Siberian Journal of Numerical Mathematics

Volume 19, 2016


Number 1, pp. 1-123
Number 2, pp. 125-233
Number 3, pp. 235-342
Number 4, pp. 343-467

Number 1, pp. 1-123  


On the anniversary of Anatoly Konovalov (in Russian), pp. 1-2

On the 90th anniversary of Gury Marchuk (in Russian), pp. 3-4

Aleksandrov Vladimir Mikhailovich

A singular solution to the problem of minimizing resource consumption (in Russian), pp. 5-18


An iterative method of finding a singular solution to the problem of minimizing resource consumption has been developed. This method is based on the information about the finite control structure. A condition for existence of a singular solution is obtained. The limit value for transferring the time between the normal and the singular solutions is found. A relation between the variations of the control switching moments and the variations of the initial conditions of the adjoint system is determined. A system of linear algebraic equations relating the variations of the initial conditions of the adjoint system to the variations of the phase coordinates from a given final state of the system is obtained. The computational algorithm, the modeling results and

the numerical calculations are presented. 

Keywords: optimal control, finite control, speed, transfer time, resource consumption, control start moment, control stop moment, iterative process, adjoint system, phase trajectory.

Ambos Andrey Yurevich

Numerical models of mosaic homogeneous isotropic random fields and problems of radiative transfer (in Russian), pp. 19-32


The new algorithms of statistical modeling of radiative transfer through different types of stochastic homogeneous isotropic media have been created. To this end a special geometric implementation of «the maximum cross-section method» has been developed. This implementation allows one to take into account the radiation absorption by the exponential multiplier factor. The dependence of a certain class of solution functionals of the radiative transfer equation on the correlation length and the field type is studied theoretically and by means of numerical experiments. The theorem about the convergence of these functionals to the corresponding functionals for an average field with decreasing the correlation length up to zero has been proved.

Keywords: Poisson ensemble, random field, correlation function, radiative transfer, maximum cross-section method.

Artemiev Sergey Semenovich, Yakunin Mikhail Aleksandrovich

Analysis of the accuracy of estimates of the first moments of solving SDE with Wiener and Poisson components by Monte Carlo method  (in Russian), pp. 33-45


In this paper, we investigate the accuracy of estimates of the first moments of a numerical solution to SDE with the Wiener and the Poisson components by the generalized Euler explicit method. The exact expressions for the mathematical expectation and variance of the test SDE solution are obtained. These expressions allow us to investigate the dependence of the accuracy of estimates obtained by Monte Carlo method on the values of SDE parameters, the size of an integration step, and the size of an ensemble of simulated trajectories of the solution. The results of the numerical experiments are presented.

Keywords: stochastic differential equations, Wiener and Poisson components, Monte Carlo method, generalized Euler method, ensemble of trajectories, integration step, estimates of moments.

Blatov Igor Anatolevich, Kitaeva Elena Victorovna 

Convergence of the adapting grid method of Bakhvalov's type for singularly perturbed boundary value problems (in Russian), pp. 47-59


We consider the Galerkin finite element method for non-self-adjoint boundary value problems on Bakhvalov's grids. Using the Galerkin projections method the convergence of a sequence of computational grids with an unknown boundary of the boundary layer has been proved. Numerical examples are presented.

Keywords: singularly perturbed boundary value problem, Galerkin projection, Bakhvalov's grid, adaptation algorithms.

Vabishchevich Petr Nikolaevich, Grigoriev Alexander Vissarionovich

Numerical modeling of a fluid flow in anisotropic fractured porous media (in Russian), pp. 61-74


A model of double porosity in the case of an anisotropic fractured porous medium is considered (Dmitriev, Maksimov; 2007). The function of the exchange flow between fractures and porous blocks, which depends on the direction of a flow, is investigated. The flow function is based on the difference between pressure gradients. This feature enables one to take into account anisotropic filtering properties in a more general form. The results of the numerical solution of the model two-dimensional problem are presented. The computational algorithm is based on the finite element spatial approximation and the explicit-implicit temporal approximation.

