Home
• Editorial Board •
Contents •
Russian page
Sibirskii Zhurnal Industrial'noi Matematiki
2014, vol. 17, No 1 (57)Contents
UDC 514.745.82
N.B. Ayupova, V.P. Golubyatnikov
On the uniqueness of a cycle in an asymmetric 3-dimensional model of a molecular repressilatorWe obtain sufficient conditions for the uniqueness of cycles in some nonlinear dynamical systems considered as models for the functioning of a molecular repressilator. A constructive method for the determination of the invariant surface containing this cycle is described as well.
Keywords: nonlinear dynamical system, phase portrait, invariant domain, molecular repressilator, cycle, projective transformation.
Pp. 3–7.Ayupova Natalia Borisovna
Golubyatnikov Vladimir Petrovich
Sobolev Institute of Mathematics, 4 Koptyug av., 630090 Novosibirsk;
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk. E-mail: ayupova@math.nsc.ru ; glbtn@math.nsc.ru
UDC 519.688
Balakina E.Yu.Numerical realization of the algorithm of reconstruction of an inhomogeneous medium for an X-ray tomography problem
We study an X-ray tomography problem which is an inverse problem for the differential transport equation. We take into account the absorption of particles by the medium and their single scattering. The statement of the problem corresponds to multiple probing. The medium is unknown; the densities of the output flux averaged over energy are given. The object in question is the discontinuity surfaces of the coefficients of the equation. This corresponds to searching for the boundaries between various substances contained in the medium that we probe. The solution is constructive, and a version of numerical realization of the algorithm is presented.
Keywords: free boundary, transport equation, inverse problem, multiple probing, tomography.
Pp. 8–17.Balakina Ekaterina Yur'evna
Sobolev Institute of Mathematics, 4 Koptyug av., 630090 Novosibirsk;
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk. E-mail: balakina@math.nsu.ru
UDC 517.958:531.32
Baranovskii E.S.
An optimal control problem for a stationary flow of a Jeffreys medium with slip boundary conditionWe study an optimal control problem for the stationary motion equations of a Jeffreys viscoelastic medium with a Navier slip boundary condition. The control parameter is provided by an external force. We prove the existence of a weak solution minimizing a given cost functional and establish some properties of the solution set to the optimization problem.
Keywords: optimal control, flow control, non-Newtonian fluid, viscoelastic medium, Jeffreys model, Navier–Stokes equations, Navier slip boundary condition, weak solution, Galerkin method.
Pp. 18–27.Baranovskii Evgenii Sergeevich
Voronezh State University of Engineering Technologies, 19 Revolutsii av. 394036 Voronezh. E-mail: esbaranovskii@gmail.com
UDC 548.1
Garipov R.M., Kotel'nikova M.S.
4-Crystals and quasicrystals4-Crystals are periodic material structures in the 4-dimensional space-time. The classical crystallographic relations are generalized to this case. The quasicrystals are solids whose X-ray photograph and electronogram contain regular pentagons. It is shown that the quasicrystals obtained experimentally by Shechtman et al. are 4-crystals.
Keywords: crystal, polarizability, symmetry.
Pp. 28–35.Garipov Ravil Mukhametzyanovich
Kotelnikova Maria Stanislavovna
Lavrent'ev Institute of Hydrodynamics, 15 Lavrent'ev av., 630090 Novosibirsk. E-mail: R.M.Garipov@mail.ru; kmaria@ngs.ru
UDC 519.63
Gorbenko N.I.
Numerical modeling of the integro-differential Korteweg–de Vries–Burgers equationWe develop explicit and implicit schemes that are based on the finite volume method for the Korteweg–de Vries–Burgers equation which become multisymplectic for the Korteweg–de Vries equation. The resulting schemes have better stability on long-term calculations. An algorithm for calculating the Duhamel integral with a fixed amount of memory is proposed. The results of numerical experiments are presented.
Keywords: integral-differential equation, Hamiltonian, multisymplectic, explicit and implicit schemes, numerical experiment
Pp. 36–45.Gorbenko Nikolay Ivanovich
Institute of Computational Mathematics and Mathematical Geophysics, 15 Lavrent'ev av., 630090 Novosibirsk E-mail: nikolay.gorbenko@gmail.com
UDC 517.95
Kazakov A.L., Kuznetsov P.A.
