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Sibirskii Zhurnal Industrial'noi Matematiki
2012, vol. 15, No 4 (52)
Contents
UDC 517.95
Alekseev G.V. and Shepelov M.A.
On the stability of solutions to coefficient inverse extreme problems for the stationary convection-diffusion equationWe study the coefficient inverse problem for the extreme stationary convection-diffusion, considered in a bounded domain with mixed boundary conditions on the boundary. The role of control is played by the velocity vector of the medium and the functions involved in the boundary conditions for the temperature. The solvability of extremal problems is proved for arbitrary weak lower semicontinuous quality functional as well as for specific quality functionals. On the basis of the analysis of the optimality system, we establish sufficient conditions on the initial data that guarantee the uniqueness and stability of optimal solutions under small perturbations of the quality functional as well as of one of the functions outside the initial boundary value problem.
Keywords: convection-diffusion equation, temperature, velocity vector, multiplicative control, coefficient inverse problems, existence, uniqueness, stability.
Pp. 3–16.Alekseev Gennadii Valentinovich
Institute of Applied Mathematics FEB RAS, 7 Radio str., Vladivostok 690041;
Vladivstok State University of Economic and Service, 41 Gogol' str., Vladivostok 690014, E-mail: alekseev@iam.dvo.ru
Shepelov Mikhail Alekseevich
Far East Federal University, 8 Sukhanov str., Vladivostok 690060;
Institute of Applied Mathematics FEB RAS, 7 Radio str., Vladivostok 690041 E-mail: root@iam.dvo.ru
UDC 517.9
Anikonov Yu. E. and Neshchadim M. V.
Representations for the solutions and coefficients of second-order differential equationsNew representations are given for the solutions and coefficients of second-order evolution differential equations in linear and nonlinear cases. The formulas obtained for linear equations have wide arbitrariness, which can be used in identification problems. We study questions of running-wave type for nonlinear one-dimensional equations.
Keywords: second-order evolution differential equations, Poisson formula, algebraic-analytic identities.
Pp. 17–23.Anikonov Yurii Evgen'evich
Neschadim Mikhail Vladimirovich
Sobolev Institute of Mathematics SDRAS, 4 Koptyug av., Novosibirsk 630090;
Novosibirsk State University, 2 Pirogova st., Novosibirsk 630090, E-mail: anikon@math.nsc.ru ; neshch@math.nsc.ru
UDC 519.624:534.1
Akhtyamov A.M. and Muftakhov A.V.
Tikhonov well-posedness of the identification problem for fixing conditions for mechanical systemsThe problem of the identification of fixing conditions is considered for distributed mechanical systems from three natural frequencies of their oscillations. Basing on the Plucker condition, which appears in the reconstruction of a matrix from its minors of maximal order, we construct the well-posedness set of the problem and prove its Tikhonov well-posedness. For a wide class of problems, we find an explicit solution to the identification problem for the matrix of boundary conditions written down in terms of the characteristic determinant of the corresponding spectral problem. We give examples of the solution of specific problems of mechanics and a counterexample showing that two natural frequencies are not enough for the uniqueness in the identification of the boundary conditions.
Keywords: Tikhonov well-posedness, inverse problem, identification of boundary conditions, Plucker condition, distributed mechanical systems, natural frequencies, eigenvalues.
Pp. 24–37.Akhtyamov Azamat Mukhtarovich
Mavlyutov Institute of Mechanics of the Ufa Scientific Center of RAS, 71 Oktyabrya av., Ufa 450054;
Bashkir State University, 32 Validi str., Ufa 450074 E-mail: AkhtyamovAM@mail.ru
Muftakhov Artur Vil'evich
Sami Shamoon College of Engineering, Basel/Bialik strs., Beer Sheva 84100, ISRAEL E-mail: muftahov@yahoo.com
UDC 517.956.32:539.3:517.956:512.816
Bel'metsev N. F. and Chirkunov Yu. A.
