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Sibirskii  Zhurnal  Industrial'noi  Matematiki
2016,  vol. 19,  No 1 (65)

Contents

UDC 517.941.1:532.529.5
DOI 10.17377/sibjim.2016.19.101

Andreev V. K. and Cheremnykh E. N.
A joint creeping motion of three fluids in a flat layer: a priori estimates and convergence to a stationary regime

We study a partially invariant solution of rank~2 and defect~3e to the equations of a viscous heat-conducting fluid. It is interpreted as a two-dimensional motion of three immiscible fluids in a flat channel bounded by solid walls for which the distribution of temperature is known. From a mathematical point of view, the resulting initial boundary value problem is nonlinear and inverse. Under some assumptions (often fulfilled in practical applications), the problem
is replaced by a linear one. We obtain a priori estimates  as well as the exact stationary solution and prove that,  the solution tends  to a stationary regime if the temperatures of the walls stabilize with  time.

Keywords: thermocapillarity, a priori estimate, conjugate boundary value problem, asymptotic behavior.
Pp. 3-17.

Andreev Viktor Konstantinovich
Cheremnykh Elena Nikolaevna
Institute of Computational Modeling SD RAS
50/44 Akademgorodok
660036 Krasnoyarsk
Siberian Federal University
79 Svobodnyi av.
660041 Krasnoyarsk
E-mail: andr@icm.krasn.ru; elena_cher@icm.krasn.ru

UDC 517.958
DOI 10.17377/sibjim.2016.19.102

Anikonov D. S. and Kipriyanov Ya. A.
An underdetermined problem of integral geometry for the generalized Radon transform

Under study is some new problem of integral geometry. All planes are considered in the three-dimensional Euclidean space. The  data are given by the integrals over all such planes of an unknown piecewise-smooth function depending both on the spatial variables and the variables characterizing the planes. The sought object is an integrant discontinuity surface of the first kind. The uniquiness theorem of of the desired surface is proved. The research od the papert presents an aspects of the theory of probing  an  unknown medium by various physical signals.

Keywords: integral geometry, generalized Radon transform, probing, unknown boundaries.
Pp. 18-26.

Anikonov Dmitrii Sergeevich
Sobolev Institute of Mathematics SD RAS
4 Koptyug av.
Novosibirsk State University
2 Pirogova st.
Kipriyanov Yaroslav Andreevich
Novosibirsk State University
630090 Novosibirsk
E-mail: anik@math.nsc.ru; yaroslav.kipriyanov@gmail.com

UDC 539.3:517.958
DOI 10.17377/sibjim.2016.19.103

Annin B. D. and  Ostrosablin N. I.
Reflection of plane waves from a rigid wall and a free surface in a transverse isotropic medium

We provide a representation of the general solution to the two-dimensional equations of the dynamics of a transverse isotropic medium
with the Carrier—Gassmann condition. The representation of the solution is based on two solving functions that satisfy two separate wave equations.
The problem of the reflection of plane waves from a rigid wall and a free surface is solved. The reflection coefficients and transformations of plane waves
are found. The obtained formulas imply the solution for an isotropic medium as well. Special cases are considered, where the forms (amplitudes) of the reflected waves are not determined uniquely but related to the form of a falling wave through a linear relation.

Keywords: transverse isotropy, plane wave, Carrier—Gassmann condition, coefficients of reflection and transformation.
Pp. 27-36.

Annin Boris Dmitrievich
Lavrent'ev Institute of Hydrodynamics SD RAS
15 Lavrent'ev av.
630090 Novosibirsk
Novosibirsk State University
2 Pirogova st.
Ostrosablin Nikolay Il'ich
Lavrent'ev Institute of Hydrodynamics SD RAS
630090 Novosibirsk
E-mail: annin@hydro.nsc.ru; abd@hydro.nsc.ru

UDC 517.925
DOI 10.17377/sibjim.2016.19.104

Bagderina Yu. Yu.
Group classification of second-order projective-type ODEs

Group classification with respect to admitted point transformation groups is implemented for second-order ordinary differential equations with cubic nonlinearity in the first-order derivative. The result is obtained with the use of invariants of the equivalence transformation group of the family of equations under consideration. The corresponding Riemannian metric is found for the equations that are the projection of a system of geodesics to a two-dimensional surface.

Keywords: transformation group, symmetry, equivalence, invariant, group classification.
Pp. 37-51.

Bagderina Yulia Yur'evna
Institute of Mathematics with Computer Center of the Ufa Scientific Center of RAS
112 Chernyshevskii st.
Ufa 450008 Russia
E-mail: yulya@mail.rb.ru

UDC  517.956.3:532.135:532.527
DOI 10.17377/sibjim.2016.19.105

Blokhin A. M. and  Semenko R. E.
On one model of a vortex motion of an incompressible polymeric fluid in the axial zone

We construct a nonstationary mathematical model of a vortex motion of an incompressible polymeric fluid. Some partial solutions to this model are found in the stationary case. We deduce a variant of this model when the pressure is time-independent along the axis.

Keywords: rheological model, nonstationary mathematical model, vortex motion.
Pp. 52-61.

