

2021 year, number 2
Sangita Jha^{1}, A.K.B. Chand^{1}, M.A. Navascues^{2}
^{1}Indian Institute of Technology Madras, Chennai, India ^{2}Universidad de Zaragoza, Zaragoza, Spain
Keywords: fractals, fractal interpolation, hermite interpolation, fractal surface, convergence
Abstract >>
Fractal interpolation provides an efficient way to describe the smooth or nonsmooth structure associated with nature and scientific data. The aim of this paper is to introduce a bivariate Hermite fractal interpolation formula which generalizes the classical Hermite interpolation formula for two variables. It is shown here that the proposed Hermite fractal interpolation function and its derivatives of all orders are good approximations of the original function even if the partial derivatives of the original functions are nonsmooth in nature.

V.P. Zhitnikov^{1}, N.M. Sherykhalina^{1}, R.R. Muksimova^{2}
^{1}Ufa State Aviation Technical University, Ufa, Bashkortostan, Russia ^{2}Saint Petersburg State University of Civil Aviation, Saint Petersburg, Russia
Keywords: heat equation, implicit scheme, Laplace equation, biharmonic equation, iteration method, numerical filtration
Abstract >>
The error caused by the inaccuracy of the equation system solution by iterative methods has been investigated. The upper error estimate for the axially symmetric heat equation is found in the accumulation process in several time steps. The upper estimate shows the linear dependence of the error on the threshold value of the limiting criterion for the iterations number, the quadratic error growth from the range partitions number, and its independence of the time partitions number. The computing experiment shows a good correspondence of the obtained estimate to real errors with boundary and initial conditions of various types. The quadratic error growth for the Laplace equation, caused by the accuracy limitation for applying the iteration method, on the number of range partitions n, is empirically found. A similar error growth for the biharmonic equation is found in proportion to n^{4}.

M.I. Ivanov, I.A. Kremer, Yu.M. Laevsky
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia
Keywords: fluid filtration, fractured porous media, double porosity, porous blocks, fractures, conservation laws, mixed finite element method, upwind scheme, maximum principle
Abstract >>
The paper discusses a computational 3D double porosity model of a twophase incompressible fluid filtration in a fracturedporous medium. Conservation laws are formulated in the integral form, and for their spatial approximation, a combination of the mixed finite element method to determine the total flow and pressure velocities is used and the finite volume method to determine the saturations in porous blocks and in fractures. The approximation of equations for saturations according to an explicit scheme with upwinding to eliminate unphysical oscillations is carried out. The model under consideration includes the injection and production wells with total flow rates. For the total velocities and pressures, the Neumann problem is formulated, for which the condition of unique solvability is indicated and a method for solving it without additional conditions is proposed. For an explicit upwind scheme for solving equations for saturations, a weak maximum principle is established, illustrated by computational experiments.

Kh.D. Ikramov^{1}, A.M. Nazari^{2}
^{1}Lomonosov Moscow State University, Moscow, Russia ^{2}University of Arak, Arak, Iran
Keywords: congruence, unitoid matrix (unitoid), cosquare, similarity, Toeplitz decomposition, indices of inertia, Pythagorean triples, Maple, circulants
Abstract >>
A matrix is said to be unitoid if it can be brought to diagonal form by a congruence transformation. We say that an algorithm is rational if it is finite and uses the arithmetic operations only. There exist rational methods designed for checking congruence of particular classes of unitoid matrices, for example, Hermitian, accretive, or dissipative matrices. We propose a rational algorithm for checking congruence of general unitoid matrices. The algorithm is heuristic in the sense that the user is required to set the values of two integral parameters M and N. The choice of these values depends on the available a priori information about the proximity of neighboring canonical angles of the matrices under checking.

M.N. Nikitenko, V.N. Glinskikh, D.I. Gornostalev
The Trofimuk Institute of Petroleum Geology and Geophysics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
Keywords: electromagnetic soundings, transient method, numericalanalytical solution, computational algorithm, layered homogeneous geoelectric model, Bazhenov formation
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This paper concerns the development of fundamental theoretical foundations and the creation of algorithms and software for pulsed electromagnetic soundings in relation to studying an unconventional source of hydrocarbons with hardtorecover reserves  the Bazhenov formation. We carry out the mathematical substantiation of a new geophysical technology for the localization of oilprospective zones, using a spatially distributed system of highly deviated wells. For the fast mathematical simulation, we obtained a solution to the problem of pulsed electromagnetic soundings in layered homogeneous models of media for an arbitrary current pulse in the electromagnetic source, which allows deep parallelization. Based on the created computational algorithm, a parallel one was developed, as well as a fast computer program for numerical simulation of the signals of the new system on multiprocessor devices of the Siberian Supercomputer Center, SB RAS. We carried out a largescale numerical simulation and analysis of the signals in realistic geoelectric models of the Bazhenov formation to estimate an applicable scope of the new pulsed electromagnetic sounding installation. The calculations show that determining spatial locations of the formation boundaries is possible when logging the wells over a wide range of the sonde spacings. We analyzed the applicability of the diagonal and offdiagonal field components to ensure high sensitivity for mapping the reservoir boundaries and evaluating its internal heterogeneities. The results obtained form a basis for the further design of the optimal configuration of the new electromagnetic sounding system.

M.H. Rashid^{1,2}
^{1}Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 55 Zhongguancun east road, Haidian district, Beijing100190, China ^{2}University of Rajshahi, Rajshahi6205, Bangladesh
Keywords: setvalued mappings, lipschitzlike mappings, generalized equations, variant of Newton's method, semilocal convergence
Abstract >>
Let X and Y be Banach spaces. Let f: Ω → Y be a Frèchet differentiable function on an open subset Ω of X and F be a setvalued mapping with closed graph. Consider the following generalized equation problem: 0 in f(x)+ F(x). In the present paper, we study a variant of Newton's method for solving generalized equation (1) and analyze semilocal and local convergence of this method under weaker conditions than those considered by JeanAlexis and Piètrus [13]. In fact, we show that the variant of Newton's method is superlinearly convergent when the Frèchet derivative of f is (L,p)Hölder continuous and (f+F)^{1} is Lipzchitzlike at a reference point. Moreover, applications of this method to a nonlinear programming problem and a variational inequality are given. Numerical experiments are provided which illustrate the theoretical results.

J.R. Sharma^{1}, H. Arora^{2}
^{1}Sant Longowal Institute of Engineering and Technology, Longowal, Punjab, India ^{2}D.A.V. University, Sarmastpur, Jalandhar, India
Keywords: nonlinear equations, iterative methods, fast algorithms, multiple roots, attraction basins
Abstract >>
We present a family of fifth order iterative methods for finding multiple roots of nonlinear equations. Numerical examples are considered to check the validity of theoretical results. The results show that the new methods are competitive to other methods for multiple roots. Basins of attraction for new methods and some existing methods are drawn to observe the dynamics in the complex plane.

