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Sibirskii  Zhurnal  Industrial'noi  Matematiki
2017,  vol. 20,  No 1 (69)

Contents

UDC 519.624:534.1
DOI 10.17377/sibjim.2017.20.101

Aitbaeva A. A.,  Ahtyamov A. M.
Identification of the fixedness and loadedness of an end of an Euler—Bernoulli beam from its natural vibration frequencies

We consider a uniform Euler—Bernoulli beam whose left end is fixed and  whose right end has a load fixed by two springs. Being hit, the beam starts
vibrating. The aim of the paper is to determine the fixedness parameters (the spring stiffness coefficients) and the loadedness parameters (the mass and moment of inertia of the load) of the right end of the beam from the natural frequencies of its bending vibrations. We show that the four unknown parameters of the boundary conditions on the right end of the beam are uniquely determined by five natural frequencies of  bending vibrations. A counterexample is exhibited demonstrating that four natural frequencies are insufficient for the unique identification of these four nonnegative parameters.

Keywords: eigenvalue, natural frequency, beam, inverse problem, point inertia element, spring stiffness coefficient.
Pp. 3-10.

Akhtyamov Azamat Muhtarovich
Bashkir State University
str. Zaki Validi, 32, 450076  Ufa
Aitbaeva Aigul Azamatovna
Mavlyutovs Institute of Mechanics, Ufa Scientific Center RAS
October prosp., 71, 450054 Ufa
E-mail: akhtyamovam@mail.ru; phunakoshi@mail.ru

UDC 519.865.3
DOI 10.17377/sibjim.2017.20.102

Aizenberg N. I.,  Zorkaltsev V. I.,  Mokryi I. V.
A study into unsteady oligopolistic markets

We study an oligopolistic market by a simulation model in continuous time. The suppliers (oligopolists) use two rules of behavior which are determined from systems of ordinary differential equations. The scenario determines the stategy of  a supplier including the possibility of the change of behavior during  interaction. We shown that there are possible stationary states when one of the suppliers gets benefits from changing  behavior, after which his income will increase as compared with the incomes of the other actors.

Keywords: simulation model, oligopoly, supplier, consumer, market equilibrium.
Pp. 11-20.

Aizenberg Natalia Ilinichna
Zorkaltsev Valeriy Ivanovich
Mokry Igor Vladimirovich
Melentiev Energy Systems Institute SB RAS
Lermontov str., 130, 664033 Irkutsk
E-mail: ayzenberg.nata@gmail.com; zork@isem.irk.ru; ygr@isem.irk.ru

UDC 510.5
DOI 10.17377/sibjim.2017.20.103

Gasenko V. G.
A differential Fourier method

We propose two new discrete sine and cosine differential Fourier transforms of a complex vector which are based on the finite-difference solution
of inhomogeneous harmonic differential equations of the first order with complex coefficients and of the second order with real coefficients respectively. In basic form, the differential Fourier methods need  less arithmetic operations as compared to the classical discrete Fourier transform method. The matrix of the sine differential Fourier transform is a complex matrix with alternating real and imaginary entries, and the matrix of the cosine transform is real. As in the classical case, both matrices transform into cyclic convolution matrices, and to them we can apply all fast convolution algorithms including the Winograd and Rader algorithms.
The differential Fourier methods are compatible with the Good—Thomas fast Fourier transform algorithm and, if combined with fast convolution algorithms, it
can potentially be faster than all known methods of acceleration of the fast Fourier transform.

Keywords: discrete Fourier transform, fast Fourier transform, harmonic differential equation, Good—Thomas algorithm, Winograd method.
Pp. 21-30.

Gasenko Vladimir Georgievich
Institute of Thermophysics SB RAS
Acad. Lavrentyev ave., 1, 630090 Novosibirsk

E-mail: gasenko48@mail.ru

UDC 629.12:539.3:532.5
DOI 10.17377/sibjim.2017.20.104

Greshilov A. G.,  Sukhinin S. V.
Chladni figures of a circular plate floating in bounded and unbounded water bodies with securing support at the center

In the framework of circular symmetry, we carry out numerical and analytical research into the Chladni modes an elastic plate that floats on the surface of a fluid and is cantilever fitted at the center to a vertical support. Using the theory of long waves in shallow water and the approximation of the vibrations of an Euler beam for bounded and unbounded water bodies, we obtain the expressions for the dependency of the natural and quasi-natural frequencies of the Chladni figures on the geometric parameters of the plate and the vibration domain with the bottom irregularity taken into account.

Keywords: flexural-gravity vibration, natural vibration, hydroelasticity, shallow water, circular plate, Chladni figures of a supported floating plate.
Pp. 31-40.

Greshilov Aleksei G.
Sukhinin Sergey V.
Lavrent'ev Institute of Hydrodynamics SB RAS
Acad. Lavrentyev ave., 15, 630090 Novosibirsk
E-mail: algreshilov@mail.ru; sukhinin@hydro.nsc.ru

UDC 517.9
DOI 10.17377/sibjim.2017.20.105

Zhalnina A. A.,  Kucher N. A.
Dependence on the domain of solutions to a boundary value problem for the equations of mixtures of compressible viscous fluids

We study the dependence of solutions to an inhomogeneous boundary value problem for the equations of mixtures of compressible viscous fluids on the shape of the flow domain. The results can be used to prove the differentiability of the solutions and functionals of the solutions (for example, the drag functional) with respect to the parameter defining the shape variations of an obstacle in the flow.

