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Sibirskii  Zhurnal  Industrial'noi  Matematiki
2013,  vol. 16,  No 2 (54)

Contents
 

  

UDC 519.633
Aida-zade K.R.,  Rahimov A.B.
Solution of a coefficient inverse problem

We study the numerical solution of a coefficient inverse problem for a parabolic equation. Boundary value problems with nonlocal (integral) boundary conditions are reduced to such a problem. We propose an approach to the solution of the  problem that bases on the method of lines. The results of numerical experiments on test problems are given.

Keywords: inverse problem, nonlocal conditions, method of lines, parabolic equation.
Pp. 3–13.

Aida-zade Kamil Rajab
 Azerbaijan State Oil Academy, 20 Azadlyg av., AZ1010 Baku, Azerbaijan;
Insitute of Cybernetics, NAS of Azerbaijan E-mail: kamil_aydazade@rambler.ru
Rahimov Anar Beybala
Institute of Cybernetics, NAS of Azerbaijan, 9 B. Vahabzade st. AZ1141 Baku, Azerbaijan. E-mail: anar_r@yahoo.com

 

UDC 517.95
Alekseev G.V., Lobanov A.V.
Stability estimates for solutions to inverse extremal problems for the Helmholtz equation

Inverse problems for the Helmholtz equation of the acoustic scattering on a three-dimensional inclusion are considered. Using an optimization method, we reduce these problems to inverse extremal problems in which the role of controls is played by a variable refraction index and boundary source density. Solvability of these problems is proved and some optimality systems are obtained that describe necessary optimality conditions. Basing on the analysis of the optimality systems, sufficient conditions on the input data are deduced that guarantee the uniqueness and stability of optimal solutions.

Keywords: Helmholtz equation, scattering problem, inhomogeneous medium, multiplicative control, inverse problem, uniqueness, stability.
Pp. 14–25.

Alekseev Gennadii Valentinovich
 Far Eastern Federal University, 8 Sukhanov st., 690950, Vladivostok, Russia;
Vladivostok State University of Economics and Service, 41 Gogol st., 690014, Vladivostok,  Russia E-mail: alekseev@iam.dvo.ru
Lobanov Alexey Victorovich
Institute of Applied Mathematics 7 Radio st., 690041, Vladivostok, Russia E-mail: AleksLobanov1@mail.ru

 

UDC 517.911.5
Anikonov D.S., Kazantsev S.G., Konovalova D.S.
Differential properties of a generalized solution to a hyperbolic system of first-order differential equations

We study some questions of the qualitative theory of solutions to differential equations. A Cauchy problem is considered for a hyperbolic system of two first-order differential equations. The right-hand sides of these equations contain discontinuous functions. A generalized solution is defined as a continuous solution to the corresponding system of integral equations. We prove the existence and uniqueness of a generalized solution and study the differential properties of the obtained solution. It is in particular established that its first-order partial derivatives are unbounded near certain parts of the characteristic lines. We observe that this property contradicts a common approach of investigation which uses the reduction of a system of two first-order equations to a single second-order equation.

Keywords:  hyperbolic equations, discontinuous functions, generalized solutions, differential properties.
Pp. 26–39.

Anikonov Dmitrii Sergeevich
Kazantsev Sergey Gavrilovich
Konovalova Dina Sergeevna
Sobolev Institute of Mathematics of SB RAS, 4 Koptyug av., 630090 Novosibirsk. E-mail: anik@math.nsc.ru; kazan@math.nsc.ru; dsk@math.nsc.ru

 

UDC 517.9
Anikonov Yu.E.,  Neshchadim M.V.
Representations for the solutions and coefficients of evolution equations

We give new representations for the solutions and coefficients of evolution equations in the linear case. The obtained formulas contain functional arbitrariness, which can be used in identification problems. We also give classes of hyperbolic equations  admitting generalized functional-invariant solutions.

Keywords: second-order evolution differential equations, Poisson  formula, generalized functional-invariant solutions.
Pp. 40–49.

Anikonov Yurii Evgen'evich
Neshchadim Mikhail Vladimirovoch
 Sobolev Institute of Mathematics of SB RAS, 4 Koptyug Av.; 
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk. E-mail: anikon@math.nsc.ru; neshch@math.nsc.ru

 

UDC 532.542.86
Basok B.I., Davydenko B.V., Gotsulenko V.V.
Auto-oscillations in a Rijke tube under the location of an electric heater directly at its input

We establish theoretically  that the self-excitation of the auto-oscillatons of the Rijke phenomenon when a constant-power heat supply is located at the entrance to the tube is caused by the zone of the reduction of the viscous resistance along the pipe. We obtain a mathematical model and determine the forms of auto-oscillations in this problem.

