation of small perturbations in a liquid with bubbles filled with vapor and gas insoluble in the liquid phase in an one-dimensional and one-velocity approximation. In order to take into account interfacial heat and mass transfer, we use the heat and diffusion equations inside the bubble and the heat equation in the fluid around the bubble. A dispersion equation was written from the existence condition of the solution in the form of a damped traveling wave, taking into account the effects of acoustic unloading of bubbles, and numerical calculations were carried out for water with vapor-gas bubbles. We investigate the features of the reflection of harmonic waves from the interface of "pure'' liquid and liquid with vapor-gas bubbles. Also, we carry out a numerical analysis of the effect of the initial volume gas content ag0 with two initial bubble sizes a0=10-6 m and 10-3 m. Finally, we study the effect of disturbance frequencies and temperature of the media on the attenuation coefficient of sound.
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Active parametric identification of Gaussian linear discrete system based on experiment design
Статья научная
The application of methods of theory of experiment design for the identification of dynamic systems allows the researcher to gain more qualitative mathematical model compared with the traditional methods of passive identification. In this paper, the authors summarize results and offer the algorithms of active identification of the Gaussian linear discrete systems based on the design inputs and initial states. We consider Gaussian linear discrete systems described by state space models, under the assumption that unknown parameters are included in the matrices of the state, control, disturbance, measurement, covariance matrices of system noise and measurement. The original software for active identification of Gaussian linear discrete systems based on the design inputs and initial states are developed. Parameter estimation is carried out using the maximum likelihood method involving the direct and dual procedures for synthesizing A- and D- optimal experiment design. The example of the model structure for the control system of submarine shows the effectiveness and appropriateness of procedures for active parametric identification.
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Статья научная
In this paper we consider a problem of two bodies bonded through a thin adhesive layer (a third material) of thickness δ. Leting δ go to zero, one obtains a boundary value transmission problem set on a fixed domain. We then give new results for the study of this problem in the framework of Hölder spaces: an explicit representation of the solution and necessary and sufficient conditions at the interface for its optimal regularity are obtained using the semigroups theory and the real interpolation spaces.
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An inference algorithm for monotone Boolean functions associated with undirected graphs
Статья научная
Boolean functions are a modelling tool useful in many applications; monotone Boolean functions make up an important class of these functions. For instance, monotone Boolean functions can be used for describing the structure of the feasible subsystems of an infeasible system of constraints, because feasibility is a monotone feature. In this paper we consider monotone Boolean functions (MBFs), associated with undirected graphs, whose upper zeros are defined as binary tuples for which the corresponding subgraph of the original undirected graphs is either the empty graph, or it has no edges. For this class of MBFs, we present the settings of problems which are related to the search for upper zeros and maximal upper zeros of these functions. The notion of k-vertices and (k,m)-vertices in a graph is introduced. It is shown that for any k-vertices of the original graph there exists a maximal upper zero of an MBF associated with the graph, in which the component xi corresponding to this k-vertex takes the value . Based on this statement, we construct an algorithm of searching for a maximal upper zero, for the class of MBFs under consideration, which allows one to find, under certain conditions, the solution to the problem of searching for a maximal upper zero, or to substantially reduce the dimension of the original problem. The proposed algorithm was extended for the case of (k,m)-vertices. This extended algorithm allows one to fix a bound on the deviation of an upper zero of the MBF from the maximal upper zeros, in the sense of the number of units in these tuples. The algorithm has the complexity O(n2p), where n is a number of vertices and p is a number of edges of the original graph.
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An integral method for the numerical solution of nonlinear singular boundary value problems
Статья научная
We discuss the numerical treatment of a nonlinear singular second order boundary value problem in ordinary differential equations, posed on an unbounded domain, which represents the density profile equation for the description of the formation of microscopic bubbles in a non-homogeneous fluid. Due to the fact that the nonlinear differential equation has a singularity at the origin and the boundary value problem is posed on an unbounded domain, the proposed approaches are complex and require a considerable computational effort. This is the motivation for our present study, where we describe an alternative approach, based on the reduction of the original problem to an integro-differential equation. In this way, we obtain a Volterra integro-differential equation with a singular kernel. The numerical integration of such equations is not straightforward, due to the singularity. However, in this paper we show that this equation may be accurately solved by simple product integration methods, such as the implicit Euler method and a second order method, based on the trapezoidal rule. We illustrate the proposed methods with some numerical examples.
