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Sibirskii Zhurnal Industrial'noi Matematiki
2009, vol. 12, No 3 (39)
Contents
The Measures of Science (To the 80th Anniversary of Yu. G. Reshetnyak)
UDC 539.3
B. D. Annin
A Transversally Isotropic Elastic Model of GeomaterialsWe study the choice of parameters in a transversally isotropic elastic model describing linear deformation of the geomaterials, as well as the analytical and numerical methods for solving the corresponding dynamic equations.
Keywords: linear elasticity, transversally isotropic medium, Lam\'e equations, Gassman condition, Friedrichs t-hyperbolicity.
Pp. 5–14.Annin Boris Dmitrievich
Lavrent'ev Institute of Hydrodynamics SB RAS, pr. Akad. Lavrent'eva 15, Novosibirsk 630090.
E-mail: annin@hydro.nsc.ru
UDC 519.688:531.36
A. V. Banshchikov
Analysis of the Dynamics of Mechanical Systems of Large Dimension with Computer Algebra SystemsUsing some developed software package, we analyze the dynamics of a system of $18$ linked solids with $32$ degrees of freedom in a Newtonian central force field. Using a personal computer for performing symbolic or symbolic-numerical computations, we model the conservative mechanical system (by construction of nonlinear and linearized differential equations of motion) and study both the stability and gyroscopic stabilization of the unperturbed motion.
Keywords: dynamics of systems of solids, computer algebra, software package, stability of motion, degree of instability, gyroscopic stabilization, solution to systems of inequalities.
Pp. 15–27.Banshchikov Andreui Valentinovich
Institute of System Dynamics and Control Theory SB RAS, ul. Lermontova 134, P. O. Box 292, Irkutsk 664033.
E-mail: bav@icc.ru
UDC 517.958
D. K. Durdiev
An Inverse Problem for Determining Two Coefficients in an Integrodifferential Wave EquationFor the equation of wave propagation in the half-space filled with some medium, we consider the problem of determining the wave propagation velocity which depends only on the variable y and the memory functions of the medium. There is a point-like pulse source on the boundary of the half-space.
We show that both unknown functions of one variable are uniquely determined by the Fourier image with respect to x of the solution to the direct problem on the boundary of the half-space. We estimate the stability of the solution to the problem.
Keywords: inverse problem, Fourier transform, stability, uniqueness.
Pp. 28–40.Durdiev Durdimurat Kalandarovich
Bukhara State University, ul M. Ikbala 11, Bukhara 706018, Uzbekistan.
E-mail: durdiev65@mail.ru
UDC 519.852.6
G. I. Zabinyako
Re-Construction of Inverse MatricesWe consider algorithms for re-constructing the inverses to basis matrices, in which the advance determination of the pivots is based on solving assignment problems. Then, in order to memory saving, by symmetric permutations, we set the order in which the pivots are used. The corresponding routines are designed for the software packages for solving mathematical programming problems.
Keywords: linear programming, systems of linear algebraic equations, sparse matrices, assignment problem.
Pp. 41–51.Zabinyako Gerard Idel'fonovich
Institute of Computational Mathematics and Mathematical Geophysics SB RAS pr. Akad. Lavrent'eva, 6, Novosibirsk 630090
E-mail: zabin@rav.sscc.ru
UDC 517.9
N. A. Kucher and D. A. Prokudin
Stationary Solutions to the Equations of a Mixture of Compressible Viscous FluidsThe existence of renormalized solutions is proved of the first boundary value problem for the equations of stationary motion of a two-component mixture of compressible viscous fluids for all values of the adiabatic exponent in (3,+¥).
Keywords: boundary value problem, mixture dynamics, solution to Navier–Stokes equations.
Pp. 52–65.Kucher Nikolay Alekseevich
Prokudin Dmitriy Alekseevich
Kemerovo State University ul. Tereshkovou 40, Kemerovo 650036
E-mail: diffur@kemsu.ru
UDC 519.9
G. S. Lbov, V. M. Nedel'ko, and S. V. Nedel'ko
An Adaptive Search Method for a Logical Solving FunctionA method is proposed for seeking the optimal logical solving function in the problems of data analysis, which is a modification of adaptive random search. The method uses a metric in the space of logical solving functions and seeks solutions in accordance with an adaptable probability measure.
Keywords: pattern recognition, machine learning, solution trees, global extremum search, optimization, adaptive random search, genetic algorithms.
Pp. 66–74.Lbov Gennadi Sergeevich
Nedel'ko Viktor Michailovich
Sobolev Institute of Mathematics SB RAS, pr. Akad. Koptyuga 4, Novosibirsk 630090
E-mail: lbov@math.nsc.ru; nedelko@math.nsc.ruNedel'ko Svetlana Valer'evna
Novosibirsk State Technical University, pr. K. Marksa 20, Novosibirsk 630092
UDC 519.866.6:519.873
E. O. Rapoport
The Distribution of an Indivisible Resource: Optimal Control and PricesWe study a dynamical system of economics with discrete time, whose states at every moment are characterized by integer nonnegative points in a two-dimensional vector space. There are two types of goods and two different production facilities, in each of which the state of the system can change by some random integer vector with various collections of probabilities. By a control we understand choosing at every moment one of the available collections of probabilities. The aim of control is to minimize the probability of leaving the positive quadrant. We study the existence of prices for the goods which agree with the optimal control.
Keywords: optimal control, prices, Markov chain.
Pp. 75–84.Rapoport Ernest Osherovich
Sobolev Institute of Mathematics SB RAS, pr. Akad. Koptyuga 4, Novosibirsk 630090.
