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Sibirskii Zhurnal Industrial'noi Matematiki
2014, vol. 17, No 4 (60)Contents
UDC 519.622.2
Aida-zade K.R., Ashrafova Y.R.
Solving a system of differential equations of block structure with unseparated boundary conditionsWe study the solution of a system of higher-dimensional ordinary differential equations of block structure. Separate subsystems are connected with each other by unseparated boundary conditions caused by an arbitrary relation between the boundary values of the solutions to the subsystems. For numerical solution, we propose a scheme of the method of transfer of boundary conditions taking into account some specific characteristics of systems under consideration. The results of numerical experiments are given.
Keywords: system of differential equations, block structure of the system, unseparated boundary condition, method of transfer of conditions, Cauchy problem, Runge–Kutta method.
Pp. 3–13.Aida-zade Kamil Rajab
Azerbaijan State Oil Academy,
20 Azadlig av., AZ 1010, Baku, Azerbaijan
E-mail: kamil_aydazade@rambler.ru
Ashrafova Yegana Ramiz
Institute of Cybernetics of the Azerbaijan National Academy of Sciences, 9 B.Vahabzade st., AZ 1141 Baku, Azerbaijan
E-mail: y_aspirant@yahoo.com
UDC 517.946
Kozhanov A.I., Amirov Sh.
Solvability of a mixed problem for some of higher-order Sobolev-type strongly nonlinear equationsWe study the solvability of a mixed initial-boundary value problem for Sobolev-type strongly nonlinear equations of order 2m in the space variable. The existence and uniqueness of regular solutions are proved.
Keywords: Sobolev-type equation, strongly nonlinear divergent and nondivergent equations, mixed problem, regular solution, existence, uniqueness.
Pp. 14–30.Amirov Sharif
Karabuk University, 78050 Karabuk, Turkey
Kozhanov Aleksandr Ivanovich
Sobolev Institute of Mathematics SB RAS, 4 Koptyug av., 630090 Novosibirsk;
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk.
E-mail: kozhanov@math.nsc.ru
UDC 517.9
Anikonov Yu.E., Ayupova N.B.The Hopf–Cole transformation and multidimensional representations of solutions to evolution equations
We consider new identities and representations of solutions to second-order differential equations connected with the Hopf–Cole transformation.
Keywords: inverse problem, evolution equation, Hopf–Cole transformation.
Pp. 31–37.Anikonov Yurii Evgen'evich
Ayupova Natalia Borisovna
Sobolev Institute of Mathematics SB RAS, 4 Koptyug av., 630090 Novosibirsk;
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk. 630090 Novosibirsk
E-mail: anikon@math.nsc.ru; ayupova@math.nsc.ru
UDC 517.956.3:532.135
Blokhin A.M., Semisalov B.V.
A stationary flow of an incompressible viscoelastic polymeric fluid through a channel with elliptical cross sectionSome boundary-value problem for a quasilinear elliptic equation is posed. The solution to this problem defines the velocity profile for a stationary flow of an incompressible viscoelastic polymeric fluid through a tube having elliptical cross section. The problem is solved numerically with the use of a nonlocal algorithm without saturation.
Keywords: rheological model, boundary value problem, quasilinear elliptic equation, nonlocal numerical method.
Pp. 38–47.Blokhin Aleksandr Mikhailovich
Sobolev Institute of Mathematics SB RAS, 4 Koptyug av., 630090 Novosibirsk;
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk.
E-mail: Blokhin@math.nsc.ru
Semisalov Boris Vladimirovich
Design Technological Institute of Digital Techniques SB RAS, 6 Rzhanov st., 630090 Novosibirsk.E-mail: ViBiS@ngs.ru
UDC 517.98:519.677
Derevtsov E.Yu., Maltseva S.V., Svetov I.E.
Approximate recovery of a function given in a domain with low refraction from the ray integrals of the functionWe suggest an approach to the recovery of a function given in a Riemannian domain with low refraction from the ray integrals of the function. We construct an inversion algorithm with the use of the back-projection operator and the fast Fourier transform. The algorithm is investigated by numerical methods.