Key words: Rijke tube, auto-oscillation, distributed system, mechanisms of auto-oscillations, negative resistance.
Pp. 50–61.

Basok Boris Ivanovich
Davydenko Boris Viktorovich
Gotsulenko Vladimir Vladimirovich
 Institute of Engineering Thermophysics of the NAS of Ukraine, 2a Zhelyabov st., 03057 Kiev, Ukraine. E-mail: gosul@ukr.net

 

UDC 519.6
Bondarenko A.N., Dedok V.A., Kozinkin L.A., Tokarev M.P.
 Estimation of the efficiency of the method of hierarchical reconstruction for the problem of the reconstruction of discrete scattering centers from  projections

We study the actual problem of determining the maximal set for the set of points in R3 from the set of two-dimensional projections. This problem arises naturally in the applications of physical hydrodynamics for the optical diagnostics of real flows of liquid and gas by measuring the instantaneous rate fields in the volume of flow. We propose a method for reconstructing the given set and determining the sufficiency of the measurements for the uniqueness of a solution to the inverse problem for parallel and perspective projections. A statistical evaluation for the efficiency of the reconstruction method is obtained.

Keywords: 3D reconstruction of point objects, particle image velocimetry, 3D PTV, inverse problems.
Pp. 62–71.

Bondarenko Anatoly Nikolaevich
Dedok Vasily Aleksandrovich
 Sobolev Institute of Mathematics of  SB RAS, 4 Koptyug Av., 630090 Novosibirsk.
Kozinkin Leonid Alekseevich
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk.
Tokarev Mikhail Petrovich
Kutateladze Institute of Thermophysics of SB RAS, 1 Lavrent'ev Av., 630090 Novosibirsk. E-mail: bondarenkoan1953@mail.ru,; dedok@math.nsc.ru; leon7archer@gmail.com; mtokarev@itp.nsc.ru

 

UDC 517.958
Durdiev D.K., Totieva Zh.D.
The problem of determining the one-dimensional kernel of the viscoelasticity equation

We consider the integrodifferential system of viscoelasticity equations. The direct problem consists in determining the displacement vector from the initial boundary value problem for this system. Under the assumption that the coefficients of the equation depend only on one space variable x3, the system is reduced to an equation for one component u1(x3,t). For this equation, we investigate the problem of finding the kernel belonging to the integral part of the equation. For its determination, an additional condition is given on u1(x3,t) for x3=0. The inverse problem is replaced by an equivalent system of integral equations for unknown functions. To this system, we apply the contraction mapping principle. A theorem of global unique solvability is proved and a stability estimate of a solution to the inverse problem is obtained.

Keywords: inverse problem, stability, delta-function, Lam'e coefficients, kernel.
Pp. 72–82.

Durdiev Durdimurod Qalandarovich
 Bukhara State University, 11 Mukhammad Iqbol st., 200177  Bukhara,  Uzbekistan
Totieva Zhanna Dmitrievna
Center of Geophysical Investigations of the Vladikavkaz Scientific Center of the Russian Academy of Sciences and Republic of North Ossetia–Alania, 93a Markov st., 362002  Vladikavkaz. E- mail:  durdiev65@mail.ru; jannatuaeva@inbox.ru

 

UDC 519.632
Il'in V.P.
DELAUNAY: a technological environment for grid generation

We consider the concept of technological environment for multidimensional grid generation for solving problems of mathematical modeling in computation domains with complicated configuration of piecewise smooth boundaries including direct and inverse interdisciplinary statements described by systems of differential and/or integral equations. In general, the computation grid domain that is constructed consists of subdomains in each of which grids can be of different types (for example, structured or nonstructured) and discretization at the inner boundaries can be consistent or nonconsistent. The methodology of such quasistructured grids assumes the possibility of using various algorithms and codes in subdomains as well as the plurality of the formats of grid data structure and their convertation. The proposed technologies include control of the grid quality, generation of dynamical grids adapted to the singularities of the input geometric data structure, and multigrid approaches, local refinements and account taken of the a priori and/or a posteriori information about the solution. Scalable parallelization is supported by a balanced decomposition of the grid domain.

Keywords: multidimensional boundary value problems, grid computation domain, adaptive quasi-structured grids, grid generation method, grid data structures, scalable parallelezation.
Pp. 83–97.