On a boundary value problem for a nonlinear heat equation in the case of two space variablesThe article deals with a boundary value problem with degeneration for a nonlinear heat equation in the case of two space variables. Solving this problem makes it possible to study heat conduction in a neighborhood of a closed cylindrical surface. The theorem is proved of the existence and uniqueness of an analytic solution to the problem.
Keywords: nonlinear heat equation, boundary value problem, analytic solution
Pp. 46–54.Kazakov Aleksandr Leonidovich
Institute for System Dynamics and Control Theory, 134 Lermontov st. 664033 Irkutsk. E-mail: kazakov@icc.ru
Kuznetsov Pavel Alexandrovich
Irkutsk State University, 1 Marx st., 664003 Irkutsk. E-mail: pav_ku@mail.ru
UDC 519.688
Kandryukova T.A, Laevsky Yu. M.
On the numerical simulation of filtration combustion of gases on multi-core computing systemsWe analyze the performance of some computational models of filtration gas combustion on multi-core computers. The analysis is limited to models based on explicit difference schemes. In particular, an explicit two-level parallel algorithm using an adaptive mesh is constructed The two ways of shared memory parallelization are applied: direct application of OpenMP directives and a special data distribution between the threads. It is shown numerically that the latter method provides a significant performance advantage.
Key words: combustion wave, explicit scheme, adaptive mesh, parallelization on shared memory.
Pp. 55–64.Kandryukova Tat'yana Aleksandrovna
Laevskii Yurii Mironovich
Institute of Computational Mathematics and Mathematical Geophysics, 6 Lavrent'ev av., 630090 Novosibirsk E-mail: kandryukova@labchem.sscc.ru; laev@labchem.sscc.ru
UDC 539.3
Karaul'nyi P.V.
Optimal control of the rigidity of an inclusion in an elastic bodyA three-dimensional elastic body with an inclusion is considered. There is a crack part of which is situated on the boundary of the inclusion. On the crack edges, there are given boundary conditions of the type of equalities and inequalities. We consider optimal control problem that allows to choose the safest inclusion from the standpoint of Griffith's criterion. An existence theorem of a solution to the optimal control problem is proved.
Keywords: crack, elastic inclusion, optimal control, Griffith's criterion.
Pp. 65–77.Karaul'nyi Pavel Vladimirovich
North-Eastern Federal University, 48 Kulakovskogo st., 677000 Yakutsk. E-mail: paulkaraulny@gmail.com
UDC 517.95
Nazarova L.A., Nazarov L.A., Karchevsky A.L., Vandamme M.
Estimation of the diffusion-capacitive parameters of a coal bed by the data of gas pressure measuring in a borehole based on inverse problem solvingA geomechanical model of gas emission from a block-structure coal bed is proposed. The inverse coefficient problem is formulated of finding the initial gas content, diffusion and mass-transfer coefficients by the gage data of the borehole pressure, the solvability of the problem is studied. It is shown that, for the uniqueness of a solution to the inverse problem, additional information is required on the gas-kinetic characteristics of the coal bed.
Keywords: geomechanical model, coal bed, inverse problem, cost function, gas content, diffusion and mass transfer coefficients, degassing.
Pp. 78–85.Nazarova Larisa Alekseevna
Nazarov Leonid Anatol'evich
Chinacal Institute of Mining; 54 Krasnyi av., 630091 Novosibirsk. E-mail: naz@misd.nsc.ru
Karchevskii Andrey Leonidovich
Sobolev Institute of Mathematics, 4 Koptyug av., 630090 Novosibirsk. E-mail: karchevs@math.nsc.ru
Vandamme Matthieu
Ecole des Ponts ParisTech-Laboratoire Navier, 6-8 Av. B. Pascal, 77420 Champs-sur-Marne, France. E-mail: matthieu.vandamme@enpc.fr
UDC 519.246
Nedel'ko V.M.
Regression models in the classification problemWe carry out a comparative analysis of the efficiency of several methods of classification (pattern recognition) based on regression models, in particular, logistic regression and its modifications. A new method for constructing a decision function is proposed. The method is based on the maximization of the area under the error curve over the linear functions in the space of the decision functions obtained by a special transformation. The performance of the method is illustrated by solving an applied problem.
