Home • Editorial Board • Contents • Russian page

Sibirskii  Zhurnal  Industrial'noi  Matematiki
2010,  vol. 13,  No 2 (42)

Contents
 

Heavenly Bodies and Satellites
 (on the Occasion of the 70th Birthday of Yuri Ershov)

UDC 517.95
Alekseev G. V., Khludnev A. M.
The Stability of Solutions to Extremal Problems of Boundary Control for Stationary Heat Convection Equations

We study the extremal problems of boundary control for the stationary heat convection equations with Dirichlet boundary conditions on the velocity and temperature. As cost functionals we choose the mean square deviation of the required temperature field from the temperature field  measured in some part of the flow region.  The controls are functions involved in  the Dirichlet conditions.  We prove the stability of solutions under certain perturbations of both the quality functional and one of the known functions involved in the original equations of the model.

Keywords:  heat convection, extremal problems, uniqueness,  stability estimates.
Pp. 5–18. 

Alekseev Gennady Valentinovich
Applied Mathematics Institute, FEB RAS, Radio str., 7, Vladivostok 690041, E-mail:  alekseev@iam.dvo.ru
Khludnev Aleksandr Mikhauilovich
 Lavrent'ev Institute of Hydrodynamics, SB RAS,  Acad. Lavrent'ev str., 15, Novosibirsk 630090, E-mail: khlud@hydro.nsc.ru

UDC 533.72

Andreev A. P., Latyshev A. V., Popov V. N., Yushkanov A. A.
Rarefied Gas Flow Near a Rotating Cylinder

In the isothermic approximation with the second order correction relative to the Knudsen number, we construct an analytic solution to the problem of a circular cylinder rotating in a rarefied gas. As the basic equations we take a generalization of  the BGK model of kinetic Boltzmann equations to the case of rotational degrees of freedom of gas molecules, and as a microscopic boundary value on the surface of the cylinder we take an accommodation two-moment boundary value.  We graph the mass velocity profile for the rarefied gas carried along by the rotating cylinder and make a comparison with the analogous results obtained using the diffuse reflection model, as well as the results of classical hydrodynamics.

Keywords: kinetic Boltzmann equation, analytic solutions to model kinetic equations, models of boundary values,  accommodation coefficients.
Pp. 19–29.

Andreev Aleksey Pavlovich
Popov Vasiliy Nikolaevich
 Pomor State University, Lomonosov prospekt, 4, Arkhangel'sk 163002,
Latyshev Anatoliy Vasil'evich
Yushkanov Aleksandr Alekseevich
Moscow Region University, Radio str., 10a, Moscow 107005, E-mail: popov.vasily@pomorsu.ru

UDC 517.9

Anikonov Yu. E., Krivtsov Yu. V., Neshchadim M. V.
Constructive Methods in Nonlinear Control Theory Problems

We consider nonlinear control problems of transferring a substance from one state to the other given boundary data. We define second order differential operators with three time-independent coefficients.  Our constructive analytical methods use in particular the Burman–Lagrange inversion formula for analytic functions. We give formulas for solutions to control problems in some particular cases of initial and boundary data. In addition to theoretical study we attempt, basing on a symbolic computation system, to implement some algorithms and programs to automatically generate the formulas that yield solutions to control problems.

Keywords: inverse problems,  control theory.
Pp. 30–45.

Anikonov Yuriy Evgen’evich
Neshchadim Mikhail Vladimirovich
 Sobolev Institute of Mathematics, SB RAS, Acad. Koptyug prospekt, 4, Novosibirsk 630090, E-mail: anikon@math.nsc.ru ; neshch@math.nsc.ru ;
 Krivtsov Yuriy Vasil’evich
Siberian State University of Telecommunications and Informatics, Kirova str., 86, Novosibirsk 630102, E-mail: yuri.krivtsov@rambler.ru

UDC 519.865.3
Bredikhin S. V.,  Khutoretskiy A. B.
Equilibrieum Allocation of Processor Time with Linear Utility Functions

We consider a model for the processor time market in a multiprocessor system. The processes differ in speed and expences. Each user has one job, whose volume and budget are known, and a utility function depending linearly on the share of the job completed during the period to be allocated. The budget of each job coincides with a monetary estimate of its utility for the user.  We show that under these assumptions the equilibrium prices and distribution of processor time between the jobs can be obtained from certain linear programming problems.

