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Sibirskii Zhurnal Industrial'noi Matematiki
2013, vol. 16, No 4 (56)Contents
UDC 517.911.5
Anikonov D.S., Kazantsev S.G., Konovalova D.S.
An inverse problem of location type for a hyperbolic systemWe consider an inverse problem for a hyperbolic system of two first-order partial differential equations with two independent variables. The right-hand sides of the system are assumed discontinuous functions. The inverse problem consists in the determination of a certain hull which contains the discontinuity line of the right-hand sides. We first study the corresponding direct problem. The existence and uniqueness of a generalized solution to the direct problem are established, and the differential properties of this solution are studied. In particular, we prove that its first-order partial derivatives are unbounded near some rays directed along characteristics. This property is the base of the algorithm for solving the inverse problem. The inverse problem is considered in the two versions: in the first version the coefficients of the equations are given, and in the second, they are unknown.
Keywords: inverse problem, hyperbolic equation, discontinuous function, generalized solution, differential property.
Pp. 3–20.Anikonov Dmitrii Sergeevich
Kazantsev Sergey Gavrilovich
Konovalova Dina Sergeevna
Sobolev Institute of Mathematics SB RAS, 4 Koptyug av., 630090 Novosibirsk. E-mail: anik@math.nsc.ru ; kazan@math.nsc.ru ; dsk@math.nsc.ru
UDC 519.2
Belyavskii G.I., Danilova N.V., Nikonenko N.D.
Random walks with missing summandsWe consider a new model of the behavior of the cost of a risky asset in which a random walk with missing summands is used, formulas for the computation of the process of fair prices for a financial commitment in the stationary and nonstationary cases are deduced.
Keywords: random walk, martingale measure, Fourier integral, Esscher transform.
Pp. 21–28.Belyavskii Grigorii Isaakovitch
Danilova Natalia Victorovna
South Federal University, 105/42 Bolshaya Sadovaya st., 344006 Rostov-on-Don
Nikonenko Natalya Dmitrievna
The South Russian Institute of the Russian Presidential Academy of National Economy and Public Administration 70, Pushkinskaya st. 344002 Rostov-on-Don. E-mail: danilova198686@mail.ru
UDC 517.946
Bondarenko A.N., Ivashchenko D.S.
Application of the finite element method for inverse problems of anomalous diffusionWe consider some aspects of the application of the finite element method for the numerical solution of initial boundary value problems for a multidimensional time-fractional diffusion equation. A survey of the existing results is made, efficient algorithms for constructing meshes are discussed, and a number of numerical examples is exposed.
Keywords: finite element method, anomalous diffusion, fractional derivative, automatic mesh generation.
Pp. 29–37.Bondarenko Anatoly Nikolaevich
Sobolev Institute of Mathematics SB RAS, 4 Koptyug av., 630090 Novosibirsk. E-mail: bondarenkoan1953@mail.ru
Ivashchenko Dmitrii Sergeevich
RN-UfaNIPIneft, 3/1 Bekhterev st., 450103 Ufa. E-mail: stanger.dmitry@gmail.com
UDC 517.925.51
Demidenko G.V.
Systems of differential equations with periodic coefficientsWe consider linear systems of differential equations with periodic coefficients. We prove the solvability of nonhomogeneous systems in the Sobolev space $W^1_2(R)$ and establish the estimates for the solutions. Using the result, we prove a theorem on a perturbation for the exponential dichotomy of systems of differential equations with periodic coefficients.
Keywords: exponential dichotomy, periodic coefficients, perturbation, projection, Green matrix, Sobolev space.
Pp. 38–46.Demidenko Gennadii Vladimirovich
Sobolev Institute of Mathematics SB RAS, 4 Koptyug av., 630090 Novosibirsk;
Novosibirsk State University, 2 Pirogov st., 630090 Novosibirsk. E-mail: demidenk@math.nsc.ru
UDC 519.23
Denisov V.I., Timofeeva A.Yu., Khailenko E.A., Buzmakova O.I.
