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# Home – Home – Jornals – Numerical Analysis and Applications 2021 number 3

## Numerical Analysis and Applications

### 2021 year, number 3

#### 1.Control of effects in the right-hand sides of a large ODE system of a block structure and optimization of sources in unseparated boundary conditions

1Institute of Control Systems of Azerbaijan National Academy of Sciences, Baku, Azerbaijan
2Institute of Mathematics and Mechanics of Azerbaijan National Academy of Sciences, Baku, Azerbaijan
3Baku State University, Baku, Azerbaijan
Keywords: block structure, large system of ODE, unseparated conditions, functional gradient, optimality conditions, sweep method

Abstract >>
In this paper, we investigate the problem of control of a complex object, described by a large ODE system of a block structure with unseparated boundary conditions between blocks. The controls in the right-hand sides of the equations and the values of the source parameters in the boundary conditions are to be optimized. We propose to apply the first order optimization methods for the numerical solution to the optimal control problem, using functional gradient formulas participating in the obtained necessary optimality conditions. Special schemes of the sweep method for the solution to the direct and conjugate boundary value problems, having a block structure, and unseparated non-local boundary conditions are offered. This method takes into account special features of ODE systems and boundary conditions, allows the transfer of boundary conditions for each block and each boundary condition in the block independent of each other. The obtained results of numerical experiments in solving the test problem and their analysis are given.

#### 2.About sampling of the two-dimensional Laplace operator in a smooth two-dimensional area

S.D. Algazin
Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow, Russia
Keywords: numerical methods without saturation, problems on eigenvalues, Laplace operator

Abstract >>
Currently the finite element method is most widespread technique for solving problems of mechanics of a deformable solid body. Its shortcomings are well-known: approximating a displacement by a piecewise-linear function, we obtain the tension to be discontinuous. At the same time, it is necessary to notice that most problems of mechanics of a deformable solid body are described by the elliptic type equations which have smooth decisions. It seems to be relevant to develop algorithms which would take this smoothness into account. The idea of such algorithms belongs to K.I. Babenko. This idea was stated in the early seventies of the last century. A long-lasting application of this technique in elliptic tasks to eigenvalues has proved their high performance to the author of this study. However, in this technique the matrix of the finite-dimensional task turns out to be not symmetric but only close to that to be symmetrized. Below, the application when sampling the Bubnov-Galyorkina method, this defect is eliminated. Let us note that the symmetry of the matrix of the finite-dimensional task is important when studying the stability. Unlike classical difference methods and the finite element method where the dependence of the convergence ratio on the number of nodes of the grid is power, we have an exponential decrease of the error.

#### 3.Parameter identification of constrained data by a new class of rational fractal function

S.K. Katiyar, A.K.B. Chand, S. Jha
1Indian Institute of Technology Madras, Chennai, India
2Indian Institute of Technology Madras, Ченнаи, Индия
Keywords: iterated function system, fractal interpolation functions, rational cubic fractal functions, rational cubic interpolation, constrained interpolation, positivity

Abstract >>
This paper sets a theoretical foundation for applications of fractal interpolation functions (FIFs). We construct rational cubic spline FIFs (RCSFIFs) with a quadratic denominator involving two shape parameters. The elements of the iterated function system (IFS) in each subinterval are identified befittingly so that the graph of the resulting C1-RCSFIF lies within a prescribed rectangle. These parameters include, in particular, conditions on the positivity of the C1-RCSFIF. The problem of visualization of constrained data is also addressed when the data is lying above a straight line, the proposed fractal curve is required to lie on the same side of the line. We illustrate our interpolation scheme with some numerical examples.

#### 4.Experimental study of the efficiency of solving 2D boundary value problems on subgrids of quasistructured rectangular grids

A.N. Kozyrev1, V.M. Sveshnikov1,2
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia
2Novosibirsk State University, Novosibirsk, Russia
Keywords: subgrids of quasistructured grids, solvers of boundary value problems, iterative methods, direct methods, experimental studies

Abstract >>
An experimental study of the solvers efficiency of 2D boundary value problems on subgrids of quasistructured rectangular grids was carried out. A solver is understood as a solution method and its software implementation. Three solvers are considered: one direct solver -- the Buneman cyclic reduction method and two iterative ones: the Peaceman-Rachford method and the method of successive over relaxation. Characteristic features of the studies are: 1) the subgrids contain a small number of nodes, namely 8 х 8, 16 х16, 32 х 32, 64 х 64; 2) the efficiency is estimated not only for single calculations, but also mainly for series of calculations, in each of which several repetitions of solving the problem with different boundary conditions on the same subgrid are carried out. Based on serial calculations, a combined method is proposed, and recommendations on the use of solvers are given.

