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Sibirskii  Zhurnal  Industrial'noi  Matematiki
2012,  vol. 15,  No 3 (51)

Contents
 

  

UDC 519.621.2
Abdullaev V. M.
Solution of differential equations with nonseparated multipoint and integral conditions

We propose a numerical method for solving systems of linear nonautonomous ordinary differential equations with nonseparated multipoint and integral conditions. The method is based on the operation of the convolution of the integral conditions into local ones, which allows us to reduce the solution of the original problem to the solution of a Cauchy problem for ordinary differential equations and for a system of algebraic linear equations. We prove the stability of the computational schemes. Many numerical experiments on specially constructed test problems with the use of the formulas and schemes for numerical solution are proposed in the article. The experimental results showed a sufficiently high efficiency of this approach.

Keywords: system of ordinary differential equations, nonseparated conditions, integral conditions, nonlocal multipoint conditions, operation of successive shift.
Pp. 3–15.

Abdullaev Vaqif Maarif oglu
 Institute of Cybernetics NAS of Azerbaijan, 9 F. Agaev st., Baku, AZ1141. E-mail:  vaqif_ab@rambler.ru

 

UDC 517.958
Baranovsky E. S.
 An inhomogeneous boundary value problem for the stationary equations of the Jeffreys model for the motion of a viscoelastic medium.

We consider an inhomogeneous Dirichlet problem for the stationary equations of the motion of a viscoelastic medium of the Jeffreys type. We prove the solvability of this problem in a generalized (weak) formulation and establish the sequential weak closedness of the solution set.

Keywords: non-Newtonian fluid dynamics, Jeffreys model, viscoelastic medium, inhomogeneous boundary value problems, weak solution.
Pp. 16–23.

Baranovsky Evgeny Sergeevich
 Voronezh State University, 1 Universitetskaya Sq. Voronezh 394006. E-mail: bes220@rambler.ru

 

UDC 338.4.01
Gusev V. B.
A model for the diversification of production for a region of intensive subsoil development

We construct a dynamical model of the economic activities of a region specializing in the intensive development of mineral resources. The model takes into account the depletion of resources. Built-in mechanisms of autonomous control of the formation and raising of an alternative production can guarantee a sustainable development of the region also after the cessation of mining minerals.

Keywords:  multicommodity model, region of intensive development of mineral resources, autonomous control mechanism.
Pp. 24–36.

Gusev Vlasislav Borisovich
 Trapeznikov Institute of Control Sciences RAS, 65 Profsoyuznaya st. Moscow 117997. E-mail: gusvbr@ipu.ru

 

UDC 532.546:550.820.7
Korsakova N. K.
Numerical modeling of the ablation of impermeable inclusions of rocks in a flow of ground waters

We consider a mathematical model for the mass transfer process from the surface of impermeable inclusions in a potential flow of water. Using the finite element method, we solve the problem describing a stationary plane flow in a porous media that includes a massif of impermeable rock. Numerical computations are carried out.

Keywords: diapir, filtration, ablation, mathematical model, finite element method, numerical solutions.
Pp.  37–44.

Korsakova Nadezda Konstantinovna
 Lavrent'ev Institute of Hydrodynamics SB RAS. Pr. Lavrenteva, 15, 630090, Novosibirsk. E-mail: kors@hydro.nsc.ru

 

UDC 621.373:535
Kuzovatov I. A., Shamshurin A. V.
Numerical modeling of the process of orientation of two-level molecules in an external field using the method of exponential fitting

Numerical modeling is carried out of the process of orientation of molecules in an external field with account taken of transitions between the energy levels. An algorithm is proposed for the numerical solution of a system of nonstationary kinetic equations on the basis of exponential fitting. The possibility of an inversionless light amplification by dichroic molecules that can be oriented selectively by states by means of applying an external field.

Keywords: inversionless light amplification, molecule balance, exponential fitting.
Pp. 45–57.