Exact solutions to the equations of the dynamic asymmetric model of elasticityUsing the group stratification of the equations of the dynamic asymmetric model of elasticity effectively used in the study of elastic materials made of polymers, we obtain a system that, after renaming the functions, becomes equivalent to these equations and contains fewer additional functions than the union of the resolving and automorphic systems of the accomplished group stratification. Among first-order systems equivalent to these equations it contains the least number of additional functions and is the only such system up to a nondegenerate linear transformation of the additional functions. We find its principal Lie transformation group, an optimal system of its subgroups, and their universal invariants. Some invariant and partially invariant exact solutions are found; their physical meaning is explained.
Keywords: asymmetric elasticity, group stratification, optimal system of subgroups, invariant solutions.
Pp. 38–50.Bel'metsev Nikolay Fedorovich
Chirkunov Yurii Aleksandrovich
Novosibirsk State Technical University, 20 Karl Marx av., Novosibirsk 630092 E-mail: weqsmachine@gmail.com ; chr101@mail.ru
UDC 517.956.35:517.958
Blokhin A. M., Bychkov A.S., and Myakishev V.O.
Numerical analysis of the realizability of the conditions of neutral stability for shock waves in the problem of a flow past a wedge by a van der Waals gas.By means of numerical calculation, the question is solved of the realizability of the conditions of neutral stability for shock waves in the problem of a flow of real gases past a wedge.
Keywords: Lopatinskii condition, neutrally stable, shock waves, van der Waals gas.
Pp. 51–63.Blokhin Aleksandr Mikhailovich,
Sobolev Institute of Mathematics SDRAS, 4 Koptyug av., Novosibirsk 630090. E-mail: blokhin@math.nsc.ru
Bychkov Andrey Sergeevich,
Myakishev Vladislav Olegovich
Novosibirsk State University, 2 Pirogova st., Novosibirsk 630090 E-mail: bych.andrey@gmail.com ; myakvlad@rambler.ru
UDC 517.9:519.62
Vinogradova P. V. and Samusenko A. M.
A projection method for a third-order operator differential equation with a nonlinear monotone operatorWe study the Galerkin method for a third-order operator-differential equation with the main self-adjoint operator A and the subordinate nonlinear monotone operator K in a separable Hilbert space. The existence and uniqueness of a strong solution to the original problem are proved. Convergence estimates for the Galerkin method are obtained.
Keywords: operator-differential equation, monotone operator, strong solution, convergence rate, Galerkin method
Pp. 64–70.Vinogradova Polina Vital'evna
Samusenko Aleksandr Markovich
Far Eastern State Transport University, Serisheva str. 47, Khabarovsk 680021. E-mail: vpolina17@hotmail.com ; samusenkosasha@inbox.ru
UDC 519.8
Gal'kova E. A., Maergoiz L. S., and Khlebopros R. G.
A matematical algorithm of a ``fair'' distribution of a humanitarian resource and related topicsWe set forth an optimization mathematical model for the distribution of a limited resource between different groups. Its construction is exemplified by the distribution of a humanitarian resource in an emergency situation.
Keywords: mathematical algorithm, extremal problem, distribution of a limited resource.
Pp. 71–77.Gal'kova Elena Aleksandrovna
Krasnoyarsk Transport Institute, 89 Lado Ketskhoveli str., Krasnoyarsk, 660028. E-mail: resurs7777@yandex.ru
Maergoiz Lev Sergeevich
Institute for the Control of Business Processes and Economics, Siberian Federal University, 82 Svobodnyi av. Krasnoyarsk 660041. E-mail: bear.lion@mail.ru
Khlebopros Rem Grigor'evich
Institute of Economics and Nature Management, Siberian Federal University, 79 Svobodnyi av., Krasnoyarsk 660041;
Presidium of the Krasnoyarsk Scientific Center SDRAS, 50 Akademgorodok, Krasnoyarsk 660036. E-mail: olikru@yandex.ru
UDC 519.24:519.711
Denisov V. I., Chubich V. M., and Filippova E. V.
