Home
• Editorial Board •
Contents •
Russian page
Sibirskii Zhurnal Industrial'noi Matematiki
2015, vol. 18, No 2 (62)Contents
UDC 517.988.68
DOI 10.17377/sibjim.2015.18.201
Ageev A. L., Antonova T. V.
Methods for the approximating the discontinuity lines of a noisy function of two variables with countably many singularities
We study the localization methods for the discontinuity lines of a noisy function of two variables. The function is assumed to have countably many discontinuity lines: finitely many discontinuity lines have ``large'' jump, and the jumps at the remaining discontinuity lines satisfy some smallness condition. It is required, from the noisy function and the error in $L_2$, to determine the number and localize the position of the discontinuity lines that form the first set for the exact function. The problem under consideration belongs to the class of nonlinear ill-posed problems, and for solution we have to construct regularizing algorithms. We propose a simplified theoretical approach when conditions on the exact function are imposed in a narrow strip intersecting the discontinuity lines. We construct methods for the averaging and localization of discontinuity lines and obtain estimates for the accuracy of the localization.
Keywords: ill-posed problem, regularization algorithm, localization of singularities, equation of the first kind, discontinuity line.
Pp. 3-11.
Ageev Aleksandr Leonidovich
Antonova Tatiana Vladimirovna
Krasovskii Institute of Mathematics and Mechanics Ural Division of the Russian Academy of Sciences 16 S. Kovalevskaya st., 620990 Ekaterinburg
E-mail: ageev@imm.uran.ru; tvantonova@imm.uran.ru
UDC 519.6:622.6
DOI 10.17377/sibjim.2015.18.202
Aida-zade K. R., Ashrafova Y. R.
Calculation of transient fluid flow regimes in pipeline networks
We consider the calculation problem of flow regimes of transient processes in oil pipeline networks of complicated loopback structure. The fluid flow in each linear pipe is described by a system of two first-order linear hyperbolic partial differential equations. At the junctions of the network, there are fulfilled unseparated boundary conditions determined by the first Kirchhoff law and by the continuity of the flow. We propose a scheme for the numerical solution to the problem based on the application of the grid method and obtain formulas that are an analog of the sweep method and do not depend on the number of junctions, pipes, and the structure of the pipeline network. Numerical experiments applying the proposed approach are implemented, some analysis of the results is carried out.
Keywords: transient regime, pipeline network, system of hyperbolic equations, unseparated boundary conditions, grid method, sweep method.
Pp. 12-23.
Aida-zade Kamil Rajab
Azerbaijan State Oil Academy, 20 Azadlig av., AZ 1010, Baku, Azerbaijan;
Institute of Control Systems of the Azerbaijan National Academy of Sciences; 9 B.Vahabzade st., AZ 1141 Baku, Azerbaijan.
E-mail: kamil_aydazade@rambler.ru
Ashrafova Yegana Ramiz
Institute of Control Systems of the Azerbaijan National Academy of Sciences 9 B.Vahabzade st., AZ 1141 Baku, Azerbaijan.
E-mail: ashrafova_yegana@yahoo.com
UDC 517.95
DOI 10.17377/sibjim.2015.18.203
Alekseev G. V.
Solvability of a boundary value problem for stationary equations of magnetohydrodynamics of a viscous heat-conducting fluid
We study a boundary value problem for the stationary equations of magnetohydrodynamics of a viscous heat-conducting fluid considered under the Dirichlet condition for the velocity and mixed boundary conditions for the electromagnetic field and the temperature. Sufficient conditions are established on the initial data that guarantee the global solvability of this problem and the local uniqueness of its solution.
Keywords: magnetohydrodynamics, boundary value problem, mixed boundary conditions, solvability, uniqueness.
Pp. 24-35.
Alekseev Gennadii Valentinovich
Far Eastern Federal University, 8 Sukhanova st., 690950, Vladivostok;
Institute of Applied Mathematics FEB RAS, 7 Radio st., 690041 Vladivostok.
E-mail: alekseev@iam.dvo.ru
UDC 517.95
DOI 10.17377/sibjim.2015.18.204
Anikonov Yu. E., Neshchadim M. V.
The method of differential relations and nonlinear inverse problems
We apply the method of differential relations to the study of some inverse problems for nonlinear one-dimensional differential equations of a general type including the classical equations of soliton theory. We also consider the problem of finding a potential for an equation of continuum mechanics in the one-dimensional case in presence of some differential relation.
Keywords: inverse problems, nonlinear equations, soliton, presentations of solutions.
Pp. 36-47.