Keywords: double porosity model, anisotropic filtration, fractured porous media, gradient flow function.

Krukier Lev Abramovich, Krukier Boris L’vovich, Huang Yu-Mei. 

The skew-symmetric iterative method for solving the convection-diffusion-reaction equation with the alternating-sign reaction coefficient (in Russian), pp. 75-85


The iterative product, that is, the triangular skew-symmetric method (PTSM) is used to solve linear algebraic equation systems obtained by approximation of a central-difference scheme of the first boundary value problem of convection-diffusion-reaction and standard grid ordering. Sufficient conditions of a non-negative definiteness of the matrix resulting from this approximation have been obtained for a non-stationary sign of the reaction coefficient. This feature ensures the convergence of a sufficiently wide class of iterative methods, in particular, the PTSM. In the test problems, the compliance of the theory with computational experiments is verified, and comparison of the PTSM and the SSOR is made. 

Keywords: convection-diffusion-reaction equation, alternating sign coefficient of reaction, central difference scheme, iterative method.

Singh P., Kadalbajoo M.K., Sharma K. 

Probability density function of leaky integrate-and-fire model with Lèvy noise and its numerical approximation (in Russian), pp. 87-96


We investigate a numerical analysis of a leaky integrate-and-fire model with Lèvy noise. We consider a neuron model in which the probability density function of a neuron in some potential at any time is modeled by a transport equation. Lèvy noise is included due to jumps by excitatory and inhibitory impulses. Due to these jumps the resulting equation is a transport equation containing two integrals in the right-hand side (jumps). We design, implement, and analyze numerical methods of finite volume type. Some numerical examples are also included.

Keywords: Leaky integrate-and-fire model, transport equation, finite volume approximation, Lèvy noise.

Tanana Vitaliy Pavlovich, Vishnyakov Evgeniy Yurevich, Sidikova Anna Ivanovna 

About an approximate solution to the Fredholm integral equation of the first kind by the residual method (in Russian), pp. 97-105


The Tikhonov finite-dimensional approximation was applied to an integral equation of the first kind. This allowed us to use the variation regularization method of choosing the regularization parameter residuals from the principle of reducing the problem to a system of linear algebraic equations. The estimate of accuracy of the approximate solution with allowance for the error of the finite-dimensional problem approximation has been obtained. The use of this approach is illustrated on an example of solving an inverse boundary value problem for the heat conductivity equation.

Keywords: regularization, method of residuals, module of continuity, evaluation of inaccuracy, ill-posed problem.

Shreifel Igor Semyonovich, Eliseev Ivan Nikolaevich

Theoretical basis of the iterative process of the joint assessment of difficulties in tasks and levels of training students (in Russian), pp.107-123


In this paper, we study the iterative process of the joint numerical assessment of levels of training students and difficulties in tasks of diagnostic tools using the dichotomous response matrix A of N x M size, with allowance for the contribution of tasks of different difficulty to the assessments obtained. It is shown that not for any matrix A there exist infinite iterative sequences, and in the case of their existence, they do not always converge. A wide range of sufficient conditions for their convergence have been obtained, which are based on the following: 1) matrix A contains at least three different columns; 2) if one places the columns of A in non-decreasing order of column sums, then for any position of the vertical dividing line between the columns there exists a row, which has, at least, one unity to the left of the line, and at least one zero to the right of the line. It is established that the response matrix A obtained as a result of testing reliability satisfies these two conditions. The properties of such matrices have been studied. In particular, the equivalence of the above conditions of primitiveness of the square matrix B of order M with the entries bij =∑ N ell =1 (1-ali) alj has been proved. Using the matrix analysis we have proved that the primitiveness of the matrix B ensures the convergence of iterative sequences, as well as independence of their limits of the choice of the initial approximation. We have estimated the rate of convergence of these sequences and found their limits. 

Keywords: iterative process, an iterative sequence, difficulty of test questions, level of training students, dichotomous response matrix.