Blokhin Aleksandr Mikhailovich
Sobolev Institute of Mathematics SD RAS
4 Koptyug av.
Novosibirsk State University
2 Pirogova st.
Semenko Roman Evgen'evich
Novosibirsk State University
630090 Novosibirsk

E-mail: blokhin@math.nsc.ru; rsem86@mail.ru

UDC 517.911.5
DOI 10.17377/sibjim.2016.19.106

Konovalova D. S.
Localization for the discontinuity line of the right-hand side of a differential equation

We propose a new approach to the study of inverse problems for differential equations with constant coefficients. Its application is illustrated by the example of one partial differential equation with three independent variables. The right-hand side of the equation  is assumed to be a  discontinuous function of space
variables. The inverse problem is  to find some hull containing the discontinuity line of the right-hand side. An algorithm for constructing such a hull is obtained: it is a square whose sides are tangent to the discontinuity line.

Keywords: inverse problem, discontinuous function, weak solution, differential properties.
Pp. 62-72.

Konovalova Dina Sergeevna
Sobolev Institute of Mathematics SD RAS
4 Koptyug av.
630090 Novosibirsk
E-mail: dsk@math.nsc.ru

UDC 622.271
DOI 10.17377/sibjim.2016.19.107

Nazarova L. A. and  Nazarov L. A.
Diagnostics of anti-seepage screen at a tailings dam in permafrost based on the solution of an inverse problem by piezometric measurement data

We suggest a nonlinear geomechanical model of rock mass in the vicinity of a tailings dam in the permafrost zone at the Kumtor Mine in the Kyrgyz Republic.
The model takes into account information on the structure of the object and the data on the deformation, strength, thermophysical, and filtration
characteristics of frozen and thawed ground, as well as on the seasonal oscillations of air temperature. The numerical experiments show that,
under unchanged external conditions, rock mass properties, and the position of the neutral layer, the zero isotherm, separating the frozen and thawed rocks, attains a stationary position in 12-15 years after the filling of the tailings pond and that a slight rupture of the anti-seepage screen can result in a large damage
zone in the dam. Synthetic data are used to illustrate the solvability of the  boundary value inverse problem of finding the time and place of the rupture in the anti-seepage screen from piezometric measurements in a few observation holes.

Keywords: heat-and-mass transfer, stressed state, protection dam, permafrost, anti-seepage screen, finite element method, inverse problem
Pp. 73-81.

Nazarova Larisa Alekseevna
Nazarov Leonid Anatol'evich
Chinakal Institute of Mining SD RAS
54 Krasnyi av.
630091 Novosibirsk
E-mail: larisa@misd.nsc.ru; naz@misd.nsc.ru

UDC 517.9
DOI 10.17377/sibjim.2016.19.108

Tersenov Ar. S.
On the existence of nonnegative solutions to the Dirichlet boundary value problem for the p-Laplace equation in  presence of external mass forces

Consider the Dirichlet problem for an inhomogeneous p-Laplace equation with nonlinear source in  presence of external mass forces, we obtain new sufficient conditions for the existence of a weak nonnegative bounded solution. The  conditions are written in explicit form in terms of the data of the problem.

Keywords: p-Laplace equation, regularized equation, a priori estimate.
Pp. 82-93.

Tersenov Aris Savvich
Sobolev Institute of Mathematics SD RAS
4 Koptyug av.
Novosibirsk State University
2 Pirogova st.
630090 Novosibirsk
E-mail: aterseno@math.nsc.ru

UDC 519.62:577.218:57.023
DOI 10.17377/sibjim.2016.19.109

Fadeev S. I., Kogai V. V., Khlebodarova T. M., and Likhoshvai V. A.
On the numerical study of periodic solutions to delay equations in biological models

We present the results of a numerical study of periodic solutions of to a nonlinear delay equation in connection with mathematical models having a real biological prototype. The problem is formulated as a boundary value problem for a delay equation with periodicity and transversality conditions. We propose
a spline-collocation finite-difference scheme of the boundary value problem using the Hermite interpolation cubic spline of class $C^1$ with fourth-order error. For the numerical study of the system of nonlinear equations of the difference scheme, the parameter-extension method is used, which allows us to identify the possible nonuniqueness of a solution to the boundary value problem and hence the nonuniqueness of periodic solutions regardless of their stability. It is shown by examples that periodic oscillations arise for values of the parameters typical for real molecular-genetic systems of higher organisms, for which the principle of "delay" is rather easy to implement.

Keywords: ordinary differential equation, delay, continuation method, boundary value problem, oscillation.
Pp. 94-105.

Fadeev Stanislav Ivanovich
Kogai Vladislav Vladimirovich
Sobolev Institute of Mathematics SD RAS
4 Koptyug av.
Novosibirsk State University
2 Pirogova st.
Likhoshvai Vitalii Aleksandrovich
Institute of Cytology and Genetics SB RAS
10 Lavrent'ev av.
Novosibirsk State University
Khlebodarova Tamara Mikhailovna
Institute of Cytology and Genetics SB RAS
630090 Novosibirsk
E-mail: fadeev@math.nsc.ru; kogai@math.nsc.ru; likho@bionet.nsc.ru; tamara@bionet.nsc.ru

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