Active parametric identification of stochastic continuous-discrete systems obtained by statistical linearizationFor stochastic continuous-discrete models obtained by statistical linearization, we consider the theoretical and applied aspects of active parametric identification in the case where the parameters of mathematical models to be estimated occur in the state, control, and perturbation matrices and the covariance matrices of dynamic noise and measurement errors. The results are given. An example of optimal parameter estimation for one model structure is considered.
Keywords: parameter estimation, maximum likelihood method, design of optimal input signals, information matrix, optimality criteria.
Pp. 78–89.Denisov Vladimir Ivanovich
Chubich Vladimir Mikhailovich
Filippova Elena Vladimirovna
Novosibirsk State Technical University, 20 Karl Marx av. Novosibirsk 630092. E-mail: videnis@nstu.ru ; chubich_62@ngs.ru ; alena-filippova@mail.ru
UDC 517.9:519.6
S. I. Kabanikhin and O. I. Krivorot'ko
A numerical method for solving the Dirichlet problem for the wave equationA numerical method for solving the Dirichlet problem for the wave equation in the two-dimensional space is constructed. An analysis of the ill-posedness of the problem is carried out and a reguralization algorthm is constructed. The first step in the regularization of the problem consists in expansion in a Forier series with respect to one of the variables and passage to a finite sequence of Dirichlet problems for the wave equation in the one-dimensional space. Each of the Dirichlet problems obtained for the wave equation in the one-dimensional space is reduced to the inverse problem Aq=f to some direct (correct) problem. We accomplish an analysis of the ill-posedness degree of the inverse problem on the basis of the study of the nature of the decay of the singular values of A and its discrete analog Amn. For relatively small values m and n, we develop a numerical algorithm for constructing r-solutions to the inverse problem. For the general case, we apply an optimization method for solving the inverse problem. The results of numerical calculations are given.
Keywords: Dirichlet problem, wave equation, ill-posedness degree, singular value decomposition.
Pp. 90–101.Kabanikhin Sergey Igorevich
Institute of Computational Mathematics and Mathematical Geophysics SD RAS, 6 Lavrent'ev av., Novosibirsk 630090. E-mail: kabanikhin@sscc.ru
Krivorot'ko Ol'ga Igorevna
Novosibirsk State University, 2 Pirogova st., Novosibirsk 630090. E-mail: krivorotko.olya@mail.ru
UDC 539.3:517.95
Nazarov L. A., Nazarova L. A., Karchevskii A. L., and Panov A. V.
Estimation of stresses and deformation properties of rock masses based on the solution of an inverse problem by the data of the measurement of free surface displacementWe propose a method for the interpretation of the measurement data of the variations of the displacements and strains on free surfaces induced by underground mining. The method makes it possible to estimate the components of the natural stress field and the elastic properties of the constructive elements of technologies of mineral mining. The method is based on the solution of inverse problems for the system of elasticity equation in domains of arbitrary configuration. We investigate the structure of the proposed target functions and determine the optimal values of the weight coefficients that guarantee the solvability of the inverse problem and the limit error level in the input data of different types.
Key words: inverse problem, elasticity, target function, natural stresses, mining
Pp. 90–101.Nazarov Leonid Anatol'evich
Nazarova Larisa Alekseevna
Panov Anton Vladimirovich
Chinakal Institute of Mining SD RAS, 54 Krasnyi av., Novosibirsk 630091. E-mail: naz@misd.nsc.ru
Karchevskii Andrey Leonidovich
Sobolev Institute of Mathematics SDRAS, 4 Koptyug av., Novosibirsk 630090. E-mail: karchevs@math.nsc.ru
UDC 519.62:577.218:57.017.723:004.94:57:51
Ri N. A., Khlebodarova T. M., Kogai V. V., Fadeev S. I., and Likhoshvai V. A.
The bistability of nitrite utilization by Escherichia coli: analysis of the mathematical modelA mathematical model describing nitrite utilization by E.coli cells grown in a flow chemostat is analyzed. The system is of interest from the standpoint of the analysis of the mechanisms by which a cell utilizes toxic substrates in respiration. It is demonstrated that the system admits the appearance of two stationary nitrite concentrations; thus, it becomes bistable. The range of nitrite supply rate into the chemostat for which the model is bistable is determined. The model predicts the possibility of spontaneous death of the culture in transition between the stationary states.