Number 2, pp. 125-233 


Babicheva Galina Andreevna, Kargapolova Nina Aleksandrovna, Ogorodnikov Vasilii Aleksandrovich

Special algorithms for simulation of homogeneous random fields (in Russian), pp. 125-138


In this paper, two new algorithms for the simulation of homogeneous random fields are proposed. Both algorithms are based on the widespread algorithm “in rows and columns” for the simulation of the Gaussian fields with special correlation functions. Applying the algorithms developed makes possible to efficiently simulate homogeneous random fields with non-convex correlation functions. 

Keywords: homogeneous random field, stochastic simulation, randomization.

Zhitnikov Vladimir Pavlovich, Sherykhalina Nataliya Mikhailovna, Muksimova Roza Ravilevna

The peculiarities of error accumulation in solving problems for simple equations of mathematical physics by finite difference methods (in Russian), pp. 139-152


 A mixed problem for a one-dimensional heat equation with several versions of initial and boundary conditions is considered. Explicit and implicit schemes are applied for the solution. The sweep method and the iteration methods are used for the implicit scheme for solving the implicit system of equations. The numerical filtration of a finite sequence of results obtained for different grids with an increasing number of nodal points is used to analyze errors of the method and rounding. In addition, to investigate the rounding errors, the results obtained with several lengths of the machine word mantissa are compared. The numerical solution of the mixed problem for the wave equation is studied by similar methods. 

The occurrence of deterministic dependencies of the error in the numerical method and the rounding on spatial coordinates, time and the number of nodes is revealed. The source models to describe the behavior of errors in terms of time are based on the analysis of the results of numerical experiments for different versions of conditions of problems. In accord with such models, which were verified by the experiment, the errors can increase, decrease or stabilize depending on conditions over time similar to changing the energy or mass. 

Keywords: heat equation, explicit and implicit schemes, the Courant number, model error, numerical filtration.

Kabanikhin Sergey Igorevich, Krivorotko Olga Igorevna

A numerical algorithm for computing tsunami wave amplitude  (in Russian), pp. 153-165


A numerical algorithm for computing tsunami wave front amplitude is proposed. The first step consists in solving an appropriate eikonal equation. The eikonal equation is solved by the Godunov approach and the bicharacteristic method. The qualitative comparison of the two above methods is described. Then a change in variables associated with the eikonal solution is introduced. At the last step, using the expansion of the fundamental solution of shallow water equations in the sum of singular and regular parts, we obtain the Cauchy problem for the wave amplitude. This approach allows one to reduce computer costs. The numerical results are presented. 

Keywords: shallow water equations, tsunami amplitude, fundamental solution, eikonal equation, finite difference approach.

Kansal Munish, Kanwar V., Bhatia Saurabh

Optimized mean based second derivative-free families of Chebyshev-Halley type methods (in Russian), pp. 167-181


In this paper, we present new interesting fourth-order optimal families of Chebyshev-Halley type methods free from second-order derivatives. In terms of computational cost, each member of the families requires two functions and one first-order derivative evaluation per iteration, so that their efficiency indices are 1.587. It is found by way of illustration that the proposed methods are useful in high precision computing environment. Moreover, it is also observed that larger basins of attraction belong to our methods, whereas the other methods are slow and have darker basins, while some of the methods are too sensitive to the choice of the initial guess. 

Keywords: basins of attraction, Newton's method, King's methods, optimal iterative methods, efficiency index.

Korneev Vladimir Dmitrievich, Sveshnikov Viktor Mitrofanovich

Parallel algorithms and domain decomposition technologies for solving three-dimensional boundary value problems on quasi-structured grids (in Russian), pp. 183-194


A new approach to the decomposition method of a three-dimensional computational domain into subdomains, adjoint without overlapping, which is based on a direct approximation of the Poincare-Steklov equation at the conjugation interface, is proposed. With the use of this approach, parallel algorithms and technologies for three-dimensional boundary value problems on quasi-structured grids are presented. The experimental evaluation of the parallelization efficiency on the solution of the model problem on quasi-structured parallelepipedal coordinated and uncoordinated grids is given. 