Kuzovatov Igor Anatolevich
Shamshurin Aleksey Valerevich
Siberian Federal University 26 Akad. Kirenskogo st. Krasnoyarsk 666074 E-mail: kuzovatov@yandex.ru; shamshurin.alex@mailru

 

UDC 539.311
Lazarev N. P.
The problem of equilibrium of a shallow Timoshenko-type shell containing a through-thickness crack

We study the nonlinear problem of the equilibrium of an elastic shallow Timoshenko-type shell containing a through-thickness crack. Boundary conditions in the form of inequalities are imposed on the curve defining the crack.  We establish the unique solvability of the variational statement of the nonlinear problem of the equilibrium of the shell. We prove that, for sufficient smoothness of the solution, the initial variational statement is equivalent to the differential formulation of the problem. We deduce boundary conditions on the inner boundary that describes the crack. In the case of the zero disclosure of the crack, we prove the local infinite differentiability of the solution function with additional assumptions on the functions defining the curvatures of the shell and the external load.

Keywords: elastic shallow, Timoshenko-type shell, crack, variational problem, impenetrability condition.
Pp. 58–69.

Lazarev Nyurgun Petrovich
Institute of Mathematics SEFU 48 Kulakovskogo St. Yakutsk 677000
Lavrent'ev Institute of Hydrodynamics SB RAS 15 Lavrent'ev av. Novosibirsk 630090 E-mail: nyurgun@ngs.ru

  

UDC 541.128
Leshakov O. E., Mamash E. A.
Modeling of critical phenomena in a five-step reaction of the catalytic oxidation of carbon monoxide

We construct a model of the dynamics of the five-step catalytic reaction of the oxidation of carbon monoxide with the formation of an unreactive form of an intermediate substance (buffer step). We construct the line of multiplicity and the line of neutrality in the plane of the constants of the reaction rates. It is shown that the inclusion of the mechanism of reaction buffer stage affects the variety of dynamic modes.

Keywords: chemical kinetics, critical phenomena, the multiplicity of steady states.
Pp. 70–76.

Leshakov Oleg Eduardovich
Mamash Elena Aleksandrovna
 Tuva Institute for Complex Development of Natural Resources SB RAS 117a Internatsional'naya St. 667077, Kyzyl E-mail: o-leshakov@mail.ru; elenamamash@gmail.com

 

UDC 517.958
Nazarov A. L., Romanov V. G.
A uniqueness theorem for the inverse problem for the integrodifferential electrodynamics equations

The problem of finding a kernel and the index of dielectric permeability for the system of integro-differential equations of electrodynamics with wave dispersion is studied. We consider a direct problem in which the external pulse current is a dipole located at a point y on the boundary ∂B of the unit ball B. The point y runs over the whole boundary and is a parameter of the problem. The information available about the solution to the direct problem is the trace on ∂B of the solution to the Cauchy problem given for the time moments close to the time when a wave from the dipole source arrives at the point x. The main result is a uniqueness theorem for the solution of the inverse problem.

Keywords: electrodynamics, dispersion, inverse problem, uniqueness.
Pp. 77–86.

Nazarov Aleksei Leonidovich
 Novosibirsk State University of Agriculture 160 Dobrolyubova St. Novosibirsk 630039 E-mail:  alexey.l.nazarov@gmail.com
Romanov Vladimir Gavrilovich
Sobolev Institute of Mathematics SB RAS 4 Koptyug av. Novosibirsk 630090 E-mail: romanov@math.nsc.ru

UDC 539.3:517.958
Ostrosablin N. I.
Diagonalization of the system of Lame static equations of linear isotropic elasticity

We find the simplest representation of the general solution to the system of static Lame equations of linear isotropic elasticity in the form of a linear combination of the first derivatives of three functions that satisfy three independent harmonic equations. The representation depends on 12 free parameters choosing which it is possible to obtain various representations of the general solution and simplify the boundary value conditions for the solution of boundary value problems as well as the representation of the general solution for dynamic Lame equations. The system of Lame equations diagonalizes, i.e., is reduced to the solution of three independent harmonic equations. The representation implies three conservation laws and a formula for producing new solutions making it possible, given a solution, to find new solutions to the Lam\'e static equations by derivations.  In the two-dimensional case of a plane deformation, the so-found solution immediately implies the Kolosov–Muskhelishvili representation for shifts by means of two analytic functions of complex variable. Two examples are given of applications of the proposed method of diagonalization of two-dimensional elliptic systems.