Keywords: tomography, refraction, ray transform, back-projection operator, inversion formula, fast Fourier transform.
Pp. 48–59.Derevtsov Evgenii Yur'evich
Mal'tseva Svetlana Vasilievna
Svetov Ivan Evgen'evich
Sobolev Institute of Mathematics SB RAS, 4 Koptyug av., 630090 Novosibirsk;
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk.
E-mail: dert@math.nsc.ru; sv-maltseva@mail.ru; svetovie@math.nsc.ru
UDC 517.994
Imomnazarov K.K., Imomnazarov S.K., Mamatkulov M.M., Chernykh E.G.
A fundamental solution to the stationary equation for two-velocity hydrodynamics with one pressureWe construct a fundamental solution for describing the three-dimensional steady flows of viscous fluids of a two-velocity continuum with phase equilibrium pressure.
Key words: two-velocity hydrodynamics, viscous fluid, fundamental solution, potentials of single and double layer.
Pp. 60–66.Imomnazarov Kholmatzhon Khudaynazarovich
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, 6 Lavrent'ev av., 630090 Novosibirsk. E-mail: imom@omzg.sscc.ru
Imomnazarov Sherzad Kholmatzhonovich
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk.
E-mail: shirz999@mail.ru
Mamatkulov Musazhon Mashrabovich
Nizami Tashkent State Pedagogical University, 103 Usufa Khos Khodjiba Str., 100064 Tashkent
E-mail: musa-mm@mail.ru
Chernykh Evgenii Gennadievich
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, 6 Lavrent'ev av., 630090 Novosibirsk. E-mail: jenya-ch@ngs.ru
UDC 533.951:517.948
Medova Yu.A., Chesnokov A.A.
Shear Hele–Shaw flows of a weakly compressible fluidWe derive the integro-differential system of equations describing shear flows of a weakly compressible multicomponent fluid in a Hele–Shaw cell. The propagation velocities of nonlinear perturbations in the fluid are determined, and the characteristic form of the system is obtained. We formulate conditions for the hyperbolicity of the model that are necessary for a correct statement of the Cauchy problem. One-dimensional multilayered flows are considered, and the interpretation of the Saffman–Taylor instability in the framework of the two-layered fluid scheme is given.
Key words: Hele–Shaw flow, barotropic fluid, shear flow, hyperbolicity, stability.
Pp. 67–78.Medova Yulia Aleksandrovna
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk.
Chesnokov Aleksandr Aleksandrovich
Lavrent'ev Institute of Hydrodynamics SB RAS, 15 Lavrent'ev av., 630090 Novosibirsk
E-mail: chesnokov@hydro.nsc.ru
UDC 533.72
Popov V.N., Smolenskaya E.A.
Mathematical modeling of transfer processes in the problem of the thermal slip of a gas along a flat surfaceIn the framework of the kinetic approach, we construct an analytic solution (in the form of a Neumann series) to the problem of the thermal slip of a gas along a rigid flat surface. As the basic equation, we use the linearized ellipsoidal statistical model of the Boltzmann kinetic equation, and as boundary conditions on the streamlined surface we take the mirror-diffuse reflection model. For the various values of the accommodation coefficient of the tangential momentum of the molecules of the gas, we compute the velocity of the thermal slip of the gas along the surface and find the distributions of the gas velocity and the heat flow vector. A comparison with similar results available in the previously published works is made.
Keywords: Boltzmann kinetic equation, model kinetic equation, boundary condition model, exact analytical solution.
Pp. 79–87.Popov Vasilii Nikolaevich
Smolenskaya Elena Aleksandrovna
Lomonosov Northern (Arctic) Federal University, 17 Severnaya Dvina Emb., 163002 Arkhangelsk. E-mail: v.popov@agtu.ru; e.smolenskaya@agtu.ru
UDC 533.16:532.5
Sidnyaev N.I., Gordeeva N.M.