Ilin Valery Pavlovich
 Institute of Computational Mathematics and Mathematical Geophysics SB RAS 6 Lavrentiev av., Novosibirsk. e-mail: ilin@sscc.ru

 

UDC 539.375
Lazarev N.P.
The Griffith formula for a Timoshenko-type plate with a curvilinear track

We consider an equilibrium problem for an elastic transversely isotropic Timoshenko-type plate with a curvilinear crack. Nonpenetration conditions on the faces of the crack having the form of inequalities (conditions of the Signorini type) are given. It is proved that the solutions to the equilibrium problems with a perturbed crack converge to the solution to the equilibrium problem with the unperturbed crack in the corresponding space. The derivative of the energy functional with respect to the crack length is obtained.

Keywords: Timoshenko-type plate, crack, nonpenetration condition, Griffith criterion, variational inequality, derivative of the energy functional, nonsmooth domain.
Pp. 98–108.

Lazarev Nyurgun Petrovich
Institute for Mathematics of NEFU, 58 Belinskii st., 677000 Yakutsk, Republic of Sakha; Lavrent'ev Institute of Hydrodynamics SB RAS, 15 Lavrent'ev ave, 630090 Novosibirsk. E-mail: nyurgun@ngs.ru

 

UDC 519.233.22
Lisitsin D.V.,  Gavrilov K.V.
Estimation of the parameters of a compactly-supported model  stable under the violation of compact supportedness

We consider two approaches to the problem of robust estimations of the distribution parameters of a random variable whose range is bounded (from either or both sides). The traditional approach gives stable parameter estimates only when the actual and model random variables have the same range. The generalized approach introduced in the paper can be applied in the presence of observations lying outside the range of the model random variable. It is shown that, under some conditions, these approaches are asymptotically equivalent.

Keywords: compactly-supported model, parameter estimation, robustness, influence function, cosine distribution.
Pp. 109–121.

Lisitsin Daniil Valer'evich
 Novosibirsk State Technical University, 20 Marx Av., 630073 Novosibirsk. E-mail: dalis2@yandex.ru
Gavrilov Konstantin Viktorovich
Sever Federal State Unitary Enterprise, 3 Ob'edineniya st., 630075 Novosibirsk, E-mail: Qot@ngs.ru

 

UDC 539.374
Mantsybora A.A.,  Rusanov M.M.
The automodel problem of the dynamic unloading of an elasticplastic half-space

We give a solution to a planar automodel problem of multistep high-speed unloading from the boundary of an elasticplastic half-space with a considerable level of accumulated irreversible deformations. Uniqueness in the totality of the arising unload waves (simple or shock waves) is based on the thermodynamics laws and the evolution conditions for propagating deformation discontinuities.

Keywords: elasticplastic, shock waves, Riemann waves, automodel problems.
Pp. 122–129.

Mantsybora Alexander Anatol'evich
Rusanov Maxim Mikhailovich
Institute of Automation and Control Processes FEB RAS, 5 Radio st., 690041, Vladivostok; Far Eastern Federal University, 8 Suhanova st., 690091, Vladivostok. E-mail: manzubor@iacp.dvo.ru; maxprimat@mail.ru

 

UDC 523.51
Nazarov V.G.
The problem of the formation of materials with given radiation characteristics

The problem of the formation of materials with given radiation characteristics is considered on the example of the formation of a mixture that has the X-radiation attenuation and scattering coefficients close or equal to those of an a priori given material on a predetermined energy band. The posed optimization problem is reduced to solving a system of linear equations under some conditions. A geometric interpretation of the problem and its solution are given.

Key words: radiative transfer equation, X-ray tomography.
Pp. 130–143.

Nazarov Vasily Gennadievich
 Institute of Applied Mathematics FEBRAS, 7 Radio st., 690041, Vladivostok. E-mail: naz@iam.dvo.ru

 

UDC 539.3:517.958
Khludnev A.M.
On an equilibrium problem for a two-layer elastic body with a crack

We analyze a free boundary value problem describing an equilibrium state of a two-layer elastic body with a crack. The existence of a solution is proved. Invariant integrals over curves surrounding the crack tip are found. Passages to the limit as the rigidity parameter of the layer vanishes are studied.

Key words: crack, nonlinear boundary conditions, two-layer body, derivative of the energy functional, invariant integrals.
Pp. 144–153.

Khludnev A.M.
Lavrent'ev Institute of Hydrodynamics SB RAS, 15 Lavrent'ev av., 630090 Novosibirsk. E-mail: khlud@hydro.nsc.ru

 

  

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