Статьи журнала - Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование
Все статьи: 739
On a class of Sobolev-type equations
Статья научная
The article surveys the works of T.G. Sukacheva and her students studying the models of incompressible viscoelastic Kelvin-Voigt fluids in the framework of the theory of semilinear Sobolev-type equations. We focus on the unstable case because of greater generality. The idea is illustrated by an example: the non-stationary thermoconvection problem for the order 0 Oskolkov model. Firstly, we study the abstract Cauchy problem for a semilinear nonautonomous Sobolev-type equation. Then, we treat the corresponding initial-boundary value problem as its concrete realization. We prove the existence and uniqueness of a solution to the stated problem. The solution itself is a quasi-stationary semi-trajectory. We describe the extended phase space of the problem. Other problems of hydrodynamics can also be investigated in this way: for instance, the linearized Oskolkov model, Taylor''s problem, as well as some models describing the motion of an incompressible viscoelastic Kelvin-Voigt fluid in the magnetic field of the Earth.
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On a heat and mass transfer model for the locally inhomogeneous initial data
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We consider a model case of the problem of heat diffusion in a homogeneous body with a special initial state. The peculiarity of this initial state is its local inhomogeneity. That is, there is a closed domain Ω inside a body, the initial state is constant out of the domain. Mathematical modelling leads to the problem for a homogeneous multi-dimensional diffusion equation. We construct the boundary conditions on the boundary of the domain Ω, which can be characterized as "transparent" boundary conditions. We separately consider a special case - a model of redistribution of heat in a uniform linear rod, the side surface of which is insulated in the absence of (internal and external) sources of heat and of locally inhomogeneous initial state.
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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 oscillations of a thin flat plate with a variety of mounts on opposite sides
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We consider a model case of stationary vibrations of a thin flat plate, one side of which is embedded, the opposite side is free, and the sides are freely leaned. In mathematical modeling there is a local boundary value problem for the biharmonic equation in a rectangular domain. Boundary conditions are given on all boundary of the domain. We show that the considered problem is self-adjoint. Herewith the problem is ill-posed. We show that the stability of solution to the problem is disturbed. Necessary and sufficient conditions of existence of the problem solution are found. Spaces of the ill-posedness of the considered problem are constructed.
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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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On evolutionary inverse problems for mathematical models of heat and mass transfer
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This article is a survey. The results on well-posedness of inverse problems for mathematical models of heat and mass transfer are presented. The unknowns are the coefficients of a system or the right-hand side (the source function). The overdetermination conditions are values of a solution of some manifolds or integrals of a solution with weight over the spatial domain. Two classes of mathematical models are considered. The former includes the Navier-Stokes system, the parabolic equations for the temperature of a fluid, and the parabolic system for concentrations of admixtures. The right-hand side of the system for concentrations is unknown and characterizes the volumetric density of sources of admixtures in a fluid. The unknown functions depend on time and some part of spacial variables and occur in the right-hand side of the parabolic system for concentrations. The latter class is just a parabolic system of equations, where the unknowns occur in the right-hand side and the system as coefficients. The well-posedness questions for these problems are examined, in particular, existence and uniqueness theorems as well as stability estimates for solutions are exposed.
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On existence of solutions to stochastic differential equations with current velocities
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The notion of mean derivatives was introduced by E. Nelson in 60-th years of XX century and at the moment there are a lot of mathematical models of physical processes constructed in terms of those derivatives. The paper is devoted to investigation of stochastic differential equations with current velocities, i.e. with Nelson's symmetric mean derivatives. Since the current velocities of stochastic processes are natural analogues of ordinary physical velocities of deterministic processes, such a research is important for investigation of models of physical processes that take into account stochastic properties. An existence of solution theorem for those equations is obtained.
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On factorization of a differential operator arising in fluid dynamics
Статья научная
Spectral properties of linear operators are very important in stability analysis of dynamical systems. The paper studies the non-selfadjoint second order differential operator that originated from a steady state stability problem in dynamic of viscous Newtonian fluid on the inner surface of horizontally rotating cylinder in the presence of gravitational field. The linearization of the thin liquid film flow in the lubrication limit about the uniform coating steady state results into the operator which domain couples two subspaces spanned by positive and negative Fourier exponents which are not invariant subspaces of the operator. We prove that the operator admits factorization and use this new representation of the operator to prove compactness of its resolvent and to find its domain.
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On fixed point theory and its applications to equilibrium models
Статья научная
For a given set and a given (generally speaking, multivalued) mapping of this set into itself, we study the problem on the existence of fixed points of this mapping, i.e., of points contained in their images. We assume that the given set is nonempty and the given mapping is defined on the entire set. In these conditions, we give the description (redefinition) of the set of fixed points in the set-theoretic terms. This general idea is concretized for cases where the set is endowed with a topological structure and the mapping has additional properties associated with this structure. In particular, we provide necessary and sufficient conditions for the existence of fixed points of mappings with closed graph in Hausdorff topological spaces as well as in metric spaces. An example illustrating the possibilities and advantages of the proposed approach is given. The immediate applications of these results to the search of equilibrium states in game problems are also given: we describe the sets of saddle points in the minimax problem (an analogue of the Fan theorem) and of Nash equilibrium points in the game with many participants in cases where the sets of strategies of players are Hausdorff spaces or metrizable topological spaces.
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On global in time solutions for differential-algebraic equations
Статья научная
We investigate differential-algebraic equations arising in mathematical models that describe some radio-technical devises. A class of differential-algebraic equations is described, for which necessary and sufficient conditions for global in time existence of solutions are proved. As well as in many papers where sufficient conditions for such equations are obtained, we reduce them to ordinary differential equations and then apply the necessary and sufficient conditions for the latter. We deal with the systems whose matrix pencil is regular and (for simplicity) the characteristic polynomial satisfies the rank-degree condition. We also require some additional conditions that allow us to reduce the differential-algebraic system to ordinary differential one.
