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Sibirskii  Zhurnal  Industrial'noi  Matematiki
2016,  vol. 19,  No 3 (67)

Contents

UDC 517.958
DOI 10.17377/sibjim.2016.19.301

Bondarenko A. N.,  Bugueva T. V.,  Dedok V. A.
Inverse problems of anomalous diffusion theory: the artificial neural network approach

We present the results of the computer modeling of the work of a three-layer perceptron trained to solve inverse problems of anomalous diffusion theory.
Several types of inverse problems  are considered including the recovery of the Hurst exponent of a self-similar medium.

Keywords: anomalous diffusion, inverse problem, artificial neural network.
Pp. 3-14.

Bondarenko Anatolii Nikolaevich
Sobolev Institute of Mathematics SB RAS
Acad. Koptyug ave., 4
Bugueva Tatiana Vladimirovna
Sobolev Institute of Mathematics SB RAS
Novosibirsk State University
Pirogova str., 2
Dedok Vasily Aleksandrovich
Sobolev Institute of Mathematics SB RAS
630090 Novosibirsk
E-mail: bondarenkoan1953@mail.ru; bugueva@math.nsc.ru; dedok@math.nsc.ru

UDC 519.23
DOI 10.17377/sibjim.2016.19.302

Denisov V. I., Timofeeva A. Yu., Khailenko E. A.
Estimating polynomial models with errors in variables without additional information

We consider the problem of estimating a polynomial model with classical error in the input factor in the functional case. The nonparametric method
of estimation of structural dependencies does not use additional information but is extremely hard computationally and requires a large sample size. That is why we propose a number of easier approaches. The first approach is based on the preliminary estimation of the Berkson error variance under the assumption of its normality for a piecewise-linear model. The so-obtained estimate is used to calculating the parameters of the polynomial by the methods of general and adjusted least squares. In the case when the error distribution deviates from the normal distribution, we develop a second method, the adaptive estimation method, based on te universal lambda-distribution. The proposed approaches were developed for solving the problem of analyzing the level of knowledge.

Keywords: model with errors in both variables, method of generalized least squares, method of adjusted least squares, maximum likelihood method, adaptive method, generalized distribution.
Pp. 15-28.

Denisov Vladimir Ivanovich
Timofeeva Anastasia Yurievna
Khailenko Ekaterina Alekseevna
Novosibirsk State Technical University
K. Marx ave., 20
630073 Novosibirsk

E-mail: a.timofeeva@corp.nstu.ru; ekavka@yandex.ru

UDC 517.95
DOI 10.17377/sibjim.2016.19.303

Namsaraeva G.
Inverse problems of the determination of external sources in the equation of longitudinal wave propagation

We consider inverse problems for the equation of longitudinal wave propagation with overdetermination conditions of final and integral types. The main purpose of the research is the proof of the existence of regular solutions to the inverse determination problems together with the solution of the also unknown external sources. One of the proposed approaches is based on the reduction of the inverse problem to an integro-differential equation.

Keywords: equation of longitudinal wave propagation, inverse problem, integro-differential equation, a priori estimate, regular solution, existence, uniqueness.
Pp. 28-40.

Namsaraeva Gerelma Vladimirovna
East Siberian State University of Technology and Management
Klyuchevskaya str., 40V
670013 Ulan-Ude
E-mail: gerel@inbox.ru

UDC 517.95
DOI 10.17377/sibjim.2016.19.304

Popova T. S.
A contact problem for a viscoelastic plate and an elastic beam

We consider the problem of contact of a viscoelastic plate with an elastic beam. For characterizing the viscoelastic deformation of the plate, we use hereditary integrals. We present a differential statement of the problem with conditions having the form of a system of equalities and inequalities in the domain
of possible contact and prove its equivalence to a variational inequality. We establish the unique solvability and the existence of the derivative
of a solution with respect to time. The limit problem is considered with the parameter of bending rigidity of the plate tending to infinity.

Keywords: viscoelasticity, beam, plate, hereditary integral, variational inequality, nonpenetration condition.
Pp. 41-55.

Popova Tatiana Semenovna
North-Eastern Federal University
Kulakovsky str., 48
677000 Yakutsk
E-mail: ptsokt@mail.ru

UDC 517.95
DOI 10.17377/sibjim.2016.19.305

Prokudin D. A.,  Krayushkina M. V.
Solvability of a steady boundary value problem for a model system of equations of a barotropic motion of a mixture of viscous compressible fluids

We consider a boundary value problem for a model system of equations describing a steady barotropic motion of a homogeneous mixture of viscous compressible fluids in a bounded three-dimensional domain. An existence theorem is proved for weak solutions to the problem without constraints on the structure of the total viscosity matrix except the standard requirements of positive definiteness.

