Page 656 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 656
NUMERICAL ANALYSIS AND SCIENTIFIC COMPUTING
research methods. This work belongs to a series of works by the authors aimed at the study
and improvement of mathematical models in computed tomography. To date, tomography has
developed many computational methods, algorithms and software tools aimed at restoring the
internal properties of an object. They perform well when restoring objects with smooth prop-
erties, but give unsatisfactory results for objects with discontinuous characteristics. Therefore,
there is a need to create mathematical methods for approximating discontinuous functions for
a more accurate idea of the structure of the studied object. The mathematical foundations of
tomography were laid at the beginning of the last century in the works of the German scientist J.
Radon, who developed the theory of the transformation of functions of many variables (Radon
transformation). According to these transformations, the function of many variables can be
characterized not only by its values at points of multidimensional space, but also by integrals
from this function taken over an infinite set of lines or planes. A series of works by authors
[18-20] devoted to solving the flat problem of radon computed tomography using the hetero-
geneity of the internal structure of a two-dimensional body. For this purpose, it is advisable
to use function interlination operators, since these operators restore (possibly approximated)
functions on their known traces on a given system of lines. They provide an opportunity to
construct operators whose integrals from these lines (linear integrals) will be equal to integrals
from the most renewable function. That is, interlination is a mathematical apparatus, naturally
related to the task of restoring the characteristics of objects according to their known projec-
tions. This article is a continuation of this article series. Paper is devoted to the development
of a method for approximating two variables discontinuous functions by discontinuous interli-
nation splines using arbitrary triangular elements. Experimental data are one-sided traces of a
function along a system of given lines, such data are used in remote methods, in particular in
tomography. The paper is also devoted to the development a method for approximating of two
variables discontinuous functions by triangular elements that comprise one curved side. These
methods make it possible to approximate the discontinuous function, using its more complex
domains of definition and avoiding the Gibbs phenomenon
Numerical computation of the complex zeros of Bessel and Hankel
functions
Diego Ruiz-Antolín, ruizantolind@unican.es
University of Cantabria, Spain
Coauthors: Amparo Gil, Javier Segura
The complex zeros of cylinder functions appear in several problems of applied mathematics
and theoretical physics. For example, the complex zeros of Hankel functions are involved in
quantum scattering problems by spheres and cylinders. An algorithm (with a Matlab imple-
mentation) for computing the complex zeros of the Bessel function of first kind Jν(z), second
kind Yν(z), Hankel functions Hν(1)(z), Hν(2)(z) and general combinations of Bessel functions
αJν(z) + βYν(z) and Hankel functions αHν(1)(z) + βHν(2)(z) is described in this presentation.
The algorithm, based on the results obtained in [1], [2], allows to obtain with certainty and ac-
curacy all the zeros of the selected function inside a box in the complex plane. The performance
of the algorithm is illustrated with numerical examples. This is a joint work in collaboration
with Amparo Gil and Javier Segura.
654
research methods. This work belongs to a series of works by the authors aimed at the study
and improvement of mathematical models in computed tomography. To date, tomography has
developed many computational methods, algorithms and software tools aimed at restoring the
internal properties of an object. They perform well when restoring objects with smooth prop-
erties, but give unsatisfactory results for objects with discontinuous characteristics. Therefore,
there is a need to create mathematical methods for approximating discontinuous functions for
a more accurate idea of the structure of the studied object. The mathematical foundations of
tomography were laid at the beginning of the last century in the works of the German scientist J.
Radon, who developed the theory of the transformation of functions of many variables (Radon
transformation). According to these transformations, the function of many variables can be
characterized not only by its values at points of multidimensional space, but also by integrals
from this function taken over an infinite set of lines or planes. A series of works by authors
[18-20] devoted to solving the flat problem of radon computed tomography using the hetero-
geneity of the internal structure of a two-dimensional body. For this purpose, it is advisable
to use function interlination operators, since these operators restore (possibly approximated)
functions on their known traces on a given system of lines. They provide an opportunity to
construct operators whose integrals from these lines (linear integrals) will be equal to integrals
from the most renewable function. That is, interlination is a mathematical apparatus, naturally
related to the task of restoring the characteristics of objects according to their known projec-
tions. This article is a continuation of this article series. Paper is devoted to the development
of a method for approximating two variables discontinuous functions by discontinuous interli-
nation splines using arbitrary triangular elements. Experimental data are one-sided traces of a
function along a system of given lines, such data are used in remote methods, in particular in
tomography. The paper is also devoted to the development a method for approximating of two
variables discontinuous functions by triangular elements that comprise one curved side. These
methods make it possible to approximate the discontinuous function, using its more complex
domains of definition and avoiding the Gibbs phenomenon
Numerical computation of the complex zeros of Bessel and Hankel
functions
Diego Ruiz-Antolín, ruizantolind@unican.es
University of Cantabria, Spain
Coauthors: Amparo Gil, Javier Segura
The complex zeros of cylinder functions appear in several problems of applied mathematics
and theoretical physics. For example, the complex zeros of Hankel functions are involved in
quantum scattering problems by spheres and cylinders. An algorithm (with a Matlab imple-
mentation) for computing the complex zeros of the Bessel function of first kind Jν(z), second
kind Yν(z), Hankel functions Hν(1)(z), Hν(2)(z) and general combinations of Bessel functions
αJν(z) + βYν(z) and Hankel functions αHν(1)(z) + βHν(2)(z) is described in this presentation.
The algorithm, based on the results obtained in [1], [2], allows to obtain with certainty and ac-
curacy all the zeros of the selected function inside a box in the complex plane. The performance
of the algorithm is illustrated with numerical examples. This is a joint work in collaboration
with Amparo Gil and Javier Segura.
654
