CITE THIS NOTEBOOK: Symbolic Computation Applied to Cauchy Type Singular Integrals by Ana Conceição and Jessica Pires. Wolfram Community FEB 14 2023.
ORIGINAL ARTICLE: Conceição AC, Pires JC. Symbolic Computation Applied to Cauchy Type Singular Integrals. Mathematical and Computational Applications. 2022; 27(1):3. https://doi.org/10.3390/mca27010003.
ORIGINAL ARTICLE: Conceição AC, Pires JC. Symbolic Computation Applied to Cauchy Type Singular Integrals. Mathematical and Computational Applications. 2022; 27(1):3. https://doi.org/10.3390/mca27010003.
Article Abstract: 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.
Introduction
Introduction
In recent years, several software applications with extensive capabilities of symbolic computation were made available to the general public. These applications, known as computer algebra systems (CAS), allow the delegation to a computer of all, or a significant part, of the symbolic and numeric calculations present in many mathematical algorithms. In our work, we use the computer algebra system Mathematica (Wolfram Mathematica is a symbolic mathematical computation program, conceived by Stephen Wolfram, used in many scientific, engineering and computing fields) to implement for the first time on a computer analytical algorithms developed by us and others within the operator theory. The design of our algorithms is focused on the possibility of implementing on a computer all, or a significant part, of the extensive symbolic and numeric calculations present in the analytical algorithms. The methods developed rely on innovative techniques of operator theory and have a potential for extension to more complex and general problems. 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 [1-5]. Singular integrals are classic mathematical objects with a vast array of applications in the main scientific research areas, and the importance of their study is globally acknowledged. There are several numerical algorithms and approximation methods for evaluating some classes of singular integrals. There exist also several analytical techniques that allow the exact computation of particular classes of singular integrals. However, the [SInt] and [SIntAFact] algorithms [2] are the only analytical algorithms, to our knowledge, written and implemented for computing Cauchy type singular integrals with general functions, defined in the unit circle. Although the [SInt] algorithm was designed to be efficient for a wide class of singular integrals, it can be improved in terms of implementation. For instance, this first version does not identify whether a given function has poles in the unit circle or if the user inputs non-valid functions, nor can it work with the generality of fifth degree or higher polynomials. Here, we present an improved and fully efficient version of the algorithm, the algorithm, designed with the Root object concept (to represent solutions to one-variable algebraic equations) available in Wolfram Mathematica. This conception of an improved version became possible after the design of the [ASPPlusPMinus] algorithm, created for the rational case, which also calculates the projections associated with the integral of type Cauchy, even in cases involving polynomials of the fifth degree or higher. The output provided by these algorithms can be used for several other algorithms to solve singular integral equations, to factorize functions, to compute the dimension of the kernel of some classes of singular integrals and to study the spectra of some operators. 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 [1,5-8], at least for the rational case. The corresponding source code of the algorithms has been made available as a Supplementary Material to the online edition of the original article (https://www.mdpi.com/article/10.3390/mca27010003/s1 ).
[SInt]
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Operator Theory Basic Concepts
Operator Theory Basic Concepts
Let denote the unit circle in the complex plane. Let and denote the open unit disk and the exterior region of the unit circle (∞ included), respectively. As usual, () denotes the space of all essentially bounded functions defined on and () the class of all bounded and analytic functions in . Let R() be the algebra of rational functions without poles on and () the subsets of R() whose elements have no poles in , respectively. The study of singular integral operators has applications in different research areas, such as the theory of diffraction of acoustic and electromagnetic waves, theory of scattering and of inverse scattering and factorization theory (see, for instance, [9–16]). It is well known that the singular operator with Cauchy kernel, , defined almost everywhere on , by φ(t)=τ,t∈, where the integral is understood in the sense of its principal value, represents a bounded linear operator in the Lebesgue space (). In addition, is a self-adjoint and unitary operator in () [17]. Thus, we can associate with two complementary Cauchy projection operators =(I±)/2, where represents the identity operator. The projectors allow us to decompose the algebra in the topological direct sum , where ()=R() and ()=R(). We also have ()=()⊕.
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[ARoots] - Theory and Examples
[ARoots] - Theory and Examples
This algorithm identifies, after calculating the roots of a polynomial p(t), the location of the roots relative to , , and . It is also possible to ask for an approximate value of a desired root.
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Algorithm: ARoots Authors: Ana C. Conceição (aconcei@ualg.pt) & Jéssica Pires (jessicaccpires@gmail.com) Year: 2021 Institution: Center for Functional Analysis, Linear Structures and Applications (CEAFEL), Universidade do Algarve Article/Journal: Symb
olic comput
ation applied to Cauchy type singular integrals. Math. Comput. Appl. License: 
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[AZeros] - Theory and Examples
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[APoles] - Theory and Examples
[APoles] - Theory and Examples
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Supplementary Material of the Article
Supplementary Material of the Article
The following algorithms are available online at https://www.mdpi.com/article/10.3390/mca27010003/s1, Algorithm S1: ARoots; Algorithm S2: AZeros; Algorithm S3: APoles; Algorithm S4: ASPPlusPMinus; Algorithm S5: SInt2.0.
Funding
Funding
This research was funded by FCT/MCTES (PIDDAC) within the project CEAFEL UIDB/04721/2020–IST-ID, grant 1801P.00956.1.01.
References
References
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2. Conceição, A.C.; Kravchenko, V.G.; Pereira, J.C. Computing some classes of Cauchy type singular integrals with Mathematica software. Adv. Comput. Math. 2013, 39, 273–288.
3. Conceição, A.C.; Kravchenko, V.G.; Pereira, J.C. Rational functions factorization algorithm: A symbolic computation for the scalar and matrix cases. In Proceedings of the 1st National Conference on Symbolic Computation in Education and Research, Lisboa, Portugal, 2–3 April 2012.
4. Conceição, A.C.; Kravchenko, V.G.; Pereira, J.C. Factorization Algorithm for Some Special Non-rational Matrix Functions. In Operator Theory: Advances and Applications; Birkhäuser Verlag: Basel, Switzerland, 2010; Volume 202, pp. 87–109.
5. Conceição, A.C.; Pereira, J.C. Exploring the spectra of some classes of singular integral operators with symbolic computation. Math. Comput. Sci. 2016, 10, 291–309.
6. Conceição, A.C.; Kravchenko, V.G. About explicit factorization of some classes of non-rational matrix functions. Math. Nachr. 2007, 280, 1022–1034.
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8. Conceição, A.C.; Marreiros, R.C.; Pereira, J.C. Symbolic computation applied to the study of the kernel of a singular integral operator with non-Carleman shift and conjugation. Math. Comput. Sci. 2016, 10, 365–386.
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