Keywords: existence theorem, steady boundary value problem, viscous compressible fluid, homogeneous mixture with two velocities, effective viscous flux.
Pp. 55-67.

Prokudin Dmitry Alekseevich
Lavrent'ev Institute of Hydrodynamics SB RAS
Acad. Lavrent'ev ave., 15
630090 Novosibirsk
Krayushkina Marina Vladimirovna
Kemerovo State University
Krasnaya str., 6
650043 Kemerovo

E-mail: prokudin@hydro.nsc.ru; krayushkinamv@mail.ru

UDC 532.542.3
DOI 10.17377/sibjim.2016.19.306

Proskurin A. V., Sagalakov A. M.
Mathematical modeling of one flow in a pipe using the method of $R$-functions

The article deals with the flow in a pipe of rectangular cross section with an inner circular cylindrical element. We use a numerical method that is based
on $R$-functions. This method is meshfree and therefore more efficient than the finite element method which requires remeshing when the geometry of the problem is changed. We consider the dependence of the flow on the diameter of the central cylinder and its position in the square pipe under constant pressure gradient. It is found that the resistance decreases when the inner element is displaced from the center of the pipe.

Keywords: $R$-function, pipe flow, meshfree method.
Pp. 68-74.

Sagalakov Anatoly Mikhailovich
Proskurin Alexander Viktorovich
Altai State University
Lenin ave., 61
656049 Barnaul

E-mail: amsagalakov@mail.ru; k210@list.ru

UDC 539.3:517.97
DOI 10.17377/sibjim.2016.19.307

Pyatkina E. V.
Optimal control of the shape of a layer shape in the equilibrium problem of elastic bodies with overlapping domains

We consider the equilibrium problem for a two-layer elastic body. One of the plates contains a crack. The second is a disk centered
at one of the crack tips. The spherical layer is glued by its edge to the first plate. The unique solvability  is proved of the problem
in the nonlinear setting. An optimal control problem is also considered. The radius of the second layer $a$ is chosen as the control function. It is assumed that $a$ is positive and takes values in a closed interval. We show that there exist a value of $a$ minimizing the functional that characterizes the change of the potential energy as the crack length increases and a value of $a$ that characterizes the opening of the crack.

Keywords: elastic plate, overlapping domain, crack with nonpenetration, optimal control problem.
Pp. 75-84.

Pyatkina Evdokia Vladimirovna
Lavrent'ev Institute of Hydrodynamics SB RAS
Acad. Lavrent'ev ave., 15
630090 Novosibirsk
E-mail: dusya_pyatkina@mail.ru

UDC 621.224.35
DOI 10.17377/sibjim.2016.19.308

Skorospelov V. A., Turuk P. A.
Generation of a multiparameter family of surfaces of the runner blade of a Kaplan turbine for subsequent shape optimization

The paper proposes a method for the creation of a multiparameter family of surfaces of the runner blade of a Kaplan turbine for finding the best options by varying the initial prototype of the surface of the blade.

Keywords: Kaplan turbine, runner blade, variation of a surface, optimization.
Pp. 85-89.

Skorospelov Vladimir Anatol'evich
Turuk Polina Aleksandrovna
Sobolev Institute of Mathematics SB RAS
Acad. Koptyug ave., 4
630090 Novosibirsk
E-mail: vskrsp@math.nsc.ru; turuk@math.nsc.ru

UDC 539.3:517.958
DOI 10.17377/sibjim.2016.19.309

Fankina I. V.
A contact problem for an elastic plate with a thin rigid inclusion

An equilibrium problem for a plate under the influence of external forces is investigated. It is assumed that the plate contains a thin rigid inclusion that reaches the boundary under zero angle and is in partial contact with an undeformable solid. There is a delamination at one of the faces of the inclusion. A complete Kirchhoff—Love model is considered, where the unknown functions are the vertical and horizontal displacements of the points of the middle surface of the plate. We give a differential statement and a variational statement of the problem and prove the existence and uniqueness of a solution.

Keywords: plate, rigid inclusion, contact problem, fictitious domain.
Pp. 90-98.

Fankina Irina Vladimirovna
Novosibirsk State University
Pirogova str., 2
Lavrent'ev Institute of Hydrodynamics  SB RAS
Acad. Lavrent'ev ave., 15
630090 Novosibirsk
E-mail: fankina.iv@gmail.com

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