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Sibirskii  Zhurnal  Industrial'noi  Matematiki
2011,  vol. 14,  No 3 (47)

Contents
 

UDC 519.626.1
Aleksandrov V. M., Dykhta V. A.
 An approximate solution to the minimization problem for expended resources. II. Control proximity estimate

We estimate the proximity of a control quasioptimal in terms of  resource consumption to the optimal control. We present a method for dividing a bounded region of initial conditions into subregions, which makes the quasioptimal control  closer to the optimal control in terms of  resources.

Keywords: optimal control, quasioptimal control, resource consumption, linear system, switching moments, dual system, control proximity estimate, admissible region
Pp. 3–13.

Aleksandrov Vladimir Mikhauilovich
 Sobolev Institute of Mathematics SB RAS, 4 Acad. Koptyuga ave., Novosibirsk 630090,  , E-mail: vladalex@math.nsc.ru
Dykhta Vladimir Aleksandrovich
Institute of Systems Dynamics and Control SB RAS, 134 Lermontova str. Irkutsk 664033, , E-mail: dykhta@icc.ru

 

UDC 573.2.57.017
Bukharina T. A., Golubyatnikov V. P., Golubyatnikov I. V., Furman D. P.
A mathematical modeling of the first phase of morphogenesis of the mechanoreceptors

In their development (morphogenesis) the mechanoreceptors, in particular macrochaetae,  of drosophila pass three stages whose genetic support is described in terms of gene networks. A key object in the gene networks of macrochaetae morphogenesis  is the gene complex it achaetescute (it AS-C). Each mechanoreceptor develops out of one parent cell, distinguished by a high content of the AS-C protein. The activity of this complex, which guarantees a protein level critical for initiating morphogenesis, is determined by the central regulatory contour, including a system of interactions among certain objects of the networks (genes and their products).
Pp. 14–19.
Bukharina Tatyana Anatol'evna
Furman Dagmara Pavlovna
Institute of Cytology and Genetics SB RAS, 10 Acad. Lavrent'eva ave. Novosibirsk 630090 E-mail: bukharina@bionet.nsc.ru; furman@bionet.nsc.ru
Golubyatnikov Vladimir Petrovich
Golubyatnikov Ivan Vladimirovich,
Sobolev Institute of Mathematics SB RAS, 4 Acad. Koptyuga ave. Novosibirsk 630090, E-mail: glbtn@math.nsc.ru ; ivan.golubyatnikov@gmail.com

 

UDC 519.237.7
Gol'tyapin V. V.
The use of the pseudoinverse matrix of the factor mapping in measuring factors

We consider the problem of finding factors and prove theorems that lead to the pseudoinverse atrix  found for the factor mapping obtained by principal component analysis as well as factor analysis methods. We develop the corresponding numerical algorithms for finding factors.

Keywords: pseudoinverse matrix, factor structure, problem of finding factors.
Pp. 20–30.

Gol'tyapin Viktor Viktorovich
Omsk Branch

Sobolev Institute of Mathematics SB RAS, 13 Pevtsova str.Omsk 640099. E-mail: goltyapin@mail.ru

 

UDC 519.622:553.982
Kabanikhin S. I., Cheremisin A. N., Shishlenin M. A.
The inverse problem of determining stream watering and discharge in a vertical flowing well

We consider direct and inverse problems for a vertical flowing well. The direct problem consists in finding the pressure and temperature of a two-phase flow along the well given the temperature and pressure at its bottom. The inverse problem consists in finding the discharge (the amount of fluid coming out) and well stream watering (the water content in the total fluid volume) given the temperature and pressure measured at the well mouth. We assume that the temperature and pressure at the bottom are known. We propose an algorithm for solving the direct and inverse problems in the case that the temperature and pressure are measured on the surface (at the mouth). We study the character of temperature and pressure variation on the surface as functions of discharge and stream watering. We describe the equivalence classes of solutions to the inverse problem within a given measurement error. We present a simulation of the inverse problem in a neighborhood of the exact solution and analyze stability.

Keywords: vertical flowing well, system of ordinary differential equations, Adams method, inverse problem.
Pp. 31–36.

Kabanikhin S. I.
 Institute of Computational Mathematics and Mathematical Geophysics SB RAS, 6 Acad. Lavrent'ev ave.
Sobolev Institute of Mathematics SB RAS, 4 Acad. Koptyug ave. Novosibirsk 630090. E-mail: kabanikhin@sscc.ru
Cheremisin Aleksandr Nikolaevich
Tyumen' Branch Institute of Theoretical and Applied Mechanics SB RAS 74 Tauimyrskaya str. Tyumen' 625026. E-mail: ancheremisin@tnk-bp.com
Shishlenin Maksim Aleksandrovich
Sobolev Institute of Mathematics SB RAS, 4 Acad. Koptyug ave. Novosibirsk 630090. E-mail: mshishlenin@ngs.ru

 

UDC 519.622.2
Kalinina E. A., Samarina O. N.
The error of Euler's method for floating-point arithmetic computations

We present an algorithm that enables us to find the optimal number of steps in Euler's method, in the sense of computational precision, while solving a Cauchy problem for a system of linear differential equations with constant coefficients. We include numerical examples of applications of this method for evaluating a solution to the Cauchy problem at a point and constructing solutions to systems of nonlinear ordinary differential equations.

