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Sibirskii  Zhurnal  Industrial'noi  Matematiki
2016,  vol. 19,  No 2 (66)

Contents

UDC 517.95
DOI 10.17377/sibjim.2016.19.201

Alekseev G. V.,  Brizitskii R. V.,  Saritskaya Zh. Yu.
Stability estimates of solutions to extremal problems for a nonlinear convection-diffusion-reaction equation

We consider an identification problem for a stationary nonlinear convection-diffusion-reaction equation in which the reaction coefficient
depends nonlinearly on the concentration of the substance. This problem is reduced to an inverse extremal problem by  optimization. Solvability is proved of the boundary value problem and the optimization problem. In the case that the reaction coefficient is quadratic when the equation
acquires cubic nonlinearity, we deduce the conditions of  optimality. Analyzing the latter, we establish estimates of the local stability of solutions to the eoptimization problem under small perturbations both of the cost functional and of the given velocity vector that occurs multiplicatively in the convection-diffusion-reaction equation.

Keywords: nonlinear convection-diffusion-reaction equation, Dirichlet problem, optimal control problem, solvability, optimality conditions, stability estimate.
Pp. 3-16.

Alekseev Gennadii Valentinovich
Brizitskii Roman Victorovich
Institute of Applied Mathematics FEB RAS
7 Radio str.
690041 Vladivostok
Far Eastern Federal University
8 Sukhanova str.
690950 Vladivostok
Saritskaya Zhanna Yurievna
Far Eastern Federal University

E-mail: alekseev@iam.dvo.ru; mlnwizard@mail.ru; zhsar@icloud.com

UDC 517.946
DOI 10.17377/sibjim.2016.19.202

Aliev R. A.
Finding the coefficients of a linear elliptic equation

We study the inverse problems of finding the coefficients of a linear elliptic equation for various boundary conditions in a prescribed rectangle.
Existence, uniqueness, and stability theorems are proved for solutions to the inverse problems for the particular statements studied in the paper. An iterative method is employed to construct a regularization algorithm for solving the inverse problems.

Keywords: inversion problem, elliptic equation.
Pp. 17-28.

Aliev Ramiz Atash ogly
Azerbaijan Cooperation University
8v Narimanov str.
AZ1106 Baku, Azerbaijan

E-mail: ramizaliyev3@rambler.ru

UDC 548.1
DOI 10.17377/sibjim.2016.19.203

Garipov R. M., Kotelnikova M. S.
Symmetry of finite crystalline solids

The symmetry of finite crystalline solids is described by generalized Fourier series invariant under finite groups. The electronogram of the Shechtman icosahedral phase is calculated.

Keywords: group, electron, material wave, icosahedral phase.
Pp. 29-36.

Garipov Ravil' Mukhamedzyanovich
Kotelnikova Maria Stanislavovna
Lavrent'ev Institute of Hydrodynamics SB RAS
15 Lavrent'ev ave.
630090 Novosibirsk

E-mail: R.M.Garipov@mail.ru; Kotelnikova@hydro.nsc.ru

UDC 517.958:531.33
DOI 10.17377/sibjim.2016.19.204

Gerus A. A., Gritsenko S. A., Meirmanov A. M.
The deduction of the homogenized model of isothermal acoustics in a heterogeneous medium in the case of two different poroelastic domains

We consider a mathematical model of isothermal acoustics in a composite medium consisting of two different porous soils (poroelastic domains) with common boundary. Each of the domains has its own characteristics of the solid skeleton; the fluid filling the pores is the same for both domains. The differential
equations of the accurate model contain rapidly oscillating coefficients. We deduce homogenized equations (i.e. equations not containing rapidly oscillating coefficients).

Keywords:  composite medium, periodic structure, isothermal Stokes equations, acoustics equations, poroelasticity, homogenization of periodic structures, two-scale convergence.
Pp. 37-46.

Gerus Artur Aleksandrovich
Gritsenko Svetlana Aleksandrovna
Meirmanov Anvarbek Mukatovich
Belgorod State University
85 Pobedy str.
308015 Belgorod

E-mail: artur-gerus@mail.ru; sv.a.gritsenko@gmail.com; anvarbek@list.ru

UDC 517.925.5:517.929
DOI 10.17377/sibjim.2016.19.205

Demidenko G. V., Uvarova I. A.
On a class of systems of ordinary differential equations of large dimension

We consider the Cauchy problem for a class of systems of ordinary differential equations of large dimension. We prove that, for a sufficiently many equations, the last component of the solution to the Cauchy problem is an approximate solution to an initial value problem for a delay differential equation. Estimates of the approximation are obtained.