The asymptotic theory of flows for the near wake of an axisymmetric bodyWe obtain the solution of the problem of the near wake of a thin cylinder in an incompressible laminar flow in the form of an asymptotic series. For finding the solution, we use the method of series expansion for solutions for the interior and exterior parts of the flow. The influence of the circulating current is neglected. The solution is applicable only at a bounded distance from the aft of the body. There are significant differences of the axisymmetric problem from the corresponding problem of the two-dimensional wake of a flat plate.
Keywords: asymptotic method, near wake, flow over, axisymmetric body, differential equation, series, boundary layer.
Pp. 88–97.Sidnyaev Nikolay Ivanovich
Gordeeva Nadezhda Mikhailovna
Bauman Moscow State Technical University, Baumanskaya 2ya st., 105005 Moscow. E-mail: sidnyaev@bmstu.ru; nmgordeeva@bmstu.ru
UDC 519.632
Sorokin S.B.
Justification of a discrete analog of the conjugate-operator model of the heat conduction problemFor the conjugate-operator model of the heat conduction problem, we construct and justify a discrete analog preserving the structure of the initial model. The justification of convergence is carried out for a difference scheme in the conjugate-operator form. It is shown that the difference scheme converges with second-order accuracy for the cases of discontinuous medium parameters in the Fourier law and nonuniform grids.
Key words: heat conductivity problem, mathematical model, discrete analog, approximation, stability, convergence, difference scheme
Pp. 98–110.Sorokin Sergey Borisovich
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, 6 Lavrent'ev av.;
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk.
E-mail: sorokin@sscc.ru
UDC 517.95
Tereshko D.A.
Numerical reconstruction of the boundary heat flow for stationary heat convection equationsA numerical algorithm is proposed for solving the inverse problem for the stationary heat convection equations based on the constrained minimization methods. This algorithm is applied for estimating the heat flow on a part of the boundary from the measured values of the temperature or the velocity vector in the flow domain.
Keywords: heat convection, inverse problem, minimization problem, numerical method.
Pp. 111–119.Tereshko Dmitrii Anatol'evich
Institute of Applied Mathematics FEB RAS, 7 Radio st., 690041 Vladivostok
E-mail: ter@iam.dvo.ru
UDC 519.63:621.38
Fadeev S.I., Kostsov E.G., Pimanov D.O.
Numerical study of mathematical models of MEMS resonators of different typesWe present the results of a numerical study of mathematical models of MEMS resonators (microelectromechanical systems). The mathematical models of the devices are given by the statements of initial boundary value problems that describe the oscillations of the mobile electrode. Oscillations occur during the launch under the influence of the electrostatic field between the mobile and fixed electrodes in the micro gap, and then, after the launch, become free oscillations. The conditions for the appearance of oscillations of the mobile electrode are numerically determined in the form of a connection between the launch parameters and the parameters of the free oscillations.
Keywords: electrostatic attraction, method of lines, self-oscillation, free oscillation.
Pp. 120–135.Fadeev Stanislav Ivanovich
Sobolev Institute of Mathematics SB RAS, 4 Koptyug av., 630090 Novosibirsk;
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk.
E-mail: fadeev@math.nsc.ru
Kostsov Eduard Gennadievich
Institute of Automation and Electrometry SB RAS, 1 Koptyug av., 630090 Novosibirsk.
E-mail: kostsov@iae.nsk.su
Pimanov Daniil Olegovich
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk.
E-mail: pimanov-daniil@yandex.ru
UDC 622.272:516.02
Cherdantsev S.V., Cherdantsev N.V.
Pontoon rolling on regular waves in the sump of a coal quarryIn the framework of the linear theory of hydromechanics, we consider the problem of the rolling of a pontoon on regular waves in the sump of a coal quarry, in solving which we find the basic characteristics of the rolling, construct the graphs of their dependences on some parameters of the pontoon, and find the dangerous mode of its rolling.
Keywords: pontoon, velocity potential, Laplace equation, Lagrange integral, fluid waves, rolling, stability of a pontoon on waves
Pp. 136–146.Cherdantsev Sergey Vasil'evich
Cherdantsev Nikolay Vasil'evich
Gorbachev Kuzbass State Technical University, 28 Vesennyaya st., 650000 Kemerovo.
E-mail: sych01@yandex.ru; nych2014@yandex.ru