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Sibirskii  Zhurnal  Industrial'noi  Matematiki
2017,  vol. 20,  No 2 (70)

Contents

UDC 519.676
DOI 10.17377/sibjim.2017.20.201

Artemiev S. S., Yakunin M. A.
Parametric analysis of the oscillatory solutions to SDEs with Wiener and Poisson components by a Monte Carlo method

We investigate the influence of Wiener and Poisson random noises on the behavior of oscillatory solutions to systems of stochastic differential equations (SDEs) with the use of a Monte Carlo method. For linear and Van der Pol oscillators, we investigate the accuracy of the estimates of the functionals of numerical solutions to SDEs obtained by the generalized Euler explicit method. For the linear oscillator, the exact analytical expressions of the mathematical expectation and the variance of the solution to the SDE are obtained. These expressions allow us to investigate the dependence of the accuracy of the estimates of the moments of the solution on the values of the parameters of the SDE, the size of the integration step, and the size of the ensemble of the simulated trajectories of the solution. For the Van der Pol oscillator, the dependence of the frequency and the decay rate of the oscillations of the mathematical expectation of solution to the SDE on the values of the parameters of the Poisson component is investigated. The results of numerical experiments are presented.

Key words: stochastic differential equation, Poisson component, Monte Carlo method, generalized Euler method, stochastic oscillator.
Pp. 3-14.

Artemiev Sergey Semyonovich
Institute of Computational Mathematics and Mathematical Geophysics SB RAS
Akad. Lavrentyev ave., 6
Novosibirsk State University
Pirogova str., 2
630090 Novosibirsk
Yakunin Mikhail Aleksandrovich
Institute of Computational Mathematics and Mathematical Geophysics SB RAS

E-mail: ssa@osmf.sscc.ru; yma@osmf.sscc.ru

UDC 573.2:57.017.64
DOI 10.17377/sibjim.2017.20.202

Ayupova N. B., Golubyatnikov V. P.
A three-cell model of the initial stage of the development of one proneural cluster

We construct a 9-dimensional nonlinear dynamical system modeling the initial stage of the interaction of three adjacent cells in a proneural cluster of Drosophila melanogaster. Conditions of existence of three stable equilibrium points in the phase portrait of this system are obtained, other equilibrium points are listed, a biological interpretation is given.

Key words: nonlinear dynamical system, gene network, negative and positive feedback, phase portrait, equilibrium point, stability.
Pp. 15-20.

Ayupova Nataliya Borisovna
Golubyatnikov Vladimir Petrovich

Sobolev Institute of Mathematics SB RAS
Akad. Koptyuga ave., 4
Novosibirsk State University
Pirogova str., 2
630090 Novosibirsk

E-mail: ayupova@math.nsc.ru; glbtn@math.nsc.ru

UDC 517.958
DOI 10.17377/sibjim.2017.20.203

Baranovskii E. S.
On weak solutions to evolution equations of viscoelastic fluid flows

We study the system of nonlinear equations describing unsteady flows of a viscoelastic fluid of Oldroyd type in a bounded three-dimensional domain with mixed boundary conditions. On one part of the boundary, the Navier slip condition is given, while on the other one, the no-slip condition is used. We prove the theorem on the existence, uniqueness, and energy estimates for weak solutions.

Key words: initial boundary-value problem, weak solution, viscoelastic fluid, Oldroyd model, Navier slip boundary condition.
Pp. 21-32.

Baranovskii Evgenii Sergeevich
Voronezh State University
Universitetskaya pl., 1
394036 Voronezh

E-mail:esbaranovskii@gmail.com

UDC 519.688
DOI 10.17377/sibjim.2017.20.204

Vasil'ev V. I., Vasil'eva M. V., Laevskii Yu. M., Timofeeva T. S.
Numerical simulation of two-phase fluid filtration in heterogeneous media

The paper deals with the numerical solution of problems of two-phase filtration. The statement of the problem is given in terms of the velocity, pressure and saturation. To approximate the velocity and pressure, we use the mixed finite element method. For discretizating the convective term in the equation for saturation, we apply the flow schemes. The results of the numerical solution of the model problem for heterogeneous media are presented.

Key words: filtration, pressure, velocity, saturation, velońity, heterogeneous medium, finite element method, flow scheme.
Pp. 33-40.

Vasiliev Vasiliy
Vasilyeva Maria

North-East Federal University
Kulakovsky str., 48
677027 Yakutsk
Laevsky Yuri
Institute of Computational Mathematics and Mathematical Geophysics SB RAS
Akad. Lavrentyev ave., 6
630090 Novosibirsk
Timofeeva Tatiana
North-East Federal University

E-mail: vasvasil@mail.ru; vasilyevadotmdotv@gmail.com; laev@labchem.sscc.ru

UDC 539.3:517.95
DOI 10.17377/sibjim.2017.20.205

Nazarova L. A., Nazarov L. A., Vandamme M., Pereira J.-M.
Direct and inverse problems of gas emission and the sorptive deformation of coal beds

Using equations of state for fractured-porous media to describe the sorption-induced deformation of coal, we develop a geomechanical model for radial gas influx to a borehole drilled in a coal bed with the concurrent evolution of the stress field in the borehole environment. A numerical and analytical method is put forward for solving the corresponding system of equations for poroelastic media. The correlation is found between the volume of slack withdrawn from the borehole (when opening up gas-bearing seams), the sorption-and-storage capacities of coal, the permeability $k$, and the natural horizontal stress $\sigma_h$. The solvability is shown of the inverse boundary-coefficient problem on $k$ and $\sigma_h$ from the shut-in well pressure. The express-method for estimating the permeability from pressure measurements taken in a borehole operating in pressure drop mode is justified.

