

2015 year, number 2
Tatiana Vladimirovna Antonova
Institute of Mathematics and Mechanics Ural Branch of Russian Academy of Scienes, 16 S. Kovalevskaya str., Ekaterinburg, Russia, 620990
Keywords: illposed problems, localization of singularities, equation of the first kind, parameter identification
Abstract >>
In this paper, we propose a regular iterative method of identifying a numerical parameter in the kernel of the integral equation of the first kind of the convolution type. It is shown that an unambiguous identification of the parameter is possible when an exact solution has discontinuities of the first kind. The convergence theorem is proved, and an example of the equation with a parameter, for which the method constructed is applicable, is given.

Sergey Semenovich Artemiev^{1,2}, Aleksandr Aleksandrovich Ivanov^{1}
^{1}Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Lavrentieva, 6, Novosibirsk, 630090 ^{2}Novosibirsk State University, Pirogova 2, Novosibirsk, Russia, 630090
Keywords: stochastic differential equations, cumulative frequency curve, frequency phase portrait, generalized Euler's method, strange attractors
Abstract >>
In this paper, we numerically investigate the influence of random noise on the behavior of the trajectories of strange attractors defined by a system of ordinary differential equations. The resulting stochastic differential equations are solved by the generalized Euler method. The results of numerical experiments conducted on a cluster of NKS30T Siberian Supercomputer Center at ICMMG using the program package PARMONC. For the analysis of the numerical solutions, the frequency characteristics of generalizing the integral curve and the phase portrait are used.

Kirill Vladislavovich Voronin, Yuri Mironovich Laevsky
^{1}Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Lavrentieva, 6, Novosibirsk, Russia, 630090 ^{2}Novosibirsk State University, Pirogova 2, Novosibirsk, Russia, 630090
Keywords: heat transfer, mixed formulation, finite element method, splitting scheme
Abstract >>
In this paper, we investigate the stability of some splitting schemes approximating the equations for a heat flux, obtained by a mixed finite element method. For the twodimensional problem, the splitting scheme is based on the alternating direction method, and for the threedimensional problem the splitting scheme is based on the DouglasGunn scheme.

Sergey Anatol’evich Gusev
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Lavrentieva, 6, Novosibirsk, Russia, 630090
Keywords: heat equation, discontinuous coefficients, integral averaging, diffusion process, stochastic differential equations, Euler method
Abstract >>
This paper proposes the use of the numerical solution to stochastic differential equations (SDE's) to find estimates of the solutions to boundary value problems for linear parabolic equations with discontinuous coefficients. The solution of the problem with smoothed coefficients is taken as an approximation of the generalized solution to the considered boundary value problem. The results of calculations for a thermal barrier coating comprising a composite cellular material are presented.

Evginii Alekseevich Kotel'nikov
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Lavrentieva, 6, Novosibirsk, Russia, 630090
Keywords: quadratic optimization on sphere, collinearity gradients, convex majorant, Cholesky decomposition
Abstract >>
The minimization of convex functions on a sphere reduces to a sequence of problems minimizing its convex majorants on a sphere. To build majorants, the representation of the target function as a difference of convex quadratic functions and the solutions of the problem at the previous step is used. Representation of the target function in the form of a difference of convex quadratic functions is based on a modified procedure of decomposition of the Cholesky symmetric alternatingsign matrices.

Aleksey Georgievich Nikolaev, Evgeniy Andreevich Tanchik
National Aerospace University KhAI International Relations Department, 17 Chkalova str., Kharkiv, Ukraine, 61070
Keywords: boundary value problem, multiconnected body, generalized Fourier method, resolving system, cylindrical boundary, addition theorems
Abstract >>
An efficient method for the analyticalnumerical solution to the nonaxyally symmetric boundary value problem of elasticity theory for a multiconnected body in the form of a cylinder with N cylindrical cavities is proposed. The solution is constructed as superposition of the exact basis solutions of the Lame equation for a cylinder in the coordinate systems assigned to the centers of the boundary surfaces of the body. The boundary conditions are exactly satisfied with the help of the apparatus of the generalized Fourier method. As a result, the original problem reduces to an infinite system of linear algebraic equations, which has a Fredholm operator in the Hilbert space l_{2}. The resolving system is numerically solved by the reduction. The rate of convergence of the reduction is investigated. The numerical analysis of stresses in the areas of their greatest concentration is carried out. The reliability of the results obtained is confirmed by comparing them for the two cases: a cylinder with sixteen and a cylinder with four cylindrical cavities.

Lev Yakovlevich Saveliev^{1,2}
^{1}Novosibirsk State University, Pirogova 2, Novosibirsk, Russia, 630090 ^{2}Sobolev Institute of Mathematics of the Siberian Branch, 4 Acad. Koptyug avenue, Novosibirsk, Russia, 630090
Keywords: stochastic model, binary Markov chain, distribution, generating function, mean, variance
Abstract >>
This paper derives exact and approximate formulas for the distribution, average values and variances of the number of units on the segments of binary Markov sequences. Various ways to calculate these formulas are proposed. Estimates of the errors are given. An example of the calculation for a binary Markov model of the precipitation process is presented.

Larisa Valentinovna Stepanova, Sergej Aleksandrovich Igonin
Samara State University, 1 Ak. Pavlova st., Samara, Russia, 443011
Keywords: fatigue crack growth, cyclic loading, asymptotic analysis, nonlinear eigenvalue problem, analytical solution
Abstract >>
In this paper, the asymptotic analysis of the near fatigue growing cracktip fields in a damaged material is done. The integrity parameter describing the damage accumulation process in the vicinity of a crack tip is incorporated into the constitutive law of the isotropic linear elastic material. The asymptotic solution based on the eigenfunction expansion method is obtained. It is shown that the problem is reduced to the nonlinear eigenvalue problem. The analytical solution of the nonlinear eigenvalue problem is found by the artificial small parameter method. The perturbation theory approach allows us to derive the analytical presentation of the stress and integrity distributions near the crack tip. The technique proposed permits us to find the higherorder terms of the asymptotic expansions of the stress components and the integrity parameter.

Elena Valer'evna Shkarupa
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Lavrentieva, 6, Novosibirsk, Russia, 630090
Keywords: functional algorithms of statistical modeling, biharmonic equation, error estimation, optimization
Abstract >>
Functional algorithms of the statistical modeling are designed to construct an approximation of the problem solution as function on a required domain. The approaches to construction of the upper error bound in the metrics of the space C with allowance for the degree of dependence of the estimates were devised for functional algorithms with different types of stochastic estimates in the nodes. Furthermore, there exists a universal approach applicable at any degree of dependence. The constructed upper error bound of the functional algorithm is used for choosing an optimal value of parameters, such as the number of grid nodes and the sample size. Optimality of the chosen parameters directly depends on the accuracy of the used upper error bound. The primary intent of the present paper is a comparison of universal approaches and those with allowance for the degree of dependence of the estimates.

