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Sibirskii  Zhurnal  Industrial'noi  Matematiki
2012,  vol. 15,  No 1 (49)

Contents
 

UDC 517.988.68
Ageev A. L., Antonova T. V.
Approximation of discontinuity lines of a noisy function of two variables

We construct and study methods for localizing (determining the location) a line in whose neighborhood the function of two variables in question is smooth while having  discontinuity of the first kind along the line. Instead of the exact function, an approximation in L2 is available with a known noise level. This problem belongs to the class of ill-posed nonlinear problems and, in order to solve it, we have to construct regularizing algorithms. We propose a simplified theoretical approach to the problem of discontinuity line localization for a noisy function with conditions on the exact function imposed in an arbirarily thin strip crossing the discontinuity line. We construct averaging methods and estimate the precision of localization for them.

Keywords: ill-posed problem, regularizing algorithm, localization of singularities, discontinuity of the first kind.
Pp. 3–13.

Ageev Aleksandr Leonidovich
Antonova Tat'yana Vladimirovna
Institute of Mathematics and Mechanics UB RAS 16 Kovalevskoui str., Ekaterinburg 620990 Russia. E-mail: ageev@imm.uran.ru; tvantonova@imm.uran.ru

 

UDC 622.276.031:532.5.001
Bocharov O. B., Kushnir D. Yu.
An analytical solution of the problem of wall cake growth with washing-off

We obtain an analytical solution for the thickness of incompressible wall cake and pressure drop in the well bore zone in the case of single-phase filtration in a porous medium. We assume that the deposition of clay particles on the well wall is proportional to the flux through the wall. We describe filtration basing on Darcy's law. We obtain analytic formulas for the case of cake washing-off by the drilling fluid and clogging zone with parameters independent of time. We consider the cases of linear and radial filtration.

Keywords: well bore zone, wall cake, drilling fluid filtration.
Pp. 14–21.

Bocharov Oleg Borisovich
Kushnir Dmitriui Yur'evich
 Baker Hughes Novosibirsk Technological Center, 4a Kutateladze str., Novosibirsk 630128, Russia. E-mail: Oleg.Bocharov@bakerhughes.com ; kushnir.dmitriy@gmail.com

 

UDC 517.9
Dvirnyui A.I., Slyn'ko V.I.
An analog of Kamenkov's critical case for systems of ordinary differential equations with pulse action

We propose a new approach to studying the stability of systems of nonlinear differential equations with pulse action in critical cases. This approach rests on Lyapunov functions. We obtain sufficient conditions for the asymptotic stability of critical equilibria in some case analogous to Kamenkov's critical case.

Keywords: stability in the sense of Lyapunov, Kamenkov's critical case, differential equations with pulse action, Lyapunov's direct method.
Pp. 22–33.

Dvirnyui Aleksandr Ivanovich
 Hadmark University College,
Hadmark,  Norway. E-mail: dvirny@mail.ru
Slyn'ko Vitaliui Ivanovich
Institute of Mechanics NAS of Ukraine, Kiev, Ukraine. E-mail: vistab@ukr.net

 

UDC 519.83:621.311:51.001.57
Zorkal'cev V. I., Perzhabinskiui S. M.
Models for estimating the power deficit in electric power grid

We consider versions of a model for estimating power deficit developed for the computing clusters analyzing the reliability of electric power grids. We discuss some mathematical properties and specific features of the models. We pay particular attention to the model accounting for the nonlinear character of power loss in transmission lines and indicate a method for reducing it to a convex programming problem.

Keywords: electric power grid, model for estimating power deficit, reliability, inner point methods.
Pp. 34–43.

Zorkal'cev Valeriui Ivanovich
Perzhabinskiui Sergeui Mikhauilovich
Institute of Energy Systems SB RAS, 130 Lermontova str., Irkutsk 664033. Russia. E-mail: zork@isem.sei.irk.ru; smper@isem.sei.irk.ru

 

UDC 541.124:541.126:517.9
Kononenko L. I., Volokitin E. P.
Parameterization and qualitative analysis of a singular system in a mathematical model of catalytic oxidation

We  qualitatively analyze the slow subsystem on one sheet of a parameterized slow surface and, as a result, justify the possibility of duck solutions.

Keywords: mathematical modeling, singularly perturbed systems.
Pp. 44–52.

Kononenko Larisa Ivanovna
Volokitin Evgeniui Pavlovich
 Sobolev Institute of Mathematics SB RAS, 4 Acad. Koptyuga av.,
Novosibirsk State University, 2 Pirogova str., Novosibirsk 630090, Russia. E-mail: larakon2@gmail.ru; volok@math.nsc.ru

 

UDC 539.371:532.546
Manakov A. V., Rudyak V. Ya.
An algorithm for joint modeling of filtration and geomechanical processes in the well bore zone

We design an algorithm for joint modeling of filtration and geomechanical processes developing in the well bore zone during drilling. We describe geomechanical processes in the framework of linear elasticity theory with pore pressure. For the first time we explicitly account for the formation and dynamics of wall cake. The algorithm, based on the finite element method, enables us to study the variation of the strained and deformed state of the porous medium and pore pressure as the drilling fluid permeats the medium.