Keywords: boundary value problems, domain decomposition methods, Poincare-Steklov equation, quasistructured grids, algorithms and technologies of parallelization.

Monakhov Oleg Gennad'evich, Monakhova Èmiliya Anatol'evna, Pant Millie

Application of differential evolution algorithm for optimization of strategies based on financial time series (in Russian), pp. 195-205


An approach to optimization of trading strategies (algorithms) based on indicators of financial markets and evolutionary computation is described. A new version of the differential evolution algorithm for the search for optimal parameters of trading strategies for the trading profit maximization is used. The experimental results show that this approach can considerably improve the profitability of the trading strategies. 

Keywords: trading strategy, parallel genetic algorithm, technical analysis, financial indicator, template, evolutionary computation.

Stepanova LarisaValentinovna, Yakovleva Ekaterina Michailovna

Asymptotics of eigenvalues of the nonlinear eigenvalue problem arising from the near mixed-mode crack-tip stress-strain field problems (in Russian), pp. 207-222


In the present paper, approximate analytical and numerical solutions to nonlinear eigenvalue problems arising in the nonlinear fracture mechanics in analysis of stress-strain fields near a crack tip under a mixed mode loading are presented. Asymptotic solutions are obtained by the perturbation method (the small artificial parameter method). The artificial small parameter is a difference between the eigenvalue corresponding to the nonlinear eigenvalue problem and the eigenvalue related to the linear “undisturbed” problem. It is shown that the perturbation technique gives an effective method of solving nonlinear eigenvalue problems in the nonlinear fracture mechanics. Comparison of numerical and asymptotic results for different values of the mixity parameter and hardening exponent shows good agreement. Thus, the perturbation theory technique for studying nonlinear eigenvalue problems is offered and applied to eigenvalue problems arising from the fracture mechanics analysis in the case of a mixed mode loading. 

Keywords: nonlinear eigenvalue problem, perturbation theory small parameter method, asymptotics of stress and strain fields in the vicinity of the mixed-mode crack, mixed-mode loading, power constitutive law, eigenspectrum.


Favorskaya Alena Victorovna, Petrov Igor Borisovich

The study of increased order grid-characteristic methods on unstructured grids  (in Russian), pp. 223-233


We study the grid-characteristic methods for solving hyperbolic systems using a high order interpolation on unstructured tetrahedral and triangular grids for approximation. We consider the interpolation with orders from the first to the fifth included. Also, one-dimensional finite difference schemes appropriate for the considered methods are given. We study these schemes in terms of stability. The grid-characteristic method on unstructured triangular and tetrahedral grids are successfully used for solving the seismic prospecting problems, including, seismic prospecting in the conditions of the Arctic shelf and permafrost, as well as for solving seismic problems, problems of dynamic deformation and destruction, studying anisotropic composite materials. 

Keywords: grid-characteristic method, numerical simulation, unstructured grids, high order interpolation.

Number 3, pp. 235-342


Averina Tat'yana Aleksandrovna

Using a randomized method of a maximum cross-section for simulating random structure systems with distributed transitions  (in Russian), pp. 235-247


In this paper we consider the random structure systems with distributed transitions. A theorem about the form of conditional structure distributions has been proved. To simulatethese distributions a statistical algorithm using a randomized method of maximum cross-section is constructed. Also, a modified version of this algorithm using the simulation of one random number has been constructed. The algorithms developed were used for the simulation of the numerical solution of random structure systems with distributed transitions. The theorem about a weak convergence of the numerical solution, obtained by the algorithms developed has been proved. 

Keywords: statistical simulation, systems with a random structure, stochastic differential equations, Poisson flow, numerical methods, maximum cross-section method.