#### 5.Solving of a system of linear algebraic equations with a three-diagonal matrix: a new look at the Cramer method

S.K. Kydyraliev1, S.N. Sklyar1, A.B. Urdaletova2
1American University of Central Asia, Kyrgyz Republic, Bishkek
2Kyrgyz-Turkish Manas University, Kyrgyz Republic, Bishkek
Keywords: system of linear algebraic equations with a three-diagonal matrix, Cramer method, recursive algorithm, diagonal dominance, sweep method

Abstract >>
For the numerical solution of a system of linear algebraic equations with a three-diagonal matrix, a recursive version of the Cramer method is proposed. This method does not require additional restrictions on the system matrix, similar to those formulated for the sweep method. The results of numerical experiments are presented on a large set of test problems, a comparative analysis of the effectiveness of the proposed methodology and the corresponding algorithms is given.

#### 6.On filter banks in spline wavelet transform on a non-uniform grid

A.A. Makarov1, S.V. Makarova2
1St Petersburg University, St Petersburg, Russia
2St Petersburg State University of Aerospace Instrumentation, St Petersburg, Russia
Keywords: -spline, minimal spline, spline wavelet, wavelet transform, filter banks

Abstract >>
An explicit representation of filter banks for constructing the wavelet transform of spaces of linear minimal splines on non-uniform grids on a segment is obtained. The decomposition and reconstruction operators are constructed, their mutual inverse is proved. The relations connecting the corresponding filters are established. The approach to constructing the spline wavelet decompositions used in this paper is based on approximation relations as the initial structure for constructing spaces of minimal splines and calibration relations to prove the embedding of the corresponding spaces. The advantages of the approach proposed, due to rejecting the formalism of the Hilbert spaces, are in the possibility of using non-uniform grids and fairly arbitrary non-polynomial spline wavelets.

#### 7.Solution of the inverse boundary value problem of heat transfer for an inhomogeneous ball

V.P. Tanana1,2, B.A. Markov3, A.I. Sidikova1
1South Ural State University, Chelyabinsk, Russia
2Chelyabinsk State University, Chelyabinsk, Russia
3Chelyabinsk Higher Military Aviation Red Banner School of Navigators, Chelyabinsk, Russia
Keywords: error estimation, modulus of continuity, Fourier transform, ill-posed problem

Abstract >>
This paper studies the problem of determining the boundary condition in the heat conduction equation for composite materials. Mathematically this problem is reduced to the equation of heat conduction in spherical coordinates for an inhomogeneous ball. The temperature inside the ball is assumed to be unknown for an infinite time interval. To find it, the temperature of the heat flow in the media interface is measured at the point r = r 0. An analytical study of the direct problem is carried out, which makes it possible to give a rigorous formulation of the inverse problem and to determine the functional spaces in which it is natural to solve the inverse problem. The main difficulty to be solved, is to obtain an error estimate of the approximate solution. The projection regularization method is used to estimate the modulus of conditional correctness. This allows one to obtain the order-accurate estimates.

#### 8.An approximate solution of singular integral equations using the Chebyshev series on the class of functions vanishing at one end and turning into infinity at the other end of the integration interval

Sh.S. Khubezhty
North Ossetian State University after K.L. Khetagurov, Vladikavkaz, Russia
Keywords: singular integrals, equation index, quadrature formulas, computational schemes, best uniform approximation, error estimation

Abstract >>
Computational schemes for an approximate solution of a singular integral equation of the first kind, bounded at one end and not bounded at the other end of the integration interval are constructed [- 1,1]. The solution of the equation is sought for in the form of a series in the Chebyshev polynomials of the third and the fourth kinds. The kernel and the right-hand side of the equation decompose into series with the use of the Chebyshev polynomials of the third and the fourth kinds, whose coefficients are approximately calculated by the Gaussian quadrature formulas, i.e. the highest algebraic degree of accuracy. For the coefficients of the expansion of the Chebyshev polynomials, exact values in the series are found. The coefficients of the expansion of the unknown function, i.e. solutions of the equation, are found from the solution to systems of linear algebraic equations. To justify the constructed computing schemes, the methods of functional analysis and the theory of orthogonal polynomials are used. When the existence condition for the given functions of the derivatives up to some order belonging to the Holder class is satisfied, the calculation error is estimated and the order of its turning into zero is given.