Статьи журнала - Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование
Все статьи: 729
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
Статья научная
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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On some mathematical models of filtration theory
Статья научная
The article is devoted to the study of some mathematical models arising in filtration theory. We examine an inverse problem of determining an unknown right-hand side and coefficients in a pseudoparabolic equation of the third order. Equations of this type and more general Sobolev-type equations arise in filtration theory, heat and mass transfer, plasma physics, and in many other fields. We reduce the problem to an operator equation whose solvability is established with the help of a priori estimates and the fixed point theorem. Together with the natural smoothness conditions for the data, we require also some well-posedness condition to be fulfilled which is actually reduced to the condition of nondegeneracy of some matrix constructed with the use of the data of the problem. Theorems on existence and uniqueness of solutions to this problem are stated and proven. Stability estimates are exposed. In the linear case the result is global in time, while in the nonlinear case it is local. The main function spaces used are the Sobolev spaces.
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On some methods to solve integrodifferential inverse problems of parabolic type
Статья научная
In this paper we give an overview on some methods that are useful to solve a class of integrodifferential inverse problems. Precisely, we present some methods to solve integrodifferential inverse problems of parabolic type that are based on the theory of analytic semigroups, optimal regularity results and fixed point arguments. A large class of physical models can be treated with this procedure, for example phase-field models, combustion models and the strongly damped wave equation with memory to mention some of them.
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Interest in Sobolev type equations has recently increased significantly, moreover, there arose a necessity for their consideration in quasi-Banach spaces. The need is dictated not so much by the desire to fill up the theory but by the aspiration to comprehend non-classical models of mathematical physics in quasi-Banach spaces. Notice that the Sobolev type equations are called evolutionary if solutions exist only on R +. The theory of holomorphic degenerate semigroups of operators constructed earlier in Banach spaces and Frechet spaces is transferred to quasi-Sobolev spaces of sequences. This article contains results about existence of the exponential dichotomies of solutions to evolution Sobolev type equation in quasi-Sobolev spaces. To obtain this result we proved the relatively spectral theorem and the existence of invariant spaces of solutions. The article besides the introduction and references contains two paragraphs. In the first one, quasi-Banach spaces, quasi-Sobolev spaces and polynomials of Laplace quasi-operator are defined. Moreover the conditions for existence of degenerate holomorphic operator semigroups in quasi-Banach spaces of sequences are obtained. In other words, we prove the first part of the generalization of the Solomyak - Iosida theorem to quasi-Banach spaces of sequences. In the second paragraph the phase space of the homogeneous equation is constructed. Here we show the existence of invariant spaces of equation and get the conditions for exponential dichotomies of solutions.
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Статья обзорная
This paper addresses some classes of linear and quasi-linear partial differential algebraic equations (PDAEs), i.e. systems of partial differential equations with singular matrices multiplying the higher derivatives of the desired vector-function. Such systems do not belong to the class of the Cauchy - Kovalevskaya equations, and therefore do not not comply with known existence theorems. The current research focuses on the first order evolutionary systems with one variable and investigates PDAEs depending on the parameter. The concept of index for PDAEs is introduced and various statements of initial boundary problems are considered. The results obtained are used to simulate and analyze the heat and mass exchange processes in power plants.
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On the existence of an integer solution of the relaxed Weber problem for a tree network
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The problem of finding the optimal arrangement of vertices of a tree network in the installation space representing a finite set is considered. The criterion of optimality is the minimization of the total cost of deployment and the cost of communications. Placement of different tree vertices in one point of the installation space is allowed. This problem is known as Weber problem for a tree network. The statement of Weber problem as an integer linear programming problem is given in this research. It's proved that a set of optimal solutions of corresponding relaxed Weber problem for a tree-network contains the integer solution. This fact allows to prove the existence a saddle point while proving the performance of decomposition algorithms for problems different from problems because of additional restrictions.
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