Keywords: system of ordinary differential equations of large dimension, limit theorem, delay differential equation.
Pp. 47-60.

Demidenko Gennadii Vladimirovich
Sobolev Institute of Mathematics SB RAS
4 Koptyug ave.
Novosibirsk State University
2 Pirogova str.
630090 Novosibirsk
Uvarova Irina Alekseevna
Sobolev Institute of Mathematics SB RAS

E-mail: demidenk@math.nsc.ru; sibirochka@ngs.ru

UDC 533.6
DOI 10.17377/sibjim.2016.19.206

Kovenya V. M., Kudryashov A. S.
A factorization method for  numerical solution of the Navier—Stokes equations of a  incompressible viscous fluid.

We propose an implicit differential scheme of approximate factorization for numerical solution of the Navier—Stokes equations
of an incompressible fluid in curvilinear coordinates. The algorithm is tested on the solution of the problems for Couette and Poiseuille flows. The results are presented of  numerical modeling of a flow between rotating covered cylinders.

Keywords: Navier—Stokes equations, incompressible fluid, difference scheme, splitting method.
Pp. 61-74.

Kovenya Victor Mikhailovich
Institute of Computational Technologies SB RAS
15 Lavrent'ev ave.
Kudryashov Anton Sergeevich
Novosibirsk State University
2 Pirogova str.
630090 Novosibirsk

E-mail: kovenya@ict.nsc.ru; qubabox@mail.ru

UDC 539.375
DOI 10.17377/sibjim.2016.19.207

Rudoy E. M.
Numerical solution of an equilibrium problem for an elastic body with a delaminated thin rigid inclusion

Uder cinsideration is the 2D elasticity problem for a body with a thin rigid inclusion. We assume that there is a crack between the rigid
inclusion and the elastic matrix. Some boundary conditions for the nonpenetration of crack faces in the form of inequalities are imposed. We propose a numerical method for solution that uses  decomposition of the domain and the Uzawa algorithm for variational inequalities. An example is exhibited of numerical computation by the finite element method.

Keywords: delamination crack, thin rigid inclusion, nonpenetration condition, variational inequality, domain decomposition method, Uzawa algorithm.
Pp. 74-87.

Rudoy Evgeniii Mikhailovich
Lavrent'ev Institute of Hydrodynamics SB RAS
15 Lavrent'ev ave.
Novosibirsk State University
2 Pirogova str.
630090 Novosibirsk

E-mail: rem@hydro.nsc.ru

UDC 519.63
DOI 10.17377/sibjim.2016.19.208

Savchenko À. O.,  Il'in V. P.,  Butyugin D. S.
A method of solving an exterior three-dimensional boundary value problem for the Laplace equation

We develop and experimentally study the algorithms for solving three-dimensional boundary value problems for the Laplace equation
in unbounded domains. The algoriths combinef the finite element method and the integral representation of the solution in homogeneous media. The proposed approach is based on  the Schwarz alternating method and the consecutive solution of the interior and exterior boundary value problems in subdomains with intersection such that  some iterable interface conditions are imposed on the adjacent boundaries. The convergence of the method is proved. The convergence rate of the iterative process is studied analytically in the case that  the subdomains are spherical layers with known exact representations of all consecutive approximations. In this model situation, the impact is analyzed of the parameters of the algorithm on the efficiency of the method . The above approach is implemented for solving a problem with a complicated configuration of the boundary. Also, the algorithbm uses high precision finite element methods for solving the interior boundary problems. The convergence rate of the iterations and the achieved accuracy of the computations are illustrated by a series of numerical experiments.

Keywords: Laplace equation, exterior boundary problem, Schwartz alternating method.
Pp. 88-99.

Savchenko Aleksandr Oliverovich
Il'in Valerii Pavlovich
Butyugin Dmitrii Sergeevich
Institute of Computational Mathematics and Mathematical Geophysics SB RAS
6 Lavrent'ev ave.
Novosibirsk State University
2 Pirogova str.
630090 Novosibirsk
E-mail: savch@ommfao1.sscc.ru; ilin@sscc.ru; dm.butyugin@gmail.com

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