Key words: coal bed, fracture-porous medium, filtration, permeability, intact rock mass stress, borehole, inverse problem.
Pp. 41-49.

Nazarova Larisa Alekseevna
Nazarov Leonid Anatol'evich

Chinakal Institute of Mining SB RAS
Krasny ave., 54
630091 Novosibirsk
Novosibirsk State University
Pirogova str., 2
630090 Novosibirsk
Vandamme  M.
Pereira J.-M.
Universite Paris-Est, Laboratoire Navier (UMR 8205), CNRS, ENPC, IFSTTAR
F-77455, Marne-la-Vallee, France

E-mail: naz@misd.nsc.ru; larisa@misd.nsc.ru

UDC 621.316
DOI 10.17377/sibjim.2017.20.206

Nekrasov S. A.
The method of accelerated statistical modeling and its application in problems with irremovable singularity

We study a number of known methods for solving stochastic problems on the basis of statistical tests (a Monte Carlo method). Aiming at a comparative study of the efficiency of these methods, we solve a number of problems in the theory of technical systems with inaccurately given and random parameters and characteristics.

Key words: simulation, Monte Carlo method, the method of accelerated statistical modeling, comparison of efficiency.
Pp. 50-58.

Nekrasov Sergey Aleksandrovich
South Russian State Technical University
Prosveshcheniya str., 132
346428 Novocherkassk

E-mail:nekrasoff_novoch@mail.ru

UDC 539.375
DOI 10.17377/sibjim.2017.20.207

Neustroeva N. V., Lazarev N. P.
The derivative of the energy functional in an equilibrium problem for a Timoshenko plate with a crack on the boundary of an elastic inclusion

We consider an equilibrium of a composite plate containing a through vertical crack of variable length on the separation boundary between the matrix and the elastic inclusion. The deformation of the matrix is described by the Timoshenko model, and the deformation of the elastic inclusion is described by the Kirchhoff—Love model. We obtain a formula for the derivative of the energy functional with respect to the crack length.

Key words: plate, crack, nopenetration condition, elastic inclusion, derivative of the energy functional.
Pp. 59-70.

Neustroeva Natalia Valerianovna
North-Eastern Federal University (NEFU)
Institute of Mathematics and Informatics
Kulakovskogo str., 48
677000 Yakutsk
Lavrentyev Institute of Hydrodynamics SB RAS
Acad. Lavrentyev ave., 15
630090 Novosibirsk
Lazarev Nyurgun Petrovich
North-Eastern Federal University (NEFU)
Scientific Research Institute of Mathematics
Kulakovskogo ave., 48
677000 Yakutsk

E-mail: nnataliav@mail.ru; nyurgun@ngs.ru

UDC 517.958
DOI 10.17377/sibjim.2017.20.208

Panov A. V.
Exact solutions to the equations of the dynamics of a two-phase medium. Collapse of a gas and particles in space

We consider a system of partial differential equations describing the dynamics of a two-phase medium. We find exact solutions to this system, solutions partially invariant under some four-dimensional subalgebras of rank and defect 1 for some four-dimensional subalgebras. The phenomenon of collapse (instantaneous source) in a two-phase medium is described.

Key words: two-phase medium, exact solution, symmetry Lie algebra, partially invariant solution, collapse.
Pp. 71-82.

Panov Alexandr Vasil'evich
Chelyabinsk State University
Br. Kashirinykh str., 129
454001 Chelyabinsk

E-mail: gjd@bk.ru

UDC 532.5.013.4
DOI 10.17377/sibjim.2017.20.209

Rezanova E. V.,  Shefer I. A.
Influence of the thermal load on the characteristics of a flow with evaporation

We study two-layer flows of liquid and gas in a horizontal channel under given gas flow rate. Evaporation is taken into account on the thermocapillary interface. An exact solution is constructed to Navier-Stokes equations in the Boussinesq approximation taking into account the Dufour effect in the gas-vapor layer. In the framework of the linear theory, we study the stability of the obtained solutions and the characteristics of the arising perturbations. The influence of the width of the liquid layer and the longitudinal temperature gradient on the strucuture of the principal flow and the perturbations is studied.

Key words: exact solution, Dufour effect, thermocapillary interface, stability.
Pp. 83-92.

Rezanova Ekaterina Valerievna
Altai State University
Lenina ave., 61
656049 Barnaul
Kutateladze Institute of Thermophysics SB RAS
Akad. Lavrentyev ave., 1
630090 Novosibirsk
Shefer Ilya Aleksandrovich
Institute of Mathematics and Computer Science
Siberian Federal University
Svobodny ave., 79
660041 Krasnoyarsk

E-mail: katerezanova@mail.ru; ilya.shefer@gmail.com

UDC 536.516
DOI 10.17377/sibjim.2017.20.210

Sennitskii V. L.
Predominantly unidirectional rotation of a solid body and a viscous liquid

Two problems are considered on the time periodical motion of a hydrodynamical system consisting of a viscous liquid and solid bodies bordering it. A new hydrodynamical effect is revealed.

Key words: viscous liquid, free solid body, fixed solid body, nonunidirectional oscillatory influences, stationary rotation.
Pp. 93-97.

Sennitskii Vladimir Leonidovich
Lavrentyev Institute of hydrodynamics of SB RAS
Acad. Lavrentyev ave., 15
Novosibirsk State University
Pirogova str., 2
630090 Novosibirsk

E-mail:sennitskii@yandex.ru

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