Anikonov Yurii Evgen'evich
Neshchadim Mikhail Vladimirovich
Sobolev Institute of Mathematics SB RAS, 4 Koptyug av., 630090 Novosibirsk;
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk.
E-mail: anikon@math.nsc.ru; neshch@math.nsc.ru
UDC 519.63
DOI 10.17377/sibjim.2015.18.205
Galkin V. M.
Representation of consistency conditions on characteristics
We consider an axisymmetric supersonic flow of an ideal gas. The consistency conditions on characteristics are integrated in quadratures under the condition of the constancy of the characteristic Mach number or the slope of the velocity vector. We indicate an algorithm for constructing these characteristics.
Keywords: ideal gas, method of characteristics, characteristic equation, consistency condition.
Pp. 48-51.
Galkin Vladislav Mihailovich
Tomsk Polytechnic University, 30 Lenin av., 634050 Tomsk.
E-mail: vlg@tpu.ru
UDC 517.95
DOI 10.17377/sibjim.2015.18.206
Klyachin A. A.
On the uniform convergence of piecewise linear solutions to the equilibrium capillary surface equation
We consider piecewise linear solutions to the equilibrium capillary surface equations over a given triangulation of a multifaceted domain. It is shown that, under certain conditions, the gradients of these functions are bounded in refining the triangulation, i.e., when the maximum diameter of the triangles of the triangulation vanishes. This property holds if piecewise linear functions approximate the energy integral for a smooth function with required accuracy. As a consequence of the obtained properties, we get the uniform convergence of piecewise linear solutions to the exact solution of the equation of an equilibrium capillary surface with prescribed contact angle on the boundary.
Keywords: piecewise linear functions, minimal surface equation, approximation of the energy functional.
Pp. 52-61.
Klyachin Alexei Aleksandrovich
Volgograd State University, 100 Universitetskii av., 400062 Volgograd.
E-mail: klyachin-aa@yandex.ru
UDC 517.972.5:519.651
DOI 10.17377/sibjim.2015.18.207
Mokshin P. V., Rozhenko A. I.
On search for an optimal parameter for a smoothing spline
We study the problem of choosing an optimal parameter for smoothing an abstract smoothing spline for which the norm of the deviation in the nodes of the mesh (the norm of the error) must coincide with the prescribed level of data error. The obtained equation is nonlinear in the smoothing parameter, and it can be solved iteratively, for example, by the Newton method. In using the Newton method, at each step of the iterative process, two problems of smoothing with the same smoothing parameter but with different vectors of approximated data must be solved. We propose an algorithm for solving this equation using representations of the error operator of the smoothing spline and the complementary operator in the form of the sum of power series. Its novelty consists in applying a hybrid approach depending on the relation of the next approximation of the smoothing parameter and its optimal value. In the algorithm, we use approximations of the error operator and the complementary operator as a partial sum of a series, the algorithm of linear-fractional approximation of error functions, and a refinement of approximations of the error functions by extrapolation with respect to the length of the partial sums of the series. The proposed algorithm enables us to achieve an optimal value of the smoothing parameter in fewer iterations (in practical computations, in two iterations) by solving several smoothing problems at each step.
Keywords: spline, smoothing, algorithm.
Pp. 63-73.
Mokshin Pavel Vladimirovich
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk.
Rozhenko Aleksandr Iosifovich
Institute of Computational Mathematics and Mathematical Geophysics SB RAS 6 Lavrent'ev av., 630090 Novosibirsk.
E-mail: rozhenko@oapmg.sscc.ru
UDC 517.95
DOI 10.17377/sibjim.2015.18.208
Neustroeva N. V.
An equilibrium problem for an elastic plate with an inclined crack on the boundary of a rigid inclusion
We consider an equilibrium problem for a Kirchhoff-Love elastic plate with an inclined crack on the boundary of a rigid inclusion. The nonpenetration conditions are considered at the crack faces in the form of equalities and inequalities. On the boundary of the rigid inclusion, some identity holds describing the action of the external forces on the rigid part of the plate. The variational statement of the problem is studied, and an equivalent boundary value problem is formulated. For a family of problems about a plate with inclined crack on the boundary, we analyze the passage to the limit as the rigidity parameter of the inclusion tends to infinity.
Keywords: inclined crack, rigid inclusion, plate, variational inequality.
Pp. 74-84.
Neustroeva Natalia Valerianovna
North-Eastern Federal University (NEFU) Institute of Mathematics and Informatics, 48 Kulakovskogo st., 677000 Yakutsk
E-mail: nnataliav@mail.ru
UDC 517.929:614.4
DOI 10.17377/sibjim.2015.18.209
Pertsev N. V.