Keywords: multiprocessor system, allocation of processor time, usefulness, balanced prices,  linear programming.
Pp. 46–53.

Bredikhin Sergey Vsevolodovich
 Institute of Computational Mathematics and Mathematical Geophysics, SB RAS, Lavrent’ev prospekt, 6, Novosibirsk 630090, E-mail: bred@nsc.ru;
 Khutoretskiy Aleksandr Borisovich
 Novosibirsk State Pedagogical University, Vilyuyskaya str., 28, Novosibirsk 630126, E-mail: hab@dus.nsc.ru

UDC 517.95

Gaevoy V. P.
Existence and Stability of Summable Generalized Solutions in a Mathematical Model of a Catalytic Process in a Boiling Layer

For a system of first order partial differential equations describing a catalytic process in a boiling layer, we consider a mixed problem in the half-strip 0 £ x £ h, t ³ 0. We prove the existence and uniqueness of a bounded summable generalized solution and study its stability. We prove the  stabilization as t ®¥ of the values of some physically meaningful functionals of solutions.

Keywords: mixed problem,  generalized solution,  stability.
Pp. 54–68.

Gaevoy Viktor Pavlovich
 Sobolev Institute of Mathematics, SB RAS, Acad. Koptyug prospekt, 4, Novosibirsk 630090, E-mail: gaev@math.nsc.ru

UDC 517.518.85:517.27
Klyachin V. A., Pabat E. A.
The C1-Approximation of the Level Surfaces of Functions Defined on Irregular Meshes

We consider the problem of interpolating the level surfaces of functions in some classes (Lipschitz functions, continuously differentiable functions, functions whose gradient satisfies the Hölder  condition, and twice continuously differentiable functions) given their values at the nodes of  irregular meshes. We derive geometric conditions on the triangulations of a sequence of finite collections of points which guarantee that the gradients of piecewise linear approximations converge. We illustrate the sharpness of these conditions with Schwartz's example. We propose a method for approximating level surfaces which guarantees C1-convergence without any restrictions on the location of nodes.

Keywords: triangulation, approximation of the gradient, level surface, Voronoi diagram.
Pp. 69–78.

Klyachin Vladimir Aleksandrovich
Pabat Evgeniya Aleksandrovna
 Volgograd State University, Universitetskiy prospekt, 100, Volgograd 400062, E-mail: klchnv@mail.ru ; gracheva_evg@mail.ru

UDC 517.926.7
Lomov A. A.
Controllability of Sum Linear Systems
For two linear systems S1  and  S2 described by ordinary matrix difference equations with disjoint sets of solutions  M1 and M2 we define the (disjoint) sum system, the set of whose solutions is the sum of M1 and M2. We obtain a controllability criterion for the sum  system in the form of conditions on the coefficients of S1 and S2.

Keywords:  linear difference equations, sum systems, disjunctive systems, controllability.
Pp. 79–84.

Lomov Andrey Aleksandrovich
 Sobolev Institute of Mathematics, SB RAS, Acad. Koptyug prospekt, 4, Novosibirsk State, University, Pirogova str., 2, Novosibirsk 630090, E-mail: lomov@math.nsc.ru

UDC 517.956.3
Lyul’ko N. A.
On the Instability of Hyperbolic Systems on the Plane under  Small Periodic Perturbation

For a strictly hyperbolic linear first order system in the half-strip P = {(x, t): 0 < x < 1, t > 0} we consider the mixed problem  generating a group of unitary operators. In the case of a periodic perturbation  we propose a method for finding the frequencies for which the perturbed system develops parametric resonance. We illustrate the method with a system of two equations.