Robust estimation of nonlinear structural modelsThe problem of the identification of nonlinear errors-in-variables models with large observational errors in the explanatory variable is considered. On the basis of robust estimation methods, we propose a development of the algorithms of adjusted and total least squares is suggested. This enabled us to get a better precision of the reconstruction of the response in the presence of outliers in the sample. The proposed algorithms are used in constructing the Engel curve from the data of a budget survey. As a result we managed to make more correct conclusions about the behavior of households with income variation.
Keywords: structural relation, robust estimation, least-squares method, regression penalized spline, Engel curve, budget survey
Pp. 47–60.Denisov Vladimir Ivanovich
Timofeeva Anastasia Yur'evna
Khailenko Ekaterina Alekseevna
Buzmakova Olga Ivanovna
Novosibirsk State Technical University, 20 Karl Marx av. E-mail: supernasty@mail.ru ; ekavka@yandex.ru ; buzmakovaolia@yandex.ru
UDC 517.956.223
Karachik V.V.
On solvability conditions for a Neumann problem for a polyharmonic equation in the unit ballNecessary and sufficient solvability conditions are obtained for a nonhomogeneous Neumann problem for a polyharmonic equation in the unit ball.
Keywords: Neumann problem, polyharmonic equation, solvability conditions.
Pp. 61–74.Karachik Valerii Valentinovich
South Ural State University, 76 Lenina av., 454080 Chelyabinsk. E-mail: karachik@susu.ru
UDC 519.63:621.37:621.382
Kostsov E.G., Fadeev S.I.
On the functioning of a VHF microelectromechanical resonatorWe consider a new design of a microelectromechanical resonator with micrometer dimensions operating at gigahertz frequencies, previously proposed by the authors. The article contains a description of technological problems of the realization of a microelectromechanical resonator as well as a rigorous justification of its functioning in a mathematical model. In analytical form, we establish a connection between auto-oscillations of the mobile element of the microresonator generated by electrostatic attraction during startup, and the parameters of the natural oscillations which turn into auto-oscillations at the end of the startup.
Keywords: VHF MEMS, microresonator, frequency generator, electrostatic attraction, auto-oscillation, phase portrait, rigidity of a spring.
Pp. 75–86.Kostsov Eduard Gennad'evich
Institute of Automation and Electrometry SB RAS, 1 Koptyug av., 630090 Novosibirsk. E-mail: kostsov@iae.nsk.su
Fadeev Stanislav Ivanovich
Sobolev Institute of Mathematics SB RAS, 4 Koptyug av., 630090 Novosibirsk. E-mail: fadeev@math.nsc.ru
UDC 681.5.015
Lomov A.A.
On the consistency of generalized orthoregressive parameter estimates for a linear dynamical systemWe obtain consistency conditions for generalized orthoregressive estimates for the parameters of a linear dynamical system from the observation of a large number of independent trajectories of finite length are observed. As a consequence, we obtain the average consistency of Structured Total Least Squares estimates over the trajectory ensemble.
Keywords: linear dynamical systems, parameter identification, generalized orthoregressive estimates, STLS estimates, consistency.
Pp. 87–93.Lomov Andrey Alexandrovich
Sobolev Institute of Mathematics SB RAS, 4 Koptyug av., 630090 Novosibirsk;
Novosibirsk State University, 2 Pirogov st., 630090 Novosibirsk. E-mail: lomov@math.nsc.ru
UDC 517.977
Maksimov V.I.
On one algorithm of the dynamical reconstruction of the right-hand side of a parabolic equationWe consider a parabolic equation with an unknown right-hand side. The problem of constructing an algorithm for the dynamical reconstruction of the unknown perturbation from the results of inaccurate state measurements is discussed. The problem belongs to the class of inverse problems, the desired algorithm being an algorithm of stable dynamic inversion (dynamic regularization). The existing algorithms of dynamical reconstruction of the right-hand sides of systems with distributed parameters are based on a combination of the positional control method and regularization methods. These algorithms are aimed at reconstructing perturbations over a bounded time segment. Computational and measurement errors accumulate as the time segment grows. In the present article we suggest a dynamical inversion algorithm that is free from this defect.
Keywords: parabolic equations, unknown right-hand side, dynamical inversion algorithm.
P. 94–110.Maksimov Vyacheslav Ivanovich
Ural Federal University Institute of Mathematics and Mechanics UB RAS, 16 S. Kovalevskaya st., 620990 Ekaterinburg E-mail: maksimov@imm.uran.ru
UDC 519.63:621.74.043.2
Marshirov V.V., Marshirova L.E.
