

2019 year, number 1
N. Bouteraa, S. Benaicha
University of Oran1, Ahmed Benbella, Algeria
Keywords: положительное решение, единственность и существование, итерационная последовательность, функция Грина, positive solution, uniqueness and existence, iterative sequence, Green's function
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In this paper, we obtain the existence and uniqueness of periodic solutions for a nonlinear fourthorder differential equation utilizing an explicit Green's function and fixed point index theorem combining with an operator spectral theorem. We discuss an iteration method for constant coefficient nonlinear differential equations and establish a theorem on the existence of positive solutions for fourthorder boundary value problem with variable parameter. Finally, we give an example to illustrate our results.

Sh.Kh. Imomnazarov, M.V. Urev
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva 6, Novosibirsk, 630090, Russia
Keywords: пористая среда, магнитное поле, проводящая жидкость, обобщенное решение, метод конечных элементов, porous medium, magnetic field, conductive fluid, generalized solution, finite element method
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The existence and uniqueness of the generalized solution of the boundary value problem for the system of magnetoporosity equations in the dissipative approximation have been proved. The results of the numerical solution obtained by the finite element method of the test boundary value problem of magnetoporosity in the frequency domain are presented.

E.G. Klimova
Institute of Computational Technologies of the Siberian Branch of the Russian Academy of Science, Novosibirsk, Lavrentyev Ave. 6, Russia, 630090
Keywords: усвоение данных, ансамблевый фильтр Калмана, data assimilation, Kalman ensemble filter
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The Kalman filter algorithm is currently one of the most popular approaches to solving the data assimilation problem. The major line of the application of the Kalman filter to the data assimilation is the ensemble approach. In this paper, we propose a version of the Kalman stochastic ensemble filter. In the algorithm presented the ensemble perturbations analysis is attained by means of transforming an ensemble of forecast perturbations. The analysis step is made only for a mean value. Thus, the ensemble πalgorithm is based on the advantages of the stochastic filter and the efficiency and locality of the square root filters. The numeral method of the ensemble πalgorithm realization is proposed, the applicability of this method has been proved. This algorithm is implemented for the problem in the threedimensional domain. The results of the numeral experiments with the model data for estimating the efficiency of the offered numeral algorithm are presented. The comparative analysis of the rootmeansquare error behavior of the ensemble πalgorithm and the Kalman ensemble filter by means of the numeral experiments with a onedimensional Lorentz model is made.

V.D. Liseikin^{1,2}, V.I. Paasonen^{1,2}
^{1}Institute of Computational Technologies of the Siberian Branch of the Russian Academy of Science, Novosibirsk, Lavrentyev Ave. 6, Russia, 630090 ^{2}Novosibirsk State University, Pirogova st., 2, Novosibirsk, 630090, Russia
Keywords: уравнение с малым параметром, погранслой, внутренний слой, компактная схема, схема повышенной точности, адаптивная сетка, equation with a small parameter, boundary layer, interior layer, compact scheme, scheme of high order, layerresolving grid, adaptive grid
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This paper realizes a symbiosis of two approaches to the numerical solution of second order ODEs with a small parameter having singularities such as interior and boundary layers, namely, the application of both compact schemes of high orders and layerresolving grids. The generation of layerresolving grids, based on estimates of solution derivatives and formulations of coordinate transformations eliminating solution singularities, is a generalization of the methodology early developed for the first order scheme. This paper presents the formulas of the coordinate transformations and numerical experiments for the schemes of the first, second, and third orders on uniform and layerresolving grids for the equations with boundary, interior, exponential and power layers of the first and second scales. The experiments conducted confirm the uniform convergence of the numerical solutions of equations with the help of compact schemes of high orders on the layerresolving grids. By using the transfinite interpolation methodology or numerical solutions to the Beltrami and diffusion equations in a control metric, built by the coordinate transformations eliminating the solution singularities, the developed technology can be generalized to the solution of multidimensional equations with boundary and interior layers.

A.V. Penenko^{1,2}
^{1}Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Akad. Lavrentjeva 6, Novosibirsk, 630090 ^{2}Novosibirsk State University, st. Pirogova 2, Novosibirsk, 630090
Keywords: обратная задача идентификации источников, большие данные, метод НьютонаКанторовича, сопряженные уравнения, оператор чувствительности, rпсевдообратная матрица, правая обратная, inverse source problem, big data, NewtonKantorovich method, adjoint equations, sensitivity operator, rpseudoinverse matrix, right inverse
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The algorithms for solving the inverse source problem for the productiondestruction type systems of nonlinear ordinary differential equations with measurement data in the form of time series are presented. The sensitivity operator and its discrete analogue on the basis of adjoint equations are constructed. This operator binds the perturbations in the unknown parameters of the model to those of the measured values. The operator allows one to construct a family of quasilinear operator equations linking the required unknown parameters and the data of the inverse problem. The NewtonKantorovich type method with righthand side r pseudoinverse matrices is used to solve the equations. The algorithm is applied to solving the inverse source problem for the atmospheric impurities transformation model.

A.I. Sidikova
South Ural State University Department of Computational Mathematics and High Performance Computing School of Electrical Engineering and Computer Science, Lenin Prospekt, 76, Chelyabinsk, 454080
Keywords: оценка погрешности, модуль непрерывности, преобразование Фурье, некорректная задача, error estimation, modulus of continuity, Fourier transform, illposed problem
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This paper is concerned with investigating and solving the mixed initial boundary value problem for the heat conduction equation. The statement of the problem includes the three intervals: the first one (from 0 → T_{1}) describes heating the combustion chamber, the second (from T_{1 }→ T_{2})  cooling the chamber and a slower cooling of its wall. Finally, the third interval describes natural cooling of the chamber wall when the chamber has the temperature coinciding with that of environment. The validity of the application of the Fourier transform with respect to this problem has been proved. This made possible to transform the governing equation to the ordinary differential equation. By using the resulting equation, the inverse boundary value problem for the heat conduction equation by applying the nonlinear method of projection regularization was solved and the error of approximate solution was obtained.

S.B. Sorokin^{1,2}
^{1}Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Akad. Lavrentjeva 6, Novosibirsk, 630090 ^{2}Novosibirsk State University, st. Pirogova 2, Novosibirsk, 630090
Keywords: задача Коши для уравнения Лапласа, обратная задача, численное решение, экономичный прямой метод, Cauchy problem for Laplace equation, inverse problem, numerical solution, efficient direct method
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One of widespread approaches to solving the Cauchy problem for the Laplace equation is to reduce it to the inverse problem. As a rule, an iterative procedure to solve the latter is used. In this study, an efficient direct method for the numerical solution of the inverse problem in the rectangular form is described. The main idea is based on the expansion of the desired solution with respect to a basis consisting of eigenfunctions of a difference analogue of the Laplace operator.