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Статья научная
This article is a survey devoted to inverse problems of recovering sources and coefficients (parameters of a medium) in mathematical models of heat and mass transfer. The main attention is paid to well-posedness questions of the inverse problems with pointwise overdetermination conditions. The questions of this type arise in the heat and mass transfer theory, in environmental and ecology problems, when describing diffusion and filtration processes, etc. As examples, we note the problems of determining the heat conductivity tensor or sources of pollution in a water basin or atmosphere. We describe three types of problems. The first of them is the problem of recovering point or distributed sources. We present conditions for existence and uniqueness of solutions to the problem, show non-uniqueness examples, and, in model situations, give estimates on the number of measurements that allow completely identify intensities of sources and their locations. The second problem is the problem of recovering the parameters of media, in particular, the heat conductivity. The third problem is the problem of recovering the boundary regimes, i. e. the flux through a surface or the heat transfer coefficient.
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On nonparametric modelling of multidimensional noninertial systems with delay
Статья научная
We consider the problem of noninertial objects identification under nonparametric uncertainty when a priori information about the parametric structure of the object is not available. In many applications there is a situation, when measurements of various output variables are made through significant period of time and it can substantially exceed the time constant of the object. In this context, we must consider the object as the noninertial with delay. In fact, there are two basic approaches to solve problems of identification: one of them is identification in «narrow» sense or parametric identification. However, it is natural to apply the local approximation methods when we do not have enough a priori information to select the parameter structure. These methods deal with qualitative properties of the object. If the source data of the object is sufficiently representative, the nonparametric identification gives a satisfactory result but if there are «sparsity» or «gaps» in the space of input and output variables the quality of nonparametric models is significantly reduced. This article is devoted to the method of filling or generation of training samples based on current available information. This can significantly improve the accuracy of identification of nonparametric models of noninertial systems with delay. Conducted computing experiments have confirmed that the quality of nonparametric models of noninertial systems can be significantly improved as a result of original sample «repair». At the same time it helps to increase the accuracy of the model at the border areas of the process input-output variables definition.
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On one Sobolev type mathematical model in quasi-Banach spaces
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The theory of Sobolev type equations experiences an epoch of blossoming. In this article the theory of higher order Sobolev type equations with relatively spectrally bounded operator pencils, previously developed in Banach spaces, is transferred to quasi-Banach spaces. We use already well proved for solving Sobolev type equations phase space method, consisting in reduction of singular equation to regular one defined on some subspace of initial space. The propagators and the phase space of complete higher order Sobolev type equations are constructed. Abstract results are illustrated by specific examples. The Boussinesq-Love equation in quasi-Banach space is considered as application.
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On one mathematical model described by boundary value problem for the biharmonic equation
Статья научная
In this paper mathematical model described by a generalized third boundary value problem for the homogeneous biharmonic equation in the unit ball with boundary operators up to the third order containing normal derivatives and Laplacian is investigated. Particular cases of the considered mathematical model are the classical models described by Dirichlet, Riquier, and Robin problems, and the Steklov spectral problem, as well as many other mathematical models generated by these boundary conditions. Two existence theorems for the solution of the problem are proved. Existence conditions are obtained in the form of orthogonality on the boundary of some linear combination of boundary functions to homogeneous harmonic polynomials of a particular order. The obtained results are illustrated by some special cases of the general problem.
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On one mathematical model of the extraction process of polydisperse porous material
Статья научная
We consider a mathematical model which represents the extraction process of a target component from the polydispersed porous material. The suggested model is demonstrated by the example of a flat solid material with bidispersed pores of different size in the form of a system of channels of macropores with micropores facing their walls. The macropores and the micropores in the material have homogeneous size. We model a case when micropores of the solid material (dispersed medium) are initially filled with an oil (dispersion phase), which is our target component. The macropores are filled in with a pure solvent. In the process of extraction the oil diffuses from the micropore to the macropore, and then from the micropores to the external solvent volume, wherein the ratio of concentrations in the macropore and the micropore is taken in accordance with the linear law of adsorption. The well-posedness of the formulated mathematical model has been justified.
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On perturbation method for the first kind equations: regularization and application
Статья научная
One of the most common problems of scientific applications is computation of the derivative of a function specified by possibly noisy or imprecise experimental data. Application of conventional techniques for numerically calculating derivatives will amplify the noise making the result useless. We address this typical ill-posed problem by application of perturbation method to linear first kind equations Ax=f with bounded operator A. We assume that we know the operator A and source function f only such as ||A-A||
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Статья научная
We examine inverse problems of recovering coefficients in a linear pseudoparabolic equation arising in the filtration theory. Boundary conditions of the Neumann type are supplemented with the overtermination conditions which are the values of the solution at some interior points of a domain. We expose existence and uniqueness theorems in the Sobolev spaces. The solution is regular, i.e., it possesses all generalized derivatives occurring in the equation containing in some Lebesgue space. The method of the proof is constructive. The problem is reduced to a nonlinear operator equation with a contraction operator whenever the time interval is sufficiently small. Involving the method of the proof, we construct a numerical algorithm, the corresponding software bundle, and describe the results of numerical experiments in the two-dimensional case in the space variables. The unknowns are a solution to the equation and the piezo-conductivity coefficient of a fissured rock. The main method of numerical solving the problem is the finite element method together with a difference scheme for solving of the corresponding system of ordinary differential equations. Finally, the problem is reduced to a system of nonlinear algebraic equations which solution is found by the iteration procedure. The results show a good convergence of the algorithms.
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