Zorkal'tsev Valerii Ivanovich

The search for admissible solutions by the interior point algorithms  (in Russian), pp. 249-265


 A family of interior point algorithms for the linear programming problems is considered. In these algorithms, the entering into the domain of admissible solution of the original problem is represented as optimization process of the extended problem. This extension is realized by adding just one new variable. The main objective of the paper is to give a theoretical justification of the proposed procedure of entering into the feasible domain of the original problem, under the assumption of non-degeneracy of the extended problem. Particularly, we prove that given the constraints of the original problem being consistent, the procedure leads to a relative interior point of the feasible solutions domain. 

Keywords: interior point algorithm, linear programming, techniques of arriving at the feasible solutions region.


Krukier Lev A., Martynova Tatiana S.

Preconditioning of GMRES by the skew-Hermitian iterations  (in Russian), pp. 267-279


A class of preconditioners for solving non-Hermitian positive definite systems of linear algebraic equations is proposed and investigated. It is based on the Hermitian and skew-Hermitian splitting of the initial matrix. The generalization for saddle point systems which have

semidefinite or singular (1,1) blocks is given. Our approach is based on an augmented Lagrangian formulation. It is shown that such preconditioners are effective for the iterative solution of systems of linear algebraic equations by the GMRES. 

Keywords: Hermitian and skew-Hermitian splitting, iterative methods, preconditioning, Krylov subspace method, saddle point linear system.


Owolabi Kolade M.

Mathematical study of two-variable systems with adaptive numerical methods  (in Russian), pp. 281-295


In this paper, we consider reaction-diffusion systems arising from two-component predator-prey models with Smith growth functional response. The mathematical approach used here is twofold, since the time-dependent partial differential equations consist of both linear and nonlinear terms. We discretize the stiff or moderately stiff term with a fourth-order difference operator, advance the resulting nonlinear system of ordinary differential equations with a family of two competing exponential time differencing (ETD) schemes, and analyze them for stability. A numerical comparison of these two methods for solving various predator-prey population models with functional responses is also presented. Numerical results show that the techniques require less computational work. Also in the numerical results, some emerging spatial patterns are unveiled. 

Keywords: predator-prey model, ETD methods, nonlinear, pattern formation, reaction-diffusion, stability, time-dependent PDE, Turing instability. 

Platov Gennadii Alekseevich

The influence of the shelf zone relief and the coastline geometry on coastal trapped waves  (in Russian), pp. 297-316


This paper presents the results of numerical experiments with a model of the coastal trapped waves, which made it possible to identify two features that are important in terms of the regional modeling of the shelf zone interaction with the open ocean. The first feature is the fact that the wave train of this type may be formed as a result of the wind action at a considerable distance from the place where their impact may occur. The propagation of waves along the coastline takes place without significant loss of wave energy, provided that the coastline and topography of the shelf zone contain no features comparable to the Rossby radius. However, the wave loses its energy while passing capes, submarine canyons and in the case when the width of a shelf decreases. For the regional modeling, the possibility of remote wave generation should be well understood and taken into account. The second feature is that a propagating wave is able to spend part of its energy on the formation of density anomalies on a shelf by raising the intermediate waters of the adjoining offshore areas of the open ocean. Thus, the coastal trapped waves carry the wind energy from the areas of the wind impact to other coastal areas, where it can bring about the formation of density anomalies and other types of motion. 

Keywords: coastal trapped waves, shelf zone, marginal seas.


Plotnikov Mikhail Yu., Shkarupa Elena Valer'evna

Evaluation of statistical error when calculating velocity and temperature components by the direct simulation Monte Carlo method  (in Russian), pp. 317-330


The direct simulation Monte Carlo method is now widely used to solve the problems of rarefied gas dynamics. While solving stationary problems a special feature of the method is using dependent sample values of random variables to calculate macroparameters of a gas flow. In this paper, the possibility of using the results of statistical physics to estimate the statistical error of the DSMC method is theoretically analyzed. A simple approach to approximate evaluating the statistical error while calculating components of the velocity and temperature is proposed. The approach is tested on a number of problems. 

Keywords: direct simulation Monte Carlo method, statistical error, equilibrium statistical physics.