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Analysis of socio-economic system processes performance with the help of eigenstate models
Статья научная
Analysis of the performance of business processes for socio-economic systems is considered. The analysis of the business processes efficiency is based on constructing on of a "perfect image" of their behavior. To extract the components that correspond to the "perfect image" of behavior, we propose the usage of eigenstate method, which allows to represent behavior of socio-economic system as a sum of independent components (eigenstates). The basic relationships of eigenstate method are considered. Construction of the socio-economic system models using the eigenstate method consists in calculation and selection of key eigenstates oriented towards formulated success factors. The selected eigenstates are used to form a "reference" model of the socio-economic system. The high performance of socio-economic system is evaluated with the help of comparison of model and actual values of socio-economic system variables. The eigenstate method efficiency is demonstrated by the example of analysis of sustainable development of the Chelyabinsk city. The sustainable development indicators of processes of the Chelyabinsk city are obtained.
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Статья научная
In the three-dimensional statement, we consider the Brinkman equation together with the equation of heterogeneous heat transfer for an unidirectional flow of the Newtonian fluid under laminar regime through horizontal porous channel having a constant rectangular cross-section with known thermal flows at the boundary and small values of the Darcy numbers. Due to the linearity of the formulated system of model equations, we obtain analytical solution of the system using the Laplace and Fourier integral transformation. The obtained solution allows to estimate the length of the input hydrodynamic section, the coefficient of hydraulic resistance, and the local Nusselt numbers. The results obtained for the hydrodynamic subproblem with a large porosity and thermal subproblem with a stationary temperature field agree with the classical data.
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Статья научная
Of concern is an initial-boundary value problem for the modified Boussinesq equation (IMBq equation) is considered. The equation is often used to describe the propagation of waves in shallow water under the condition of mass conservation in the layer and taking into account capillary effects. In addition, it is used in the study of shock waves. The modified Boussinesq equation belongs to the Sobolev type equations. Earlier, using the theory of relatively p-bounded operators, the theorem of existence and uniqueness of the solution to the initial-boundary value problem was proved. In this paper, we will prove that the solution constructed by the Galerkin method using the system orthornormal eigenfunctions of the homogeneous Dirichlet problem for the Laplace operator converges *-weakly to an precise solution. Based on the compactness method and Gronwall's inequality, the existence and uniqueness of solutions to the Cauchy-Dirichlet and the Showalter-Sidorov-Dirichlet problems for the modified Boussinesq equation are proved.
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Статья научная
This paper is devoted to the application of the smooth approximation of the probability function in the solution of three different stochastic optimization problems: minimization of an airstrip area under the constrained probability of successful landing, minimization of the cost of water supply system with random performance and with predefined water consumption, and determination of the set of wind speed vectors which guarantees the safe landing of an aircraft in future with the given probability. The first two problems are mathematical programming problems with probability constraint, and the third one is a problem of constructing the isoquant surface of the probability function. Smooth approximation of the probability function allows to use the gradient projection method in the constrained optimization problem and to define the isoquant surface as the solution to a partial differential equation. We provide an example for each of the considered problems and compare the results with known results previously obtained using the confidence method.
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Asymptotic estimate of a Petrov - Galerkin method for nonlinear operator-differential equation
Статья научная
In the current paper, we study a Petrov - Galerkin method for a Cauchy problem for an operator-differential equation with a monotone operator in a separable Hilbert space. The existence and the uniqueness of a strong solution of the Cauchy problem are proved. New asymptotic estimates for the convergence rate of approximate solutions are obtained in uniform topology. The minimal requirements to the operators of the equation were demanded, which guaranteed the convergence of the approximate solutions. There were no assumptions of the structure of the operators. Therefore, the method, specified in this paper, can be applied to a wide class of the parabolic equations as well as to the integral-differential equations. The initial boundary value problem for nonlinear parabolic equations of the fourth order on space variables was considered as the application.