Key words: linear elasticity, isotropic material, static Lame equation, general solution, diagonalization of an elliptic system, symmetry operators, conservation laws.
Pp. 87–98.

Ostrosablin Nikolai Il’ich
 Lavrent'ev Institute of Hydrodynamics SB RAS 15 Lavrent'ev av. Novosibirsk 630090 E-mail: abd@hydro.nsc.ru

 

UDC 519.86
Rapoport  E. O.
Discrete approximation of continuous measures and some applications

We study the best approximation (in the Kantorovich — Rubinshteuin metric) of continuous measures on the straight line by measures concentrated at finitely many points. An algorithm to obtain such measures is constructed and the questions of their existence and uniqueness are considered. Applications of the results to some problems of mathematical economics are studied.

Keywords: continuous measures, point measures, best approximation, migration resistance.
Pp. 99–110.

Rapoport  Ernest Osherovich
 Sobolev Institute of Mathematics SB RAS 4 Koptyug av. Novosibirsk 630090 E-mail: rapoport@math.nsc.ru

 

UDC 519.217.2:678.7
Savel'ev L. Ya., Balakin S. V.
Some applications of the stochastic theory of runs

We describe some natural characteristics of runs in binary Markov chains and their applications in different fileds. We calculate the distributions and the first moments of the functionals describing the characteristics. The distribution and the average of the number of runs and the maximum of its lengths enable us to forecast critical situations connected with the modeled processes. Specific examples of applications in natural sciences and humanities are given.

Key words: mathematical modeling, dynamics, Markov chains, runs, composites, crack.
Pp. 111–123.

Savel'ev Lev Yakovlevich
Institute of Mathematics SB RAS 4 Koptyug av. Novosibirsk 630090 E-mail: savelev@math.nsc.ru
Balakin Sergey Vladimirovich
Novosibirsk State University 2 Pirogova St. Novosibirsk 630090 E-mail; balakin@ngs.ru

 

UDC 532.546:533.15
Safina G. F.
Preservation of the frequencies of oscillations of a pipe with a liquid

We consider the direct and inverse problems of the flexural oscillations of a narrow pipe with an incompressible liquid. Results of the study of the influence of the parameters of the liquid on the values of the eigenfrequencies of its oscillations are given. We pose and solve the problem of the preservation of the given frequencies of the oscillations of a pipe with liquid. An algorithm is obtained for the determination of elastic fixings of the pipe that keep safe frequencies of oscillations under changes of the parameters of the liquid. We show that, for the preservation of the frequencies of the oscillations of a pipe under changes of the parameters of the liquid, it suffices to carry out the corresponding changes in the parameters of the elastic fixings of the pipe. Some examples are given.

Keywords: pipe with liquid, boundary value problem, frequencies of oscillations, diagnosing, parameters of the liquid, problem of the preservation of frequencies.
Pp. 124–134.

Safina Gul'nara Frilovna
Neftekamsk Division of Bashkir State University 1 Traktovaya st. Neftekamsk 452680 E-mail: Safinagf@mail.ru

 

UDC 517.958
Urazbakhtina  L. Z.
Integrable hydrodynamic submodels with a linear velocity field

Invariant and partially invariant solutions to equations of gas dynamics with a linear velocity field are given by a matrix satisfying a homogeneous integrable Riccati equation. The classification of solutions by the acceleration vector in the Lagrangian coordinates is carried out. We give an example of an invariant solution for which the selected volume ''collapses'' to an interval.

Keywords: gas dynamics equations, state equation, intregrable submodels with linear velocity field.
Pp. 135–145.

Urazbakhtina Liliya Zinfirovna
 Ufa State Aviation Technical University 12 Marksa st. Ufa 450000 E-mail: ylz@ya.ru

 

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