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Sibirskii Zhurnal Industrial'noi Matematiki
2013, vol. 16, No 3 (55)
Contents
UDC 514.745.82
Akin'shin A. A.
Andronov—Hopf bifurcation for some nonlinear delay equationsWe study the occurrence of Andronov–Hopf bifurcation cycles in a neighborhood of stationary points of nonlinear delay equations: we formulate conditions for the existence of a bifurcation, find the bifurcation values, and analyze the stability of the bifurcation cycles.
Keywords: Andronov—Hopf bifurcation, stationary point, delayed argument, stable cycles, first Lyapunov coefficient.
Pp.3–15.Akins'hin Andrey Alexandrovich
Altay State Technical University, 46, Lenin av., 656038, Barnaul. E-mail: andrey.akinshin@gmail.com
UDC 917.95:519.57
Akramov T.A.
Analysis of a model of a co-current packed-bed column with periodic inlet conditionsWe consider a model of a co-current packed bed column with the simplest kinetics of the rate of liquid holdup formation. The dependence of the velocities of the liquid and gas flows in neighborhoods of the stationary values of these velocities is studied. The principal terms in time of the asymptotics for the luquid holdup and the flow rates are found. We detect the growth of the amplitudes of the sinusoidal oscillations of the liquid holdup and the liquid and gas velocities with respect to the length of the column. The constructed model is used for co-current two-phase columns.
Keywords: mathematical model, co-current packed-bed column, gas and liquid velocities, asymptotics of solutions, stability, oscillation amplitude, Riemann function.
Pp. 16–27.Akramov Talgat Akramovich
Ufa Division of the Russian State University of Trade and Economics, Mendeleev Str. 177/3, 450080 Ufa. E-mail: talgataa@ hotbox.ru
UDC 517.9
Belov Yu. Ya, Korshun K. V.
On an inverse problem for a Burgers-type equationUnder consideration is the problem of the identification of the source function in a Burgers-type equation. The problem is studied in the case of Cauchy data and mixed boundary conditions in a rectangular domain. Sufficient conditions on the input data for the unique solvability of these problems in classes of smooth bounded functions are found.
Keywords: inverse problem, Burgers equation, boundary-value problem, approximation.
Pp. 28–40.
Belov Yuri Yakovlevich
Korshun Kirill Viktorovich
Institute of Mathematics and Computer Science of Siberian Federal University, 79 Svobodnyi av., Krasnoyarsk, 660041. E-mail: YBelov@sfu-kras.ru; korshun007@inbox.ru
UDC 517.956.223
Bondar' L.N.
Solvability of the second boundary value problem for one elliptic equation in a half-spaceWe consider the second boundary value problem in a half-space for a biharmonic equation with a lower-order term. We prove theorems on the solvability of the problem in a Sobolev space.
Keywords: elliptic equation, boundary value problem, Sobolev space, solvability condition.
Pp. 41–52.Bondar' Lina Nikolaevna
Sobolev Institute of Mathematics SD RAS, 4 Koptyug av.; Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk E-mail: b_lina@ngs.ru
UDC 517.929.4
Demidenko G.V., Vodopyanov E.S., Skvortsova M.A.
Estimates of solutions to linear differential equations of neutral type with several delays of argumentWe consider a system of linear differential equations of neutral type with several delays of argument. We obtain conditions on the matrix coefficients of the system under which all its solutions decay exponentially at infinity. Using functionals of Lyapunov—Krasovskii type, we establish uniform estimates of solutions.
Keywords: equations of neutral type, asymptotic stability, Lyapunov — Krasovskii functional.
Pp. 53–60.Demidenko Gennadii Vladimirovich
Sobolev Institute of Mathematics SD RAS, 4 Koptyug av., 630090 Novosibirsk;
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk. E-mail: demidenk@math.nsc.ru
Vodop'yanov Evgenii Sergeevich
Novosibirsk State University 2 Pirogova st., 630090 Novosibirsk E-mail: aclimit@gmail.com
Skvortsova Maria Aleksandrovna
Sobolev Institute of Mathematics SD RAS, 4 Koptyug av., 630090 Novosibirsk. E-mail: sm-18-nsu@yandex.ru
UDC 550.36
Duchkov A.A., Karchevsky A.L.
Determination of a terrestrial heat flow from temperature measurements in bottom sedimentsWe consider the problem of the determination of a terrestrial heat flow from temperature measurements in bottom sediments. The problem is divided into two subproblems: first we solve the one-dimensional inverse problem of estimating the thermal conductivity λ, and second we compute the heat flow value by solving the direct stationary problem, based on the already-found value of λ. We develop a sweep method for solving the direct problem which differs from the standard one. An optimization approach is used for solving the inverse problem; explicit formulas are obtained for computing the gradient of the residual functional. We analyze factors causing errors in estimating the heat flow. We show that the main contribution to the errors is given by the presence of harmonics with periods exceeding the monitoring time interval in the temperature curves. We show that if the parameters of the harmonics are known then one can calculate corrections for the found value of the heat flow. The results were applied to the data of temperature measurements carried out at the bottom of Lake Teletskoye from June 2008 to September 2010. For finding long-period harmonics, we made use of meteorological data about bottom water temperature from 1968 to 2011. This allowed us to estimate the value of the heat flow through the bottom of Lake Teletskoye as well as the thermal diffusivity in the upper layer of the sediments.
Keywords: heat flow, thermal conductivity, thermal diffusivity, inverse problem of heat conduction, Lake Teletskoye.
