Page 655 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 655
NUMERICAL ANALYSIS AND SCIENTIFIC COMPUTING
Approximate Calculation of Triple Integrals of Rapidly Oscillating
Functions using Different Types of Information about Functions
Olesia Nechuiviter, olesia.nechuiviter@gmail.com
Ukrainian Engineering and Pedagogical Academy, Ukraine
Coauthor: Kyrylo Kovalchuk
Modern methods of digital signal processing are characterized by new approaches to obtaining,
processing and analyzing information. There is a need to build mathematical models in which
information can be given not only by the values of the function at points, but also as a set of
traces of the function on the planes and as a set of traces of the function on the lines. Report is
dedicated to the improvement of mathematical models of digital signal processing and imaging
by the example of constructing cubature formulas of approximate calculation of integrals of
highly oscillatory functions of three variables. The feature of the proposed cubature formulas
is using the input information about function or as a set of traces of function on planes or a set
of traces of function on lines and of course as a set of values of the function in the points. The
cubature formulas are correlated with a formula of the Filon type. The error estimations of the
approximation of the integrals from the highly oscillating function by the cubature formulas on
the class of differential functions are obtained. The main advantages of methods are high ex-
actness of approximation, the opportunity to decrease the amount of information about function
during the calculation.
Approximation of the two variables Discontinuous Functions by
Discontinuous Interlination Splines using Triangular Elements
Iuliia Pershyna, yuliapershina78@gmail.com
Ukrainian Engineering Pedagogics Academy, Ukraine
The problem of approximation of smooth functions by continuous construction with sufficient
completeness was considered in the work of many researchers. However, as a rule, all these
problems are only pleasant exceptions or the result of excessive idealization. More often, the
objects of study are mathematically described by functions with discontinuities, kinks, and other
violations of smoothness. Such nonsmooth functions include profile shock waves generated by
powerful acoustic emitters, or the shape of dunes in the desert, which provides characteristic
sharpening. Discontinued objects also very often arise in tasks using remote methods. Thus,
the detection of cracks in industrial products using non-destructive testing is an important task
in flaw detection, as is the determination of deviations of the internal structure of the product
from the standard. In many tasks of geophysics, the determination of the location of bound aries
separating blocks with different physical properties is the first step in further studies aimed at
determining the physical quantities characterizing the internal structure of the Earth. In com-
puted tomography, when studying the internal structure of a body, it is useful to take into ac-
count its heterogeneity, that is, different densities in different parts of the body. In general,
remote methods are one of the most promising areas of environmental studies. They serve as
the most important source of objective and operational information in various phenomena oc-
curring in the geographic shell of the Earth, and are an effective tool for monitoring the state
of the environment and solving environmental management problems. It is obvious that further
progress in the development of natural sciences is inextricably linked with the use of remote
653
Approximate Calculation of Triple Integrals of Rapidly Oscillating
Functions using Different Types of Information about Functions
Olesia Nechuiviter, olesia.nechuiviter@gmail.com
Ukrainian Engineering and Pedagogical Academy, Ukraine
Coauthor: Kyrylo Kovalchuk
Modern methods of digital signal processing are characterized by new approaches to obtaining,
processing and analyzing information. There is a need to build mathematical models in which
information can be given not only by the values of the function at points, but also as a set of
traces of the function on the planes and as a set of traces of the function on the lines. Report is
dedicated to the improvement of mathematical models of digital signal processing and imaging
by the example of constructing cubature formulas of approximate calculation of integrals of
highly oscillatory functions of three variables. The feature of the proposed cubature formulas
is using the input information about function or as a set of traces of function on planes or a set
of traces of function on lines and of course as a set of values of the function in the points. The
cubature formulas are correlated with a formula of the Filon type. The error estimations of the
approximation of the integrals from the highly oscillating function by the cubature formulas on
the class of differential functions are obtained. The main advantages of methods are high ex-
actness of approximation, the opportunity to decrease the amount of information about function
during the calculation.
Approximation of the two variables Discontinuous Functions by
Discontinuous Interlination Splines using Triangular Elements
Iuliia Pershyna, yuliapershina78@gmail.com
Ukrainian Engineering Pedagogics Academy, Ukraine
The problem of approximation of smooth functions by continuous construction with sufficient
completeness was considered in the work of many researchers. However, as a rule, all these
problems are only pleasant exceptions or the result of excessive idealization. More often, the
objects of study are mathematically described by functions with discontinuities, kinks, and other
violations of smoothness. Such nonsmooth functions include profile shock waves generated by
powerful acoustic emitters, or the shape of dunes in the desert, which provides characteristic
sharpening. Discontinued objects also very often arise in tasks using remote methods. Thus,
the detection of cracks in industrial products using non-destructive testing is an important task
in flaw detection, as is the determination of deviations of the internal structure of the product
from the standard. In many tasks of geophysics, the determination of the location of bound aries
separating blocks with different physical properties is the first step in further studies aimed at
determining the physical quantities characterizing the internal structure of the Earth. In com-
puted tomography, when studying the internal structure of a body, it is useful to take into ac-
count its heterogeneity, that is, different densities in different parts of the body. In general,
remote methods are one of the most promising areas of environmental studies. They serve as
the most important source of objective and operational information in various phenomena oc-
curring in the geographic shell of the Earth, and are an effective tool for monitoring the state
of the environment and solving environmental management problems. It is obvious that further
progress in the development of natural sciences is inextricably linked with the use of remote
653
