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Sibirskii Zhurnal Industrial'noi Matematiki
2011, vol. 14, No 1 (45)
Contents
UDC 517.95
Alekseev G. V., Brizitsky R. V.
A theoretical analysis of boundary control extremal problems for the Maxwell equationsWe study the extremal problems of multiplicative boundary control for the time-harmonic Maxwell equations considered with the impedance boundary condition for the electric field. We prove the solvability of the original extremal problem and derive sufficient conditions for the original data to guarantee the stability of solutions to particular extremal problems under perturbations of both the quality functional and one of the known functions which has the meaning of the electric current density.
Keywords: Maxwell equations, impedance, boundary control, solvability, stability estimates
Pp. 3–16.Alekseev Gennadiy Valentinovich
Brizitsky Roman Viktorovich
Institute of Applied Mathematics FEB RAS, 7 Radio str., Vladivostok 690041 RUSSIA, E-mail: alekseev@iam.dvo.ru
UDC 517.9
Andreev V. K., Stepanova I. V.
On a convective flow of a binary mixture in a vertical flumeWe study an invariant solution to the heat-diffusion convection equations describing the stationary flow of a binary mixture in a vertical flume under the pressure gradient and buoyancy force depending nonlinearly on temperature and concentration. We establish a series of general properties of this solution and prove an existence theorem. We analyze a numerical solution to the problem with polynomial and exponential dependence of buoyancy on its argument.
Keywords: binary mixture, heat diffusion, invariant solution
Pp. 17–26.Andreev Viktor Konstantinovich
Stepanova Irina Vladimirovna
Institute of Computational Modelling SB RAS, 50 bldg. 44 Akademgorodok, Krasnoyarsk 660036, RUSSIA, E-mail: andr@icm.krasn.ru; stepiv@icm.krasn.ru
UDC 517.9
Anikonov Yu. E., Neshchadim M. V.
On analytical methods in the theory of inverse problems for hyperbolic equations. IWe give new representations for the solutions and coefficients of, equations of mathematical physics, in the form of differential-algebraic identities. These representations are partially used to study higher-dimensional inverse problems.
Keywords: inverse problems of mathematical physics, analytic solution methods
Pp. 27–39.Anikonov Yury Evgen'evich
Neshchadim Mikhail Vladimirovich
Institute of Mathematics SB RAS, 4 Acad. Koptyug,
Novosibirsk State University, 2 Pirogova str., Novosibirsk 630090, RUSSIA, E-mail: anikon@math.nsc.ru; neshch@math.nsc.ru
UDC 539.3
Bogan Yu. A.
On the anisotropy of multilayer nanotubes and high-modulus fibersWe try to explain the essential anisotropy of boron and carbon fibers by the presence of primary slippage planes, microscopic inhomogeneity of the material, and under contact conditions of iscous friction type between the layers of the fiber. Using Bakhvalov's method we consider the averaging problem for a multilayer carbon nanotube with cylindrical anisotropy.
Keywords: averaging, composite, nanotube
Pp. 40–45.
Bogan Yury Aleksandrovich
Lavrent'ev Institute of Hydrodynamics SB RAS, 15 Lavrent'ev prospect, Novosibirsk 630090, RUSSIA, E-mail: bogan@hydro.nsc.ru
UDC 517.938.5
Dement'ev N. P.
Quasistationary solutions in economic systems with changing technologyStationary solutions are important in the analysis of the models of economic dynamics models with constant parameters. We isolate two classes of dynamical systems with variable parameters, for which we prove the existence of special solutions preserving certain properties of stationary solutions (the uniform boundedness for instance).
Keywords: economic dynamics models, stationary solutions, variable parameters, quasistationary solutions, hyperbolic points, sustainable development
Pp. 46–55.Dement'ev Nikolai Pavlovich
Institute of Economics and Industrial Engineering SB RAS, 17 Acad. Lavrent'ev, Novosibirsk 630090, RUSSIA, E-mail: dement@ieie.nsc.ru
UDC 517.9
Denisenko V. V.