E-mail: rapoport@math.nsc.ru
UDC 519.21
G. I. Salov
To the Discord Problem for a Jump-Like Markov ProcessWe obtain the structure of the optimal moment of halting (sending an ''alarm'' signal) and the integrodifferential equation for the risk function in the discord problem for an observed jump-like Markov process in the case that the discord point coincides with one of the jumps of the process.
Keywords: Markov process, detection, discord.
Pp. 85–98.Salov Gennadiy Iosifovich
Institute of Computational Mathematics and Mathematical Geophysics SB RAS
pr. Akad. Lavrent'eva 6, Novosibirsk 630090.
E-mail: sgi@ooi.sscc.ru
UDC 519.632
V. M. Sveshnikov
Construction of Direct and Iterative Decomposition MethodsIn order to solve the boundary value problems by the method of decomposing the computation region G into subregions without overlapping and with the Dirichlet–Dirichlet type conditions, the Poincare–Steklov operator equation on the junction boundary g of the subregions, which involves the difference of the normal derivatives of the solutions on the opposite sides of g, is approximated by using the discrete Green's functions. Basing on this, we construct some direct and iterative decomposition methods which are parallel in nature. Sample computations show the precision and convergence of the proposed algorithms.
Keywords: boundary value problem, method for decomposing a region, Poincare–Steklov equation, quasistructured mesh, discrete Green's function.
Pp. 99–109.Sveshnikov Viktor Mitrofanovich
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Lavrent'eva 6,
Novosibirsk State University ul. Pirogova 2, Novosibirsk 630090.
E-mail: victor@lapasrv.sscc.ru
UDC 517.968.21:517.968.22
V. V. SmelovAn Approximate Solution to the Integral Equations with Kernels of the Form K(x – t) which Uses a Nonstandard Basis of Trigonometric Functions
Consider the integral equations with kernels of the form K(x – t). In order to find an approximate solution by the Galerkin method, we propose a nonstandard trigonometric basis. This basis possesses a high approximation quality and enables us to reduce the double integral in the Galerkin algorithm to quite simple single integration.
Keywords: Fredholm and Volterra equations, Galerkin method, nonstandard trigonometric basis.
Pp. 110–116.Smelov Vladislav Vladimirovich
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Akad. Lavrent'eva 6, Novosibirsk 630090.
E-mail: vl.smelov@gmail.ru
UDC 629.7:678.067
S. I. Snisarenko
Finite Elements for the Problems of Shock Load and Nonstationary Deformation of the Composite Shells and PlatesTo study numerically the problems of shock load and nonstationary deformation, we propose some modified finite elements for the multilayer fibrous laminated composite shells and plates, which account for transversal shears, normal squeezing, rotational inertia, and the viscoelastic behavior of the material. We state the requirements on the finite elements of this type, determine the domain of their use, and make a comparative analysis of the convergence and precision.
Keywords: composite materials, composite shells, plates, shock load, nonstationary deformation, finite elements method.
Pp. 117–129.Snisarenko Sergey Ivanovich
Novosibirsk State Technical University, pr. K. Marksa 20, Novosibirsk 630092,
E-mail: snis@craft.nstu.ru
UDC 517.948
V. P. Tanana and A. I. Sidikova
On the Order Optimality of a Method for Evaluating Unbounded Operators and Its ApplicationsWe propose an order optimal method for approximate solving the problem of calculating the values of unbounded operators in various spaces.
Keywords: inverse problem, regularization, parabolic equation, Fourier transform, unbounded operator, Hilbert space.
Pp. 130–140.Tanana Vitaliy Pavlovich
Sidikova Anna Ivanovna
South Urals State University, pr. Lenina 76, Chelyabinsk 454080.
E-mail: 7413604@mail.ru
UDC 534.14:534.2
V. S. Yurkovskiy
The Radiation of Small Amplitude Waves on the Free Surface by a Pulse Source Inside a Conical HornWe study how the geometric parameters of a horn affect raising the radiation efficiency and signal focusing for a pulse source lying on the free surface. Some dependences are obtained of the radiation intensity on the geometry of the horn, observation coordinates, and the frequencies of the source pulsations. We state criteria for determining the efficiency of the horn. Graphical and analytical dependencies are found of the distant-field radiation intensity on the parameters of the problem. An optimization technique is developed for the radiating source-horn system by adjusting its geometric parameters. The parameter values are determined that guarantee the most efficient operation of a radiating source-horn system from the viewpoint of localizing the radiation in an effective angle.
Keywords: acoustic oscillations in unbounded regions, radiation localization using a horn, source shape optimization on a free surface.
Pp. 141–150.Yurkovsky Vadim Sergeevich
Lavrent'ev Institute of Hydrodynamics SB RAS, pr. Akad. Lavrent'eva 15, Novosibirsk 630090.
E-mail: yvs2000@gmail.com
UDC 539.374
L. V. Yakhno
A Superposition of the Nadai and Prandtl Solutions to the Two-Dimensional Ideal Plasticity SystemWe give a general algorithm for transforming exact solutions to the flat ideal plasticity system of Mises using the superposition principle for solutions, which arises as a corollary to the original system admitting an infinite dimensional symmetry group. As an example we consider a relation between the known exact solutions: the Prandtl solution for a thin layer compressed by rough solid plates, and the Nadai solution for the radial distribution of stresses in a convergent channel in the shape of a flat wedge.
Keywords: flat ideal plasticity, exact solutions to differential equations, superposition principle for solutions, boundary value problem for hyperbolic systems.
Yakhno Liliya Vladimirovna
Pp. 151–156.
Siberian State Aerospace University, pr. Gazety ''Krasnoyarskiy rabochiy'' 31, Krasnoyarsk 660014.
E-mail: iakhno@kgtei.ru