Numerical simulation of alloy solidification under intensive conjugate heat transferWe consider the problem of determining the rate of cooling of a metal during solidification at the intersection with the liquidus temperature under an intense heat removal from the surface. Solving this problem is necessary for determining the process conditions, the boundary and initial conditions for which it is possible to get new alloys with microcrystalline structures. We give the necessary finite-difference equations, describe the algorithm, and, using the known experimental data, test the obtained model. We investigate the influence of the size of the casting and the heat transfer coefficient on the cooling rate of an aluminum-based alloy at the liquidus temperature.
Key words: solidification simulation, finite-difference method, cooling rate, heat transfer coefficient, microcrystalline structure.
Pp. 111–120.Marshirov Victor Victorovich
Marshirova Larisa Evgenievna
National Research University ''Higher School of Economics'', 25 Bolshaya Pecherskaya st., 603155 Nizhny Novgorod E-mail: marshirov@hse.ru; lmarshirova@hse.ru
UDC 514.8:517.983
Svetov I.E.
Properties of the ray transforms of two-dimensional 2-tensor fields given in the unit diskWe study the longitudinal, transverse, and mixed ray transforms acting on two-dimensional symmetric 2-tensor fields. Namely, the kernels of the ray transforms are described; the connection between the ray transforms and the Radon transform is established; unconditional estimates of stability for each of the ray transforms are obtained; inversion formulas for a recovery of symmetric 2-tensor field components and for a recovery of the potential are obtained; projection theorems for the ray transforms are proved.
Keywords: integral geometry, symmetric 2-tensor field, solenoidal field, potential field, longitudinal ray transform, transverse ray transform, mixed ray transform, estimate of stability, inversion formula, projection theorem.
Pp. 121–130.Svetov Ivan Evgen'evich
Sobolev Institute of Mathematics SB RAS, 4 Koptyug av., 630090 Novosibirsk;
Novosibirsk State University, 2 Pirogov st., 630090 Novosibirsk. E-mail: svetovie@math.nsc.ru
UDC 517.947:534.14:534.2
Khasanov N.A., Sukhinin S.V.
Axiradial acoustic eigenoscillations near a thin-walled obstacle in a cylindrical channel with narrowing stepsWe study the dependence of the eigenfrequencies and eigenfunctions of acoustic axiradial oscillations near a thin-walled obstacle in a channel with narrowing steps of the geometric parameters of the oscillation domain. It is discovered that, near thin-walled cylindrical obstacles, in an inhomogeneous cylindrical channel with two-sided narrowing cylindrical step, the number of the acoustic eigenfrequencies of acoustic axisymmetric oscillations of the gas can increase. We obtain the dependencies of the eigenfrequencies on the geometric parameters of the obstacle and on the inhomogeneities of the channel.
Key words: acoustic eigenoscillations in an unbounded domain, resonance phenomena, spectral properties of a Laplace operator, thin-walled obstacle in channels and tubes.
Pp. 131–141.Khasanov Nail Alfatovich
Sukhinin Sergey Viktorovich
Lavrent'ev Institute of Hydrodynamics SB RAS, 15 Lavrent'ev av., 630090 Novosibirsk. E-mail: nail_khasanov@mail.ru ; sukhinin@hydro.nsc.ru
UDC 539.95:517.977
Shcherbakov V.V.
Existence of an optimal shape for thin rigid inclusions in the Kirchhoff–Love plateThe paper deals with an optimal control problem for the elliptic system of equations describing an equilibrium of a Kirchhoff–Love plate with delaminated thin rigid inclusion. It is required to minimize the mean square integral deviation of the bending moment from the function given on the exterior boundary. The inclusion shape is considered as the control function. The solvability of the problem is established.
Keywords: Kirchhoff–Love plate model, thin rigid inclusion, crack, nonlinear boundary conditions, optimal control.
Pp. 142–151.Shcherbakov Victor Victorovich
Lavrent'ev Institute of Hydrodynamics SB RAS, 15 Lavrent'ev av., 630090 Novosibirsk. E-mail: sherbakov87@gmail.com