The energy method for calculating quasistationary atmospheric electric fieldsWe propose a new mathematical model for describing quasistationary atmospheric electric fields approximately accounting for the conductivity of the ionosphere. We use a two-dimensional model, usual for large-scale fields, of the ionospheric conducting layer under the assumption that the Earth magnetic field is vertical. In the framework of this model we can describe the ionosphere using a special boundary condition in the boundary value problem for the atmospheric electric field. We formulate a linear boundary value problem with a symmetric positive definite elliptic operator. We justify a minimum principle for a quadratic energy functional. We prove the existence and uniqueness of generalized solutions. We analyze the error of our approximate description of the ionospheric conductor.
Keywords: electric field, atmosphere, ionosphere, elliptic operator, energy functional
Pp. 56–69.
Denisenko Valery Vasil'evich
Institute of Computational Modelling SB RAS, 50 bldg. 40 Akademgorodok, Krasnoyarsk 660036 RUSSIA E-mail: denisen@icm.krasn.ru
UDC 513:517.9
Garipov R. M.
The best-on-average quasiconformal mappingsWe seek the best-on-average quasiconformal mappings as the extremals of the functional equal to the integral of the conformality squared and multiplied by a particular weight. The inverse mapping to an extremal is an extremal of the same functional. We obtain necessary and sufficient conditions for the ellipticity in the sense of Petrovsky of the system of Euler equations for the extremal. We prove the local unique solvability of boundary problems for this system in the two-dimensional case. In the general case we prove the unique solvability of boundary problems for the system linearized at the identity mapping.
Keywords: quasiconformal mapping, extremal of a functional, embedding theorem, ellipticity in the sense of Petrovsky
Pp. 70–82.Garipov Ravil' Mukhamedzyanovich
Lavrent'ev Institute of Hydrodynamics SB RAS, 15 Acad. Lavrent'ev prospect, Novosibirsk 630090 RUSSIA, E-mail: R.M.Garipov@mail.ru
UDC 517.929.4
Matveeva I. I., Shcheglova A. A.
Estimates for solutions to differential equations with delayed argument and parametersWe consider systems of quasilinear differential equations with delayed argument and periodic coefficients and parameters. We establish sufficient conditions for the asymptotic stability of zero solutions and obtain estimates for the solutions characterizing their decay rates at infinity.
Keywords: equations with delayed argument, periodic coefficients, asymptotic stability, Lyapunov–Krasovskiy functional
Pp. 83–92.Matveeva Inessa Izotovna
Institute of Mathematics SB RAS, 4 Acad. Koptyug prospekt,
Novosibirsk State University, 2 Pirogova str. Novosibirsk 630090 RUSSIA, E-mail: matveeva@math.nsc.ru;
Shcheglova Alla Arkad'evna
Institute Systems Dynamics and Control Theory SB RAS, 134 Lermontova str. Irkutsk 664033 E-mail: shchegl@icc.ru
UDC 519.958:531.327.13
Nazarov S. A.
Localization of surface waves by small perturbations of the boundary of a semisubmerged bodyUnder a symmetry assumption we show that by forming a thin groove on a flat horizontal surface of a body in a cylindric channel we can achieve the following effect in the linear problem of waves in water: on every arbitrarily short interval (0, d) of the continuous spectrum, an arbitrary given number of eigenvalues is formed, which generate ''localized'' solutions, i.e., belonging to a Sobolev space.
Keywords: surface wave, trap modes, localized solutions, singular perturbations of the boundary
Pp. 93–101.Nazarov Sergei Aleksandrovich
Institute of Machine Science Problems, 61 Sredni pr. Vasil'evsky island, Saint-Petersburg 119178 RUSSIA, E-mail: srgnazarov@yahoo.co.uk
UDC 517.977.5
Sorokin S .P.