Keywords: drilling, well bore zone, strained and deformed state, pore pressure, drilling fluid filtration, wall cake.
Pp. 53–65.

Manakov Artem Viktorovich
Rudyak Valeriui Yakovlevich
Baker Hughes Novosibirsk Technological Center, 4a Kutateladze str. Novosibirsk 630128, Russia,
Novosibirsk State University of Achitecture and Civil Engineering, 113 Leningradskaya str., Novosibirsk 630008 Russia. E-mail: artem.manakov@bakerhughes.com;  valery.rudyak@bakerhughes.com

 

UDC 550.8.05:519.651
Plavnik A. G., Sidorov A. N.
About estomation of the certainty of plotting the properties of geological objects in the framework of the spline approximation approach

We study the influence of including or excluding one or several data items on the results of plotting the properties of geological objects in the framework of spline approximation approach. We show that the ratio of approximation errors to prediction error and the ratio of  the variation of the approximating function at the points of action to the response are invariant under the specified values of the plotted parameters and can be effectively calculated for arbitrary points of the plotting region.

Keywords: geoplotting, spline approximation, approximation error, prediction error, cross-validation, stability, influence.
Pp. 66–76.

Plavnik Andreui Gar'evich
 Western Siberia Branch,
Institute of Oil and Gas Geology and Geophysics SB RAS, 56 Volodarskogo str., Tyumen' 625670 Russia.
Sidorov Andreui Nikolaevich
Scientific Analytical Center of Rational Subsurface Resource Management, 75 Malygina str., Tyumen' 625026 Russia. E-mail: plavnik@ikz.ru; sidorov@crru.tmn.ru

 

UDC 539.3
Ragozina V. E., Ivanova Yu. E.
A mathematical model of the motion of shear shock waves of nonzero curvature based on their evolution equation

We study particular features of the appearance and motion of 1-dimensional shear shock waves of nonzero curvature basing on the corresponding evolution equation. Using numerous examples of boundary value problems for axisymmetric antiplane deformation, we demonstrate the efficiency of applying solutions to the evolution equation as the frontal asymptotics in the method of matched asymptotic expansions.

Keywords: nonlinear elasticity, incompressibility, shock wave, perturbation method, evolution equation.
Pp. 77–85.

Ragozina Viktoriya Evgen'evna
Ivanova Yuliya Evgen'evna
 Institute of Automatics and Control Systems FEB RAS, 5 Radio str., Vladivostok 690041 Russia. E-mail: ragozina@vlc.ru; ivanova@iacp.dvo.ru

 

UDC 517.958
Romanov V. G.
Stability estimates for the solution in the problem of determining the kernel of the viscoelasticity equation

For the integrodifferential equation of  2-dimensional viscoelasticity we study the problem of determining the spatial part of the kernel of the integral part of the equation on assuming that the unknown function is supported on some compact region  W. As data required for solving this inverse problem, on the boundary of  W we specify the traces of the solution to the direct Cauchy problem and its normal derivative on some finite interval of time. A significant circumstance in the statement of this problem is that the solution to the direct Cauchy problem corresponds to zero initial data and time impulsive force localized on a fixed straight line disjoint from W. The main result of this article is a Lipschitz estimate for the conditional stability of the solution to this inverse problem.

Keywords: viscoelasticity, inverse problem, stability, uniqueness.
Pp. 86–98.

Romanov Vladimir Gavrilovich
Sobolev Institute of Mathematics SB RAS, 4 Acad. Koptyuga av., Novosibirsk 630090, Russia. E-mail: romanov@math.nsc.ru

 

UDC 539.375
Rudoui E. M.
Invariant integrals in the plane elasticity problem for bodies with rigid inclusions and cracks

We consider the plane elasticity problem for a body with a rigid inclusion and a crack along the boundary between the elastic matrix and rigid inclusion. We show that this problem possesses J- and M-invariant integrals. In particular, we construct an invariant integral of Cherepanov–Rice type for straight cracks.

Keywords: invariant integrals, rigid inclusion, crack, derivative of the energy functional, Cherepanov–Rice integral
Pp. 99–109.