Keywords: regression analysis, pattern recognition, machine learning, decision function, misclassification probability, logistic regression, Fisher linear discriminant.
Pp. 86–98.Nedel'ko Viktor Mikhailovich
Sobolev Institute of Mathematics, 4 Koptyug av., 630090 Novosibirsk. E-mail: nedelko@math.nsc.ru
UDC 530.12:532.5
Ostapenko V.V.
Conservation laws of shallow water theory and the Galilean relativity principleUsing an example of a model of shallow water theory, we show that the compatibility analysis of Hugoniot conditions for various basic systems of conservation laws in a coordinate system moving together with a strong discontinuity can lead to erroneous results. This is connected with the hierarchy of conservation laws of shallow water theory under a Galilean transformation, by which the law of conservation of the total energy is unconditionally noninvariant under this transformation. As a result, the Hugoniot condition corresponding to this conservation law depends on movement speed of the inertial coordinate system. It is shown that the specified defect of the classical model of shallow water theory is absent in the vortex shallow water model suggested by V. M. Teshukov.
Keywords: conservation laws of shallow water theory, Galilean relativity principle, classical and weak solutions
Pp. 99–113.Ostapenko Vladimir Viktorovich
Lavrent'ev Institute of Hydrodynamics, 15 Lavrent'ev av., 630090 Novosibirsk. E-mail: ostapenko_vv@ngs.ru
UDC 539.376:517.958
Ostrosablin N.I.
On one model of anisotropic creep of materialsThe concept of proper modules and the states, revealing the structure of generalized Hooke's law, is applied to one model of the anisotropic steady-state creep of materials. The steady-state creep equations of incompressible materials are presented in an invariant form. The matrix of anisotropy coefficients of these materials is reduced to block form with nine independent components. The special case of an ortotropic incompressible material is considered for which the matrix of anisotropy coefficients corresponds to a nonor.
Keywords: steady-state creep, anisotropy coefficients, proper anisotropy coefficients and proper states, transversal isotropy, ortotropy, incompressibility.
Pp. 114–119.Ostrosablin Nikolay Il'ich
Lavrent'ev Institute of Hydrodynamics, 15 Lavrent'ev av., 630090 Novosibirsk. E-mail: abd@hydro.nsc.ru
UDC 517.984
Sedipkov A.A.
Direct and inverse problems of acoustic sounding in a layered medium with discontinuous parametersWe consider the direct problem of finding a solution to a one-dimensional acoustic equation with discontinuous coefficients on the whole line y\in R with boundary conditions of special kind at the interior point y = 0. We prove that the direct problem is uniquely solvable in the corresponding function space and obtain a special presentation for its solution. Along with the direct problem, we study the inverse problem of recovering the acoustic impedance of the medium from known one-sided limits of the solution to the direct problem and its derivative at the point y = 0. It is shown that, with the use of the obtained special representation of the direct problem, the inverse problem can be reduced to a inverse spectral problem for a Sturm —Liouville operator with discontinuous coefficients.
Keywords: direct and inverse problems, Sturm–Liouville operator, inverse spectral problem, acoustic impedance.
Pp. 120–134.Sedipkov Aydys Alekseyevich
Sobolev Institute of Mathematics, 4 Koptyug av., 630090 Novosibirsk;
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk. E-mail: sedipkov@gmail.com
UDC 517.9
Frolenkov I.V., Romanenko G.V.
On the solvability of special systems of one-dimensional loaded parabolic equations and composite-type systems with Cauchy dataWe study a one-dimensional system of two parabolic loaded equations of special kind and one-dimensional composite-type systems of special kind. Both systems are considered with Cauchy data. We obtain sufficient conditions for the existence of solutions in the class of smooth bounded functions. The method of splitting on the differential level (the weak approximation method) is used in the proof.
Keywords: inverse problem, direct problem, loaded equation, parabolic equation, weak approximation method, system of partial differential equations, composite-type system.
Pp. 135–148.Frolenkov Igor Vladimirovich
Romanenko Galina Viktorovna
Siberian Federal University, 79 Svobodny av., 660041 Krasnoyarsk. E-mail: igor@frolenkov.ru; galina.romanencko@yandex.ru