Keywords: Euler's method, Cauchy problem, system of ordinary differential equations, floating-point arithmetic, computational error.
Pp. 37–49.

Kalinina Elizaveta Aleksandrovna
Samarina Ol'ga Nikolaevna
Saint-Petersburg State University, 35 Universitetskiui ave. Petergof, Saint-Petersburg 198504, E-mail: ekalinina69@gmail.com; samarina_o@mail.ru

 

UDC 517.958:532.591
Kovtunenko P. V., Chesnokov A. A.
Special classes of solutions to the equations of horizontal-displacement fluid motion

We construct a special class of solutions to nonlinear integrodifferential equations describing horizontaldisplacement motion of ideal fluid in an open channel in the shallow water approximation. This class of solutions is characterized by a linear relation among Riemann integral invariants and is determined by a hyperbolic system of two differential equations with two parameters. In the class of traveling waves we construct exact solutions, including arbitrary functions, to the equations of motion continuously adjacent  to a given displacement flow.
Pp. 50–57.

Kovtunenko Pavel Viktorovich
Chesnokov Aleksandr Aleksandrovich
 Lavrent'ev Institute of Hydrodynamics SB RAS, 15 Acad. Lavrent'ev ave.
Novosibirsk State University, 2 Pirogova str. Novosibirsk 630090 E-mail: pkovtunenko@gmail.com; chesnokov@hydro.nsc.ru

 

UDC 519.81
Lavlinskiui S. M., Rudnev A. S.
Problems of technological planning in oil extraction

We develop mathematical models for solving practical problems in the operational control of an oil company. We pay the principal attention to the problem of optimizing oil extraction from deposits far from a main line. For this problem we construct a decomposition algorithm that enables us to automate scheduling the work of the company's fleet of motor vehicles.

Keywords: oil extraction monitoring system, optimization of oil extraction, traveling salesman's problem, decomposition algorithm.
Pp. 58–66.

Lavlinskiui Sergeui Mikhailovich
Rudnev Anton Sergeevich
 Sobolev Institute of Mathematics SB RAS, 4 Acad. Koptyug ave. Novosibirsk 630090. E-mail: lavlin@math.nsc.ru; anton.rudnev@gmail.com

 

UDC 517.977.57:517.95
Musabekov K. S.
The existence of stationary solutions in a mathematical model of a chemical reactor

We consider a mathematical model of a nonadiabatic tubular reactor. We derive a priori estimates and prove the existence of a classical solution to a boundary value problem in a system of two quasilinear ordinary differential equations.

Keywords: mathematical model, chemical reactor, control, stationary solution.
Pp. 67–78.

Musabekov Kalimulla Sultanovich
 Novosibirsk State University, 2 Pirogova str. Novosibirsk 630090 E-mail: it.kgu@mail.ru

 

UDC 517.958:531.72:517.958:539.3/4
Nekrasova I. V.
Some models of hydraulic shock in an oil-bearing layer

We propose two mathematical models for determining pressure field distribution in a layer near a well during a hydraulic shock. The derivation of the models rests on a strict averaging of the exact equations describing the joint motion of solid ground and a viscous fluid filling the pores on a microscopic level.

Keywords: hydraulic fracturing, Stokes and Lam'e equations, two-scale convergence
Pp. 79–86.

Nekrasova Irina Viktorovna
 Belgorod State University, 85 Pobedy str. Belgorod 308015 E-mail: Nekrasova_i@bsu.edu.ru

 

UDC 517.958
Romanov V. G., Moshkalev P. S.
The one-dimensional inverse problem of finding a tsunami source

We consider in the one-dimensional version a model inverse problem of finding the initial state of the ocean surface. We prove that the initial state is uniquely determined by the oscillation regime of a sole point on the ocean surface. We give algorithms for solving the direct and inverse problems and include the results of test simulations.

Keywords: tsunami wave propagation, Cauchy problem, inverse problem, simulation algorithms.
Pp. 87–99.

Romanov Vladimir Gavrilovich
 Sobolev Institute of Mathematics SB RAS, 4 Acad. Koptyug ave. Novosibirsk 630090 E-mail: romanov@math.nsc.ru   
Moshkalev Pavel Sergeevich
Novosibirsk State University, 2 Pirogova str. Novosibirsk 630090 E-mail: moshkalev@mail.ru

 

UDC 517.9
Sazhenkov S. A.
An effective model of the dynamics of barotropic gas with fast oscillating initial data

We consider the classical three-dimensional Navier–Stokes equations of viscous compressible inhomogeneous fluid in a smooth bounded region with  adhesion on the boundary of the region and fast oscillating initial density distributions. The state equation of the medium under study is the state equation of a barotropic gas. We assume that the adiabatic constant is greater than 3. We justify rigorously the homogenization procedure as the frequencies of fast oscillations tend to infinity. This yields a limit effective model of the dynamics of compressible viscous gas with fast oscillating initial data.