Rudoui Evgeniui Mikhauilovich
Lavrent'ev Institute of Hydrodynamics SB RAS, 15 Acad. Lavrent'ev av.,
Novosibirsk State University, 2 Pirogova str., Novosibirsk 630090. E-mail: rem@hydro.nsc.ru

 

UDC 519.6
Rukavishnikov A. V.
On a precision estimate for a hydrodynamics problem with discontinuous coefficients in the norm of the space L2(Wh)

We study the 2-dimensional problem obtained by time-discretizing and linearizing the problem of flow of a 2-phase viscous fluid without mixing in the statement of incompressible Navier–Stokes equations with time-dependent interface. For an approximate solution to this problem we construct a scheme of a nonconformal finite element method. We estimate the rate of convergence of the mesh solution to the exact solution to the problem in the norm of L2(Wh), which agrees with simulations.

Keywords: discontinuous coefficients, domain decomposition, nonconformal finite element method, mortar elements.
Pp. 110–122.

Rukavishnikov Alekseui Viktorovich
Khabarovsk Branch, Institute of Applied Mathematics FEB RAS 54 Dzerzhinskogo str., Khabarovsk 680000, Russia. E-mail: 78321a@mail.ru

 

UDC 519.64
Savchenko A. O.
Calculation of the volume potential for ellipsoidal bodies

We consider quadrature formulas for calculating the integral of the product of functions, one of which has an integrable singularity admitting exact calculation of the integral. Basing on these formulas, we suggest a method for calculating numerically the potential of the ellipsoid without cutting out a region near the singularity. We approximate the inner integral by a function with a weak logarithmic singularity, and a subsequent change of variables enables us to perform further numerical integration without singularities in the integrand. For simulations we construct a quite complicated test function amounting to the exact potential of an ellipsoid of revolution with an elliptic density distribuion.

Keywords: volume potential, simulation, quadrature formulas, ellipsoid.
Pp. 123–131.

Savchenko Aleksandr Oliverovich
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, 6 Acad. Lavrent'ev av., Novosibirsk 630090, Russia. E-mail: savch@ommfao1.sscc.ru

 

UDC 517.988.7
Sidorov N. A., Sidorov D. N., Leont'ev R. Yu.
Successive approximations to solutions of nonlinear equations with vector parameter in the irregular case

We consider a nonlinear operator equation with a Fredholm operator in the principal part. The nonlinear part of the equation depends on some functionals that are defined on an open set of a normed linear space. We suggest a method of successive asymptotic approximations to branching solutions. We apply this method to study a nonlinear boundary value problem describing the oscillation of a satellite in the plane of its elliptic orbit.

Keywords: branching of solutions, Fredholm operator, asymptotics, Trenogin's regularizer, successive approximations.
Pp. 132–137.

Sidorov Nikolaui Aleksandrovich
Sidorov Denis Nikolaevich
Leont'ev Roman Yur'evich
Irkutsk State University, 1 Karl Max str., Institute of Energy Systems SB RAS 130 Lermontova str., Irkutsk 664033, Russia. E-mail: sidorov@math.isu.runnet.ru

 

UDC 539.3
Struzhanov V. V.
The determination of the deformation diagram of a material with a falling branch using the torsion diagram of a cylindrical sample

We present a method for constructing the deformation diagram of a material with a falling branch purely shearing along the torsion diagram of a solid cylindrical sample. The problem reduces to a Volterra integral equation of the first kind, and so it belongs to the class of ill-posed problems arising in attempts to determine physical quantities using measurement results. We show that for an arbitrary torsion diagram the solution to the Volterra equation leads to a sawtooth-shaped broken line in the diagram of the material. We regularize the solution using a trial-and-error method, which yields satisfactory results.

Keywords: pure shear diagram, falling branch, torsion diagram, Volterra equation of the first kind, trial-and-error method, regularization.
Pp. 138–144.

Struzhanov Valeriui Vladimirovich
Institute of Machine Engineering UB RAS, 34 Komsomol'skaya str., Ekaterinburg 620049 Russia. E-mail: stru@imach.uran.ru

 

UDC 517.948
Tanana V. P., Gauinova I. A., Sidikova A. I.
On estimating the error of an approximate solution to an overdetermined inverse problem of thermodiagnostics

Using a generalized projection regularization method, we solve an overdetermined inverse boundary value problem for the heat equation and obtain some estimates accurate in order  for the error of this solution.

Keywords: unbounded operator, Hilbert space, projection regularization operator, error estimates, approximate solutions.
Pp. 145–154.

Tanana Vitaliui Pavlovich
Sidikova Anna Ivanovna
South Urals State University, 76 Lenina av., Chelyabinsk 454080 Russia
Gauinova Irina Alekseevna
Sobolev Institute of Mathematics SB RAS, 4 Acad. Koptyuga av., Novosibirsk 630090, Russia. E-mail: tvpa@susu.ac.ru; 7413604@mail.ru; irina-gajnova@rambler.ru

  

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