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Статья научная
Camera-traps is a relatively new but already popular instrument in the estimation of abundance of non-identifiable animals. Although camera-traps are convenient in application, there remain both theoretical complications (such as spatial autocorrelation or false negative problem) and practical difficulties, for example, laborious random sampling. In the article I propose an alternative way to bypass the mentioned problems. In the proposed approach, the raw video information collected from the camera-traps situated at the spots of natural attraction is turned into the frequency of visits, and the latter is transformed into the desired abundance estimate. The key for such a transformation is the application of the correction coefficients, computed for each particular observation environment using the Bayesian approach and the massive database (DB) of observations under various conditions. The proposed method is based on automated video-capturing at a moderate number of easy to reach spots, so in the long term many laborious census works may be conducted easier, cheaper and cause less disturbance for the wild life. Information post-processing is strictly formalized, which leaves little chance for subjective alterations. However, the method heavily relies on the volume and quality of the DB, which in its turn heavily relies on the efforts of the community. Although the construction of such DB could be rather difficult and controversial, it is much easier than the solution of the initial abundance estimation problem. Moreover, such a rich DB of visits might benefit not only censuses, but also many behavioral studies.
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Boundary inverse problem for star-shaped graph with different densities strings-edges
Статья научная
The paper is devoted to the mathematical modelling of star-shaped geometric graphs with n rib-strings of different density and the solution of the boundary inverse spectral problem for Sturm-Liouville differential operators on these graphs. Earlier it was shown that if strings have the same length and densities, then the stiffness coefficients of springs at the ends of graph strings are not uniquely recovered from natural frequencies. They are found up to permutations of their places. We show, that if the strings have different densities, then the stiffness coefficients of springs on the ends of graph strings are uniquely recovered from all natural frequencies. Counterexamples are shown that for the unique recovery of the stiffness coefficients of springs on n dead ends of the graph, it is not enough to use n natural frequencies. Examples are also given showing that it is sufficient to use n+1 natural frequencies for the uniqueness of the recovery of the stiffness coefficients of springs at the n ends of the strings. Those, the uniqueness or non-uniqueness of the restoration of the stiffness coefficients of springs at the ends of the strings depends on whether the string densities are identical or different.
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Causal relations in support of implicit evolution equations
Статья научная
This is a brief exposition of dynamic systems approaches that form the basis for linear implicit evolution equations with some indication of interesting applications. Examples in infinite-dimensional dissipative systems and stochastic processes illustrate the fundamental notions underlying the use of double families of evolution equations intertwined by the empathy relation. Kisynski's equivalent formulation of the Hille-Yosida theorem highlights the essential differences between semigroup theory and the theory of empathy. The notion of K-bounded semigroups, a more direct approach to implicit equations, and related to empathy in a different way, is included in the survey.
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Commutative encryption method based on hidden logarithm problem
Статья научная
A candidate for post-quantum commutative encryption algorithm is proposed, which is based on the hidden discrete logarithm problem defined in a new 6-dimensional finite non-commutative associative algebra. The properties of the algebra are investigated in detail and used in the design of the proposed commutative cipher. The formulas describing the set of p2 different global right-sided units contained in the algebra and local left-sided units are derived. Homomorphisms of two different types are considered and used in the commutative cipher. The encrypted message is represented in the form of a locally invertible element T of the algebra and encryption procedure includes performing the exponentiation operation and homomorphism map followed by the left-sided multiplication by a randomly selected local right-sided unit. The introduced commutative cipher is secure to the known-plaintext attacks and has been used to develop the post-quantum no-key encryption protocol providing possibility to send securely a secret message via a public channel without using any pre-agreed key. The proposed commutative encryption algorithm is characterized in using the single-use keys that are selected at random directly during the encryption process.
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Статья научная
The quasi-stationary problem for modelling the process of electrochemical cutting with a plate electrode-tool is formulated. The formulation of the problem is based on the use of a stepwise function of current efficiency from the current density. Thus three areas with various conditions are formed on the machined surface. The usual stationarity condition is used in the area of high current densities. In the area of low current densities the dissolution is absent and the initial form of the boundaries remains. In the intermediate zone, the current density at each point is equal to the critical value. The presence of boundary conditions on each section of the machined surface allows to formulate a boundary problem for the analytical function of the complex variable and to find the shape of the boundary at any moment, regardless of the background. The solutions of quasi-stationary and non-stationary problems are compared, and the range of existence of quasi-stationary solutions is found.
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