Exploring the Spectra of Some Classes of Singular Integral Operators with Symbolic Computation

Conceição, Ana C.; Pereira, José C.

doi:10.1007/s11786-016-0264-2

Cited by 2 publications

(2 citation statements)

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“…Furthermore, since the majority of the concepts and results established for the unit circle within operator theory can be generalized for the real line, we are trying to make the several necessary adaptations in analytical and implementation terms so that the creation of new algorithms that use functions defined in the real line becomes possible. We believe that it will also be possible to extend the methods described in this article to other classes of singular integrals of the Cauchy type, such as those studied in [1,[5][6][7][8], at least for the rational case.…”

Section: Introductionmentioning

confidence: 99%

“…By implementing these methods on a computer, new tools are created to explore that same potential, making the results of lengthy and complex calculations available in a simple way to researchers in different areas. In the last years, we designed and/or implemented calculation techniques to compute singular integrals, analytical algorithms for solving integral equations, to study the spectrum and the kernel of several special classes of singular integral operators and to factorize functions (see, for instance [1][2][3][4][5]).…”

Section: Introductionmentioning

confidence: 99%

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Symbolic Computation Applied to Cauchy Type Singular Integrals

Conceição

Pires

2021

MCA

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The development of operator theory is stimulated by the need to solve problems emerging from several fields in mathematics and physics. At the present time, this theory has wide applications in the study of non-linear differential equations, in linear transport theory, in the theory of diffraction of acoustic and electromagnetic waves, in the theory of scattering and of inverse scattering, among others. In our work, we use the computer algebra system Mathematica to implement, for the first time on a computer, analytical algorithms developed by us and others within operator theory. The main goal of this paper is to present new operator theory algorithms related to Cauchy type singular integrals, defined in the unit circle. The design of these algorithms was focused on the possibility of implementing on a computer all the extensive symbolic and numeric calculations present in the algorithms. Several nontrivial examples computed with the algorithms are presented. The corresponding source code of the algorithms has been made available as a supplement to the online edition of this article.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Symbolic Computation Applied to Cauchy Type Singular Integrals

Conceição

Pires

2021

MCA

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Symbolic Computation Applied to the Study of the Kernel of Special Classes of Paired Singular Integral Operators

Conceição

2020

Math.Comput.Sci.

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Operator theory has many applications in several main scientific research areas (structural mechanics, aeronautics, quantum mechanics, ecology, probability theory, electrical engineering, among others) and the importance of its study is globally acknowledged. On the study of the operator's kernel some progress has been achieved for some specific classes of singular integral operators whose properties allow the use of particular strategies. However, the existing algorithms allow, in general, to study the dimension of the kernel of some classes of singular integral operators but are not designed to be implemented on a computer. The main goal of this paper is to show how the symbolic and numeric capabilities of a computer algebra system can be used to study the kernel of special classes of paired singular integral operators with essentially bounded coefficients defined on the unit circle. It is described how some factorization algorithms can be used to compute the dimension of the kernel of special classes of singular integral operators. The analytical algorithms [ADimKerPaired-Scalar], [AKerPaired-Scalar], and [ADimKerPaired-Matrix] are presented. The design of these new algorithms was focused on the possibility of implementing on a computer all the extensive symbolic and numeric calculations present in the algorithms. For the essentially bounded hermitian coefficients case, there exist some relations with Hankel operators. The paper contains some interesting and nontrivial examples obtained with the use of a computer algebra system.

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Exploring the Spectra of Some Classes of Singular Integral Operators with Symbolic Computation

Cited by 2 publications

References 34 publications

Symbolic Computation Applied to Cauchy Type Singular Integrals

Symbolic Computation Applied to Cauchy Type Singular Integrals

Symbolic Computation Applied to the Study of the Kernel of Special Classes of Paired Singular Integral Operators

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