Rozhenko Aleksandr Iosifovich, Fedorov Egor Alexandrovich

On an algorithm of bilateral restrictions smoothing with spline  (in Russian), pp. 331-342


In this paper, the problem of constructing a spline σ in the Hilbert space satisfying bilateral restrictions z- ≤ A σ ≤ z+ with a linear operator A and minimizing a squared Hilbert seminorm is studied. A solution to this problem could be obtained with the convex programming iterative methods, in particular, with the gradient projection method. A modification of the gradient projection method allowing one to reveal a set of active restrictions in a smaller number of iterations is offered. The efficiency of the modification proposed is shown on the problem of approximation with a pseudo-linear bivariate spline.

Keywords: smoothing, spline, Hilbert space, convex programming, reproducing mapping, radial basis function.

Number 4, pp. 343-467 


Voronina Tat'yana Aleksandrovna 

Recovering a tsunami source and designing an observational system based on the r-solution method   (in Russian), pp. 343-356


This study deals with the application of the r-solution method to recover the initial tsunami waveform in a tsunami source area by inverting the remote water-level measurements for a real event. The inverse problem in question is regarded as the so-called ill-posed problem and it is regularized by means of the least square inversion using the truncated Singular Value Decomposition method. The method presented allows one to control the instability of the numerical solution and to obtain an acceptable result in spite of the ill-posedness of the problem. Moreover, it is possible to make a preliminary prediction of the quality of the inversion with a given set of observational stations and to estimate further changes in the inversion result after modifying the monitoring system.

Keywords: tsunami numerical modeling, ill-posed inverse problem, regularization, singular value decomposition and r-solution.


Zhukovskaia Tatyana Vladimirovna, Zhukovskiy Evgeny Semenovich

On iterative methods for solving equations with covering mappings  (in Russian), pp. 357-369


In this paper we propose an iterative method for solving the equation ϓ (x,x)=y, where a mapping ϓ acts in metric spaces, is covering in the first argument and Lipschitzian in the second one. Each subsequent element xi+1 of a sequence of iterations is defined by the previous one as a solution to the equation ϓ (x,xi)=yi, where yi can be an arbitrary point sufficiently close to y. The conditions for convergence and error estimates have been obtained. The method proposed is an iterative development of the Arutyunov method for finding coincidence points of mappings. In order to determine xi+1 it is proposed to perform one step using the Newton-Kantorovich method or the practical implementation of the method in linear normed spaces. The obtained method of solving the equation of the form ϓ (x,u)=ψ(x)-φ(u) coincides with the iterative method proposed by A.I. Zinchenko, M.A. Krasnosel'skii, I.A. Kusakin.

Keywords: iterative methods for solving equations, covering mappings in metric spaces, approximate solution.


Leonov Aleksandr Sergeevich

Regularizing algorithms with optimal and extra-optimal quality (in Russian), pp. 371-383


The notion of a special quality for approximate solutions to ill-posed inverse problems is introduced. A posteriori estimates of the quality are studied for different regularizing algorithms (RA). Examples of typical quality functionals are provided, which arise in solving linear and nonlinear inverse problems. The techniques and the numerical algorithm for calculating a posteriori quality estimates for approximate solutions of general nonlinear inverse problems are developed. The new notions of optimal and extra-optimal quality of a regularizing algorithm are introduced. The theory of regularizing algorithms with optimal and extra-optimal quality is presented, which includes an investigation of optimal properties for estimation functions of the quality. Examples of regularizing algorithms with extra-optimal quality of solutions are given, as well as examples of regularizing algorithms without such property. The results of numerical experiments illustrate a posteriori quality estimation.

Keywords: ill-posed problems, regularizing algorithms, quality of approximate solution, a posteriori quality estimates, RA with extra-optimal quality.