Keywords: gene expression regulation, Escherichia coli, anaerobic respiration, modeling, bistability.
Pp. 110–118.Likhoshvay Vitalii Alexandrovich
Institute of Cytology and Genetics SD RAS, 10 Lavrentiev av., Novosibirsk, 630090; Novosibirsk State University, 2 Pirogova Str., Novosibirsk, 630090. E-mail: likho@bionet.nsc.ru
Ri Natalia Alexandrovna
Institute of Cytology and Genetics SD RAS, 10 Lavrentiev av., Novosibirsk, 630090, E-mail: kashev@bionet.nsc.ru
Khlebodarova Tamara Mikhailovna
Institute of Cytology and Genetics SD RAS, 10 Lavrentiev av., Novosibirsk, 630090. E-mail: tamara@bionet.nsc.ru
Fadeev Stanislav Ivanovich
Sobolev Institute of Mathematics SD RAS, 4 Koptyug av., Novosibirsk 630090. E-mail: fadeev@math.nsc.ru
Kogai Vladislav Vladimirovich
Sobolev Institute of Mathematics SD RAS, 4 Koptyug av., Novosibirsk 630090. E-mail: kogai@math.nsc.ru
UDC 519.168
Rusyak I. G. and Nefedov D. G.
Formulation and solution of the problem of optimal location for enterprises producing wood fuelWe develop a mathematical model for the optimal location of wood fuel production and show its practical use (on the example of the Udmurt Republic). A solution method is proposed that is based on the genetic algorithm with binary coding.
Keywords: location of production, mathematical model, optimization, wood fuel, genetic algorithm.
Pp. 118–123.Rusyak Ivan Grigor'evich
Nefedov Denis Gennad'evich
Izhevsk State Technical Univewrsity, 7 Studencheskaya str., Izhevsk 426069. E-mail: denisnefedov1@yandex.ru; primat@istu.ru
UDC517.958:534.18
Shamaev A. S. and Shumilova V. V.
On the spectrum of one-dimensional oscillations of a laminated composite with components of elastic and viscoelastic materialsWe construct two averaged (effective) models corresponding to transverse and longitudinal oscillations of a laminated composite. The components of such a composite are mutually alternating layers of isotropic elastic and viscoelastic materials. We prove that the study of the spectrum of each of the averaged models is reduced to finding the roots of the corresponding linear fractional equations.
Keywords: oscillation spectrum, laminated composite, elasticity, viscoelasticity, averaged model.
Pp. 124–135.Shamaev Aleksey Stanislavovich
Shumilova Vladlena Valer'evna
Instiute for Problems in Mechanics RAS, 101–1 Vernadskii av., Moscow 119526. E-mail: sham@rambler.ru; v.v.shumilova@mail.ru
UDC 519.61:577.21
Shtokalo D. N.
On passage to the limit in a multistage multiphase synthesis of substanceWe presenta scheme of the proof of passage to the limit to a delayed equation in a model of multistage synthesis of substance when the number of stages increases with the preservation of the time of the whole synthesis when the chain of the synthesis reactions splits into phases. Each phase is characterized by constants defining the intensity of the direct and reverse processes and sinks. It is shown that if the intensity of the direct process dominates the reverse process on each phase with unrestricted number of stages then, when the total number of stages increases, the function that determines the synthesized substance and is defined by the last component of the vector of the solution to the system of ordinary differential equations converges uniformly to the function of the solution to the delayed equation. Numerical estimates of the uniform convergence are obtained for specific values of the parameters.
Keywords: multistage synthesis of substance, passage to the limit, delayed equation.
Pp. 135–146.Shtokalo Dmitrii Nikolaevich
A. P. Ershov Institute of Informatics Systems SD RAS, 6 Lavrent'ev av., E-mail: shtokalod@gmail.com