Keywords: viscous compressible barotropic gas, homogenization, fast oscillating initial data.
Pp. 100–111.

Sazhenkov Sergeui Aleksandrovich
 Lavrent'ev Institute of Hydrodynamics SB RAS, 12 Acad. Lavrent'ev ave. Novosibirsk 630090 E-mail: sazhenkovs@yandex.ru

 

UDC 550.837:517.958
Soloveuichik Yu. G., Persova M. G., Abramov M. V., Tokareva M. G.
Finite-element modeling of  electric and magnetic fields of induced polarization in a three-dimensional medium

We consider methods for modeling the electric field of induced polarization (IP) and the generated distribution of current densities to simulate the magnetic field of IP. The source of the polarizing field is a horizontal current line with grounded electrodes. We discuss computational schemes for simulating the electric and magnetic fields of IP. We present examples of simulations of IP fields for some polarized objects.

Keywords: geoelectricity problems, geoelectrical prospecting, induced polarization field, electric and magnetic fields, three-dimensional finite-element modeling.
Pp. 112–124.

Soloveuichik Yuriui Grigor'evich
Persova Marina Gennad'evna
Abramov Mikhail Vladimirovich
Tokareva Marina Georgievna
Novosibirsk State Technical University, 20 Karl Marx ave. Novosibirsk 630092 E-mail: solov@fpm.ami.nstu.ru

 

UDC 519.865.3
Khutoretskiui A. B., Bredikhin S. V., Belov A. S.
An effective balanced budget and a d-nonmanipulable technique for processor time allocation

We propose a technique for processor time allocation in a grid system which is not manipulable by users and satisfies  efficiency, individual rationality, and budget balance conditions. We assume that computers in the system differ in speed and running expenses; every user has one job and a linear usefulness function; the budget of a job coincides with the financial estimate of its usefulness. The mechinque suggested in this article generates a balanced distribution of processor time. It establishes the users' payments in accordance with a generalized Vickrey auction, and payments to the suppliers so that to guarantee budget balance and individual rationality.

Keywords: grid system, processor time allocation, generalized Vickrey auction, nonmanipulability, budget balance, liner programming.
Pp. 125–131.

Khutoretskiui Aleksandr Borisovich
 Novosibirsk State Pedagogical University, 28 Vilyuuiskaya str. Novosibirsk 630126
Bredikhin Sergeui Vsevolodovich
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, 6 Lavrent'ev ave. Novosibirsk 630090
Belov Alekseui Sergeevich
Novosibirsk State University, 2 Pirogova str. Novosibirsk 630090 E-mail: hab@dus.nsc.ru; bred@nsc.ru

 

UDC 517.9:539.3
Chirkunov Yu. A.
Friedrichs systems equivalent to systems of wave equations

Up to equivalence transformations we find all evolutionary symmetric Friedrichs t-hyperbolic systems equivalent to systems of two- and three-dimensional wave equations. We obtain evolutionary symmetric Friedrichs t-hyperbolic systems describing displacement waves in a three-dimensional isotropic elastic medium both in the presence and in the absence of mass forces. We study the group properties of some of these systems. We point out a system equivalent to Maxwell's equations and describing the electromagnetic field in the vacuum, which consists of the equations of an evolutionary symmetric Friedrichs t-hyperbolic system and the Lorentz condition, which is put into involution by the law of charge conservation.

Keywords: Friedrichs systems, wave equation, equivalence transformations, equivalent systems, displacement waves in a three-dimensional isotropic elastic medium, Maxwell's equations, involution.
Pp. 132-142.

Chirkunov Yuriui Aleksandrovich
 Novosibirsk State Technical University, 20 Karl Marx ave. Novosibirsk 630092 E-mail: chr01@rambler.ru

 

UDC 539.3
Shvab A. A.
Solving elasticity problems using an integral equation method for a holomorphic vector

We consider elasticity problems with nonclassical boundary conditions. We assume that both the load and displacement vectors are specified simultaneously on a part of the boundary, either of them can be separately specified on another part of the boundary, and the conditions are absent on the remaining part (of positive measure) of the surface. We consider the questions of uniqueness of a solution. These nonclassical problems reduce to a system of singular integral equations for a holomorphic vector.

Keywords: elasticity, inverse problems, integral equations.
Pp. 143-150.
Shvab Al'bert Aleksandrovich
Lavrent'ev Institute of Hydrodynamics SB RAS, 15 Acad. Lavrent'ev ave. Novosibirsk 630090 E-mail: schwab@ngs.ru

 

 

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