Mastryukov Aleksandr Fedorovich

Optimal finite difference schemes for the wave equation  (in Russian), pp. 385-399


This paper considers the solution of the two-dimensional wave equation with the use of the Laguerre transform. The optimal parameters of finite difference schemes for this equation have been obtained. Numerical values of these optimal parameters are specified. The finite difference schemes of second order with optimal parameters give the accuracy of the solution to the equations close to the accuracy of the solution by the scheme of fourth order. It is shown that using the Laguerre decomposition, the number of optimal parameters in comparison with the Fourier decomposition can be reduced. This reduction leads to simplification of finite difference schemes and to reduction of the number of computations, i.e. the efficiency of the algorithm.

Keywords: wave equation, electromagnetic wave, finite-difference, optimal, accuracy, Laguerre method, linear system of equations.


Penenko Aleksey Vladimirovich, Penenko Vladimir Viktorovich, Tsvetova Elena Aleksandrovna

Sequential data assimilation algorithms in air quality monitoring models based on weak-constraint variational principle  (in Russian), pp. 401-418


          Data assimilation problem for non-stationary model is considered as a sequence of the linked inverse problems which reconstruct, taking into account the different sets of measurement data, the space-time structure of the state functions. Data assimilation is carried out together with the identification of additional unknown function, which we interpret as a function of model uncertainty. The variational principle serves as a basis for constructing algorithms. Different versions of the algorithms are presented and analyzed. Based on the discrepancy principle, a computationally efficient algorithm for data assimilation in a locally one-dimensional case is constructed. The theoretical estimation of its performance is obtained. This algorithm is one of the core components of the data assimilation system in the frames of splitting scheme for the non-stationary three-dimensional transport and transformation models of atmospheric chemistry.

Keywords:  data assimilation, variational principle, weak-constraint, direct and inverse problems, model as regularizer, sequential algorithms.


Plieva Lyubov' Yur'evna

Quadrature interpolation type formulas for hypersingular integrals in the interval of integration  (in Russian), pp. 419-428


          A hypersingular integral on the interval of integration with the weight function is considered. We prove the spectral ratios for hypersingular integrals on [-1, 1]. The quadrature formulas for certain integrals with the weight function are constructed. The estimation error is presented.

Keywords: hypersingular integral, quadrature formula, the estimation error.


Sorokin Sergei Borisovich

A difference scheme for a conjugate-operator model of the heat conduction problem on non-matching grids  (in Russian), pp. 429-439


           On non-matching grids discrete analogue conjugate-operator models of heat conduction, keeping the structure of the original model are constructed. Numerical experiments show that the difference scheme converges with second order of accuracy for the case of discontinuous parameters of the medium in the Fourier law and non-uniform grids. 

Keywords: problem of heat conductivity, mathematical model, discrete analog, non-matching grid, convergence, difference scheme.


Cherdantsev Sergei Vasil'evich, Cherdantsev Nikolai Vasil'evich

Analysis of the pontoon fluctuations with a seasonally changing parameter of stability on the astir surface of finite water depth (in Russian), pp. 441-456


           It is shown that due to the periodic changes in metacentric heights of a pontoon on the astir surface of liquids in the sump of an open coal mine, the pontoon is capable to produce parametric pitching, both in the longitudinal and in the transverse directions. The equation, describing parametric pitching, is transformed to the Mathieu equation, whose factors depend both on the own frequencies and the pontoon floatability features on «calm water», and on the frequency of fluctuation of a liquid, which, in turn, is defined by the sump size. The installed regularities between parameters, characterizing parametric pitching in the longitudinal and transverse directions, and areas of its instability are revealed.

Keywords: sump of an open coal mine, pontoon, potential of the velocities, frequency of the waves, waterline, metacentric heights, added masses of liquid, parametric pitching pontoon, Mathieu equation, Inc-Strutt stability diagram.


Chugunov Vadim Nikolaevich, Ikramov Khakim Dododzhanovic

Classification of real pairs of commuting Toeplitz and Hankel matrices  (in Russian), pp. 457-467


We give a complete description of the set of matrix pairs such that T is a real Toeplitz matrix, H is a real Hankel matrix, and TH = HT.

Keywords: Toeplitz matrix, Hankel matrix, circulant, skew-circulant, commutativity.