Pp. 61–85.Duchkov Anton Albertovich
Trofimuk Institute of Petroleum Geology and Geophysics SD RAS, 3 Koptuyg pr., 630090, Novosibirsk. E-mail: DuchkovAA@ipgg.nsc.ru
Karchevskii Andrey Leonidovich
Sobolev Institute of Mathamatics SD RAS, 4 Koptyug av., 630090 Novosibirsk. E-mail: karchevs@math.nsc.ru
UDC 519.85
Zabudskii G.G., Amzin I.V.
Algorithms of compact location for technological equipment on parallel linesThe two-dimensional location problem of rectangles on parallel lines is considered. For constructing a set of Pareto-optimal solutions, integer optimization and dynamic programming are applied. A computational experiment for the comparison of the approaches is carried out.
Key words: integer programming, dynamic programming, Pareto-optimal solutions, location problem.
Pp. 86–94.Zabudsky Gennadii Grigor'evich
Omsk Branch of the Sobolev Institute of Mathematics SD RAS, 13 Pevtsov st., 644043 Omsk. e-mail: zabudsky@ofim.oscsbras.ru
Amzin Igor' Viktorovich
Omsk State University, 55a Mira av., 644077, Omsk. e-mail: igor.amzin@mail.ru
UDC 51-7
Zakarin E. A., Kim D.K.
A stochastic model for the risk of biota exposure in case of accidental environmental pollutionWe describe a stochastic model for the risk of biota exposure in case of accidental environmental pollution. The model is based on the formalization of the notions of accident, environmental pollution, biomass, biota sensitivity, and their consecutive dependence on each other in case of an accident. The result of the article is the determination of a risk function which includes all elements of the considered model and quantitatively assesses the reduction of the biota in case of an accident.
Key words: accident, environmental risk, biota sensitivity, environmental pollution, stochastic model.
Pp. 95–105.Zakarin Edige Askarovich
Kim Dmitrii Konstantinovich
Kazakh National Technical University, Satpaev st. 22, 050013 Almaty, Kazakhstan. E-mail: Zakarin_Edige@mail.ru; kdk26@mail.ru
UDC 519.632
Laevskii Yu.M., Litvinenko S.A.
On a numerical algorithm for solving the Bakley—Leverett equationsWe describe a numerical algorithm for solving the problem of the two-phase filtration of an incompressible fluid in the absence of capillary forces (the Buckley—Leverett model). The usage of the conventional computational scheme for multiple injection wells leads to the ''nongluing'' of the flows of the displacing phase. We propose a way to remove this defect.
Keywords: two-phase fluid, filtration, well, difference scheme.
Pp. 106–115.Laevskii Yurii Mironovich
Litvinenko Svetlana Alekseevna
Institute of Computational Mathematics and Mathematical Geophysics SD RAS, 6 Lavrent'ev av., 630090 Novosibirsk;
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk. E-mail: laev@labchem.sscc.ru; litvin@labchem.sscc.ru
UDC 541.128
Leshakov O.E., Mamash E.A., Krasil'nikov M.P.
Modeling critical phenomena in the reaction of carbon monoxide catalytic oxidation in a reaction-diffusion systemWe construct a model of the dynamics of a five-stage mechanism of carbon monoxide catalytic oxidation reaction with diffusion. Analytical expressions for bifurcation curves are obtained. It is shown that the the inclusion of a diffusion process in the model can lead to a bifurcation.
Keywords: chemical kinetics, critical phenomena, diffusion, bifurcation.
Pp. 116–121.Leshakov Oleg Eduardovich
Krasil'nikov Mikhail Petrovich
Tuva Institute for Exploration of Natural Resourses SD RAS Internationalnaya st. 117a 667000 Kyzyl
Mamash Elena Aleksandrovna
Institut of Computational Technologies, 6 Lavrent'ev av., 630090 Novosibirsk. E-mail: o_leshakov@mail.ru; kmp000@gmail.com; elenamamash@gmail.com
UDC 517.929.4
Matveeva I.I.
Estimates for solutions to one class of nonlinear delay differential equationsWe consider systems of nonlinear delay differential equations with periodic coefficients in the linear terms. Sufficient conditions for the asymptotic stability of the zero solution are established. We obtain estimates characterizing the decay of solutions at infinity and describe the attraction sets for the zero solution.
Keywords: delay differential equations, periodic coefficients, asymptotic stability, Lyapunov–Krasovskii functional, estimates for solutions, attraction set.
Pp. 122–132.Matveeva Inessa Izotovna
Sobolev Institute of Mathematics SD RAS, 4 Koptyug av., Novosibirsk 630090.
Novosibirk State University, Pirogova st., 2, Novosibirsk 630090. E-mail: matveeva@math.nsc.ru
UDC 519.63:621.38
Fadeev S.I., Pimanov D.O.
Numerical study of mathematical models of micromechanics with periodic impulse actionWe consider the results of the numerical study of mathematical models of two microelectromechanical systems (MEMS). We formulate matematical models as initial boundary value problems describing the cylindrical flexure of the elastic beam as a movable electrode under the action of a repetitive intensity impulse of the electrostatic field between the movable and fixed electrodes in a microgap. In the first problem, both ends of the beam are rigidly fixed, and in the second problem, we consider a cantilever beam. The range of the parameters is found for a model having two periodic solutions with periods of impulse interaction one of which is stable and the other is unstable.
Keywords: nonlinear oscillation, electrostatic attraction, method of lines, boundary value problem, continuation with respect to a parameter, nonuniqueness of solutions.
Pp. 133–145.Fadeev Stanislav Ivanovich
Sobolev Institute of Mathematics SD RAS, 4 Koptyug av., 630090 Novosibirsk;
Novosibirsk State University, 2 Pirogova st., 630090, Novosibirsk. E-mail: fadeev@math.nsc.ru
Pimanov Daniil Olegovich
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk. E-mail: pimanov-daniil@yandex.ru