Analysis of solutions to mathematical models of epidemic processes with common structural properties
We present the equations of a family of mathematical models describing the process of the spread of infectious diseases among the population of one or more regions. The variables of the models are the numbers of various groups of individuals involved in the spread of the epidemic (groups of susceptible, infected, diseased individuals, etc.). The change rates in the number of groups of individuals are defined using abstract functions that take into account the current state and the history of the spread of the epidemic process. For analyzing the solutions of the models, we use the results of the theory of monotone operators and the properties of $M$-matrices. Sufficient conditions for the existence of bounded solutions of a family of models and the limit of these solutions at infinity are obtained. The results of the study of the solutions of the models of the spread of HIV-infection and tuberculosis are formulated.
Keywords: mathematical model, delay integrodifferential equations, asymptotic behavior of solutions, theory of monotone operators, $M$-matrix, epidemiology, HIV-infection, tuberculosis.
Pp. 85-98.
Pertsev Nikolai Viktorovich
Sobolev Institute of Mathematics SB RAS Omsk Branch 13 Pevtsov st., 644043 Omsk.
E-mail: homlab@ya.ru
UDC 517.98
DOI 10.17377/sibjim.2015.18.210
Prokhorov I. V., Sushchenko A. A., Kan V. A.
On a problem of reconstructing the bottom configuration of a fluctuating ocean
We formulated and investigate an inverse problem for nonstationary radiative transfer equation in application to the acoustic map making of the sea bottom with the use of side-scan sonars. In the single scattering approximation, we obtain a formula for determining the function that describes the small deviation of the surface from the bottom from a medium level.
Keywords: radiative transfer equation, inverse problems, bottom configuration, single-scattering approximation.
Pp. 99-110.
Prokhorov Igor' Vladimirovich
Sushchenko Andrei Andreevich
Institute of Applied Mathematics FEB RAS, 7 Radio st., 690041 Vladivostok;
Far Eastern Federal University, 8 Sukhanova st., 690950, Vladivostok.
E-mail: prokhorov@iam.dvo.ru; sushchenko.aa@dvfu.ru
Kan V. A.
Far Eastern Federal University, 8 Sukhanova st., 690950, Vladivostok.
E-mail: kan.va@inbox.ru
UDC 539.3
DOI 10.17377/sibjim.2015.18.211
Ragozina V. E., Ivanova Yu. E.
Ray approximations for the shock waves of an elastic deformation of axisymmetric type in a cylindrical layer
The efficiency of the version of the ray method developed directly for strong discontinuity waves (shock waves) is demonstrated by the example of an axisymmetric problem of intense deformation of a cylindrical nonlinear elastic layer under load action on its exterior border. We consider several initial stages of the wave process, namely, the motion of the created shock waves to the interior border of the layer, the reflection of a more rapid wave from the interior border, the interaction of a slow shock wave and the reflected shock wave with the formation of a new wave pattern. The solution for each deformation stage is constructed using the modified ray series method.
Keywords: nonlinear elastic medium, impact deformation, ray series, axisymmetric problem, quasi-longitudinal and quasi-transverse shock waves.
Pp. 111-123.
Ragozina Victoria Evgen'evna
Institute of Automation and Control Processes FEB RAS 5 Radio st. 690041 Vladivostok
Ivanova Yulia Evgen'evna
Institute of Automation and Control Processes FEB RAS
Far Eastern Federal University 8 Sukhanova st. 690091 Vladivostok
UDC 519.63
DOI 10.17377/sibjim.2015.18.212
Sveshnikov V. M.
Method of decomposition of the computational domain in problems of high-current electronics
The method of decomposition of the computational domain is known from solving linear problems of mathematical physics. In this article, we propose to use this method for calculating intensive beams of charged particles in nonlinear self-consistent problems of high-current electronics. The computational domain partitions into two subdomains: the cathode-full subdomain and the principal subdomain. In the cathode-full subdomain, we construct an analytic solution from known formulas while in the principal subdomain the solution is found numerically. The central question is that of coordinating the subdomains. To this end, on the conjugation condition, by analogy with linear problems, we write down the Poisson—Steklov equation, which is approximated by a system of operator nonlinear equations. It is solved by methods of quasi-Newton type, namely, by Broyden's method. As follows from the experiments, the process converges already at the fourth iteration with precision acceptable for practice.
Keywords: self-consistent problem, Poincar\'e—Steklov equation, nonlinear equation.
Pp. 124-130.
Sveshnikov Viktor Mikhailovich
Institute of Computational Mathematics and Mathematical Geophysics SB RAS 6 Lavrent'ev av., 630090 Novosibirsk;
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk.
E-mail: victor@lapasrv.sscc.ru