Keywords: mixture of viscous compressible fluids, flow around an obstacle, transposed problem, boundary value problem.
Pp. 41-52.

Zhalnina Alexandra Anatol'evna
Kucher Nikolay Alekseevich
Kemerovo State University
Krasnaya Str., 6, 650043 Kemerovo
E-mail: nakycher@rambler.ru; qwert1776@yandex.ru

UDC 539.3:539.4:001.891.573
DOI 10.17377/sibjim.2017.20.106

Karpov V. V.,  Semenov A. A.
Mathematical models and algorithms for  studying the strength and stability of shell structures

We describe several mathematical models of deformation of supported shell structures including those that take into account various properties of the material. We consider linear-elastic and physically nonlinear problems and also creep problems for structures of orthotropic and isotropic materials. All models are based on the functional of the total potential energy of the deformation of the shell. The geometric nonlinearity and transverse shifts are accounted for. Ribs are introduced as discretely as by the structural anisotropy method. We demonstrate three different algorithms for the study of the strength and stability of the shells under consideration each of which is most effective for its range of tasks.

Keywords: mathematical model, physical nonlinearity, creep, orthotropy, shell, strength, stability, algorithm, Ritz method, structural anisotropy method.
Pp. 53-65.

Karpov Vladimir Vasilevich
Semenov Alexey Aleksandrovich
Saint-Petersburg State University of Architecture and Civil Engineering
2-nd Krasnoarmeyskaya str., 4, 190005 Saint-Petersburg

E-mail: vvkarpov@lan.spbgasu.ru; sw.semenov@gmail.com

UDC 517.95
DOI 10.17377/sibjim.2017.20.107

Neshchadim M. V.
Functional-invariant solutions to the Maxwell system

We consider the problem of the existence of functional-invariant solutions to the Maxwell system. The solutions found contain functional arbitrariness which is used for dertermining the parameters of the Maxwell system (the dielectric and magnetic constants).

Keywords: Maxwell equations, functional-invariant solution.
Pp. 66-74.

Neshchadim  Mikhail Vladimirovich
Sobolev's Institute of Mathematics SB RAS
Acad. Koptyuga ave.,  4
Novosibirsk State University
Pirogova str.,  2, 630090 Novosibirsk
E-mail: neshch@math.nsc.ru

UDC 517.958
DOI 10.17377/sibjim.2017.20.108

Prokhorov I. V., Sushchenko À. À., Kim A.
An initial boundary value problem for the radiative transfer equation with diffusion matching conditions

We consider the Cauchy problem for the nonstationary equation of radiative transfer with generalized matching conditions describing the diffusion reflection and refraction on the separation boundary of the media. We prove  solvability of the initial boundary value problem and obtainh stabilization conditions for an unsteady solution.

Keywords: diffusion matching conditions, integro-differential equation, nonstationary equation, Cauchy problem, Hille—Yosida theorem.
Pp. 75-85.

Prokhorov Igor Vasilyevich
Sushchenko Àndrei Àndreyevich
Institute of Applied Mathematics Far Eastern Branch RAS
Radio str., 7, 690041 Vladivostok
Far Eastern Federal University
Sukhanova str., 8, 690950 Vladivostok
Kim Anton
Far Eastern Federal University
E-mail: prokhorov@iam.dvo.ru; sushchenko.aa@dvfu.ru; kim_a@students.dvfu.ru

UDC 536.37:538.36
DOI 10.17377/sibjim.2017.20.109

Khabibullin I. L., Nazmutdinov F. F., Vakhitova N. K.
Modeling  the heating of dielectric media by electromagnetic radiation in nonlinear mode

We consider the process of  heating  a propagating medium by high-frequency electromagnetic radiation in the presence of heat exchange with the ambient medium under the approximation of a thermally thin layer. It is shown that, due to competition between the processes of heat dissipation in the medium being heated and heat exchange with the ambient medium, the temperature profiles are realized in autowave form. Essential principles of the dynamics of temperature waves are established by inspecting the analytical and numerical solutions.

Keywords: dielectric heating, autowave mode, numerical simulation, heat exchange.
Pp. 86-92.

Khabibullin Ildus Lutfurakhmanovich
Nazmutdinov Florid Fauzievich
Vakhitova Naylya Kanzafarovna
Bashkir State University
Z. Validie str., 32, 450076 Ufa
E-mail: Habibi.bsu@mail.ru; mmx_@mail.ru; vnk-55@mail.ru

UDC 539.3:517.958
DOI 10.17377/sibjim.2017.20.110

Khludnev A. M.
Asymptotics of anisotropic weakly curved inclusions in an elastic body

We study boundary value problems that describeg the equilibrium for two-dimensional elastic bodies with thin weakly curved anisotropic inclusions. The presence of an inclusion means the existence of a crack between the inclusion and the elastic matrix. Nonlinear boundary conditions in the form of inequalities
are imposed on the crack faces to prevent their mutual penetration, which leads to formulating the problems as problems with unknown contact domain. Limit passages are investigated over the rigidity parameters of the thin inclusions. In particular, we construct the models obtained by letting the rigidity parameters tend to infinity and analyze their properties.

Keywords: thin inclusion, elastic body, crack, nonlinear boundary condition, limit model.
Pp. 93-104.

Khludnev Alexander Mikhailovich
Lavrentyev Institute of Hydrodynamics SB RAS
Acad. Lavrentyev ave., 15
Novosibirsk State University
Pirogova str., 2, 630090 Novosibirsk
E-mail: khlud@hydro.nsc.ru

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