Keywords: the mixed problem for a hyperbolic system on the plane, instability of solutions, averaging method.
Pp. 85–97.

Lyul’ko Natal’ya Al’bertovna
Sobolev Institute Mathematics, SB RAS, Acad. Koptyug prospekt, 4, Novosibirsk 630090, E-mail: natlyl@mail.ru

UDC 517.958:531.72:539.3(4)
Meyrmanov A. M.
Acoustics Equations in Elastic Porous Media

We study the acoustics equations in elastic porous media which had been obtained by the author by averaging the exact dimensionless equations that describe the joint motion of an elastic solid skeleton and a viscous fluid in the pores on the microscopic level. The small parameter of this model is the ratio e of the average pore size l to the characteristic size L of the physical region under consideration. The averaged equations (the limit regimes of the exact model as e tends to zero) depend on the dimensionless coefficients of the model, which either depend weakly on the small parameter, or are small or large as this parameter tends to zero. On assuming that the solid skeleton is periodic we analyze the particular form of acoustics equations for the simplest periodic structures.

Keywords: Stokes and Lam'e equations, two-scale convergence, acoustics equations.
Pp. 98–110.

Meyrmanov Anvarbek Mukatovich
 Belgorod State University, Pobedy str., 85, Belgorod 308015, E-mail: meirmanov@bsu.edu.ru

UDC 517.97
Nalimov V. I.
Variational Methods for Constructing Approximate Solutions to Some Problems of Continuum Mechanics

In order to find approximate solutions to continuum mechanics problems admitting a variational statement we use an approach that is based on restricting the class of functions in which we seek the extremum of the action. We demonstrate the method on some examples of the forced oscillations problem for a nonlinear elastic membrane (a string in particular), the problem of fluid flow through a porous obstacle, and the problem of stationary waves on the surface of a heavy fluid.

Keywords: variational method, approximate solutions.
Pp. 111–123.

Nalimov Viktor Ivanovich
 Lavrent’ev Institute of Hydrodynamics, SB RAS, Lavrent’ev prospekt, 15, Novosibirsk 630090

UDC 517.95
Semenko E. V.
A General Form of the Solution to the Flow Problem on the Torus

We develop the idea to use Riemann surfaces for modeling plane flows. We establish a collection of parameters which uniquely determine the solution to the flow problem on the torus and construct a homeomorphism of the flow region, in whose plane the solution is represented explicitly by irrational functions of a complex variable with coefficients depending on the parameters. Basing on that we analyze the flow schemes for some values of the parameters.

Keywords: flow scheme,  Riemann surface, torus, flow parameters.
Pp. 124–134.

Semenko Evgeniy Veniaminovich
Novosibirsk State Pedagogical University, Vilyuyskaya str., 28, Novosibirsk 630126, E-mail: semenko54@mail.ru

UDC 517.948
Serebryanskiy S. M.
On Estimating the Errors of Approximate Solution Methods for an Inverse Problem

We study an inverse boundary problem for the heat equation for an inhomogeneous rod composed of two materials. We consider three different situations of temperature measurements inside the rod, from which we must reconstruct one of the boundary values of the problem. Similar problems arise in the test benching of rocket engines, and their solutions are required to be very precise. We solve the problems by the projection regularization method, and for their solutions we obtain estimates that are precise up to the order of magnitude.

Keywords: inverse heat conduction problems, Fourier transform, projection regularization method, estimates precise up to the order of magnitude.
Pp. 135–148.

Serebryanskiy Sergey Mikhaylovich
Chelyabinsk State University, Troitsk Branch, S. Razina str., 9, Troitsk 457100, E-mail: tf_chelgu@mail.ru

 

Home • Editorial Board • Contents • Russian page