Sufficient optimality conditions in the form of the Pontryagin maximum principle in control problems for hybrid systemsWe generalize the sufficient conditions of the classical optimality theory to a class of optimal control problems for hybrid systems. For the cases of global and strong local extreme we obtain general sufficient optimality conditions and sufficient conditions in the form of the Pontryagin maximum principle. All results rest on dealing with exterior approximations of the attainability sets of controlled systems which are constructed using the solution sets to one of the Hamilton–Jacobi inequalities (strongly monotone functions of Lyapunov type).
Keywords: Hamilton–Jacobi inequality, sufficient optimality conditions, maximum principle, hybrid system
Pp. 102–126.Sorokin Stepan Pavlovich
Institute of Systems Dynamics and Control Theory SB RAS, 134 Lermontova str. Irkutsk 664033 RUSSIA E-mail: sorsp@mail.ru
UDC 539.3:517.958
Khludnev A. M.
On bending an elastic plate with a laminated thin rigid inclusionWe study the problem of bending an elastic plate with a thin rigid inclusion. We assume that the inclusion can laminate, thereby forming a crack. We find a system of boundary conditions holding on the sides of a crack. We prove the existence of solutions. We consider also the problem of bending a plate with solid rigid inclusion. We establish that the solutions to this problem converge to the solution to the original problem as the size of the inclusion tends to zero.
Keywords: plate, bending, rigid inclusion, crack, lamination.
Pp. 113–126.Khludnev Aleksandr Mikhailovich
Lavrent'ev Institute of Hydrodynamics SB RAS, 15 Acad. Lavrent'ev prospekt,
Novosibirsk State University 2 Pirogova str. Novosibirsk 630090 Russia, E-mail: khlud@hydro.nsc.ru
UDC 532.546
Khuzhaerov B. Kh., Makhmudov Zh. M., Zikiryaev Sh. Kh.
Pollutant transfer in water-bearing strata with accounting for two-site adsorptionWe consider the problem of pollutant transfer in a porous medium consisting of two zones (with moving and motionless water) taking into account convective transfer, hydrodynamic dispersion, two-site adsorption, and internal diffusive mass transfer between the two zones. Basing on a numerical solution to the problem, we determine the distribution of concentration of the substance in the moving water zone, the amount of the adsorbed substance (nonequilibirum, equilibrium, and general), and internal diffusive mass transfer for various combinations of non-equilibirum and equilibrium adsorption. Along with linear kinetics, we study the nonlinear kinetics of adsorption and internal diffusive mass transfer. We establish that with the growing share of non-equilibirum adsorption the rate and total amount of the adsorbed substance decrease, which leads to preceding advance of concentration profiles in the porous medium. In case the conditions are preserved, passage to nonlinear kinetics strengthens adsorption and internal diffusive mass transfer.
Keywords: substance adsorption, internal mass transfer, hydrodynamic dispersion, moving and otionless fluid zones, substance transfer, porous medium, two-site adsorption
Pp. 127–139.Khuzhaerov Bakhtier Khuzhaerovich
Makhmudov Zhamol Makhmud
Zikiryaev Shavkat Khudoyarovich
Complex Research Institute of Regional Problems, 3 Timur Malik str. Samarkand 703000 Uzbekistan E-mail: b.khuzhayorov@uzpa.uz
UDC 517.958
Yarovenko I. P.
Numerical experiments with an inhomogeneity indicator in positron emission tomographyUsing numerical methods, we study the applicability of an inhomogeneity indicator in positron emission tomography. The signal recorded by a tomography is described in terms of an imitation model basing on the Monte Carlo method. We demonstrate the possibility of effectively using the inhomogeneity indicator for solving the problem under consideration. We present graphically some results of solving the reconstruction problem for the boundaries of unknown activity sources.
Keywords: positron emission tomography, inhomogeneity indicator, Compton scattering, Monte Carlo method
Yarovenko Ivan Petrovich
Pp. 140–149.
Institute of Applied Mathematics FEB RAS 7 Radio str. Vladivostok 690041 Russia E-mail: yarovenko@iam.dvo.ru