Theory and Applications of Special Functions II

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Physics".

Deadline for manuscript submissions: 31 May 2024 | Viewed by 8851

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Department of Electrical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands
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Special Issue Information

Dear Colleagues,

Different applications of modern engineering and physical sciences require thorough knowledge of applied mathematics, particularly special functions. These are frequently adopted in acoustics, thermodynamics, electromagnetics, and optics, to express the approximate or exact analytical solution of complex problems, thus providing a better understanding of and meaningful insight into underlying properties and mechanisms.

In this Special Issue, we focus on the application of classical and higher-order special functions to advanced problems of mathematical physics that are characterized by specific (i.e., rectangular, cylindrical, and spherical) symmetry or, conversely, rely on more unconventional models. Attention is given, also, to the illustration of properties of novel special functions, with particular attention paid to the relevant governing differential equation; recurrence formulae; as well as efficient computational algorithms, such as those based on uniform asymptotic representations for small and large arguments.

We look forward to your contributions to review- and original-research articles dealing with the recent advances in the theory and applications of special functions.

Dr. Diego Caratelli
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Published Papers (8 papers)

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Research

17 pages, 16216 KiB  
Article
Non-Standard Finite Difference and Vieta-Lucas Orthogonal Polynomials for the Multi-Space Fractional-Order Coupled Korteweg-de Vries Equation
by Khaled M. Saad and Rekha Srivastava
Symmetry 2024, 16(2), 242; https://doi.org/10.3390/sym16020242 - 16 Feb 2024
Viewed by 588
Abstract
This paper focuses on examining numerical solutions for fractional-order models within the context of the coupled multi-space Korteweg-de Vries problem (CMSKDV). Different types of kernels, including Liouville-Caputo fractional derivative, as well as Caputo-Fabrizio and Atangana-Baleanu fractional derivatives, are utilized in the examination. For [...] Read more.
This paper focuses on examining numerical solutions for fractional-order models within the context of the coupled multi-space Korteweg-de Vries problem (CMSKDV). Different types of kernels, including Liouville-Caputo fractional derivative, as well as Caputo-Fabrizio and Atangana-Baleanu fractional derivatives, are utilized in the examination. For this purpose, the nonstandard finite difference method and spectral collocation method with the properties of the Shifted Vieta-Lucas orthogonal polynomials are employed for converting these models into a system of algebraic equations. The Newton-Raphson technique is then applied to solve these algebraic equations. Since there is no exact solution for non-integer order, we use the absolute two-step error to verify the accuracy of the proposed numerical results. Full article
(This article belongs to the Special Issue Theory and Applications of Special Functions II)
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12 pages, 392 KiB  
Article
Ulam–Hyers Stability of Linear Differential Equation with General Transform
by Sandra Pinelas, Arunachalam Selvam and Sriramulu Sabarinathan
Symmetry 2023, 15(11), 2023; https://doi.org/10.3390/sym15112023 - 05 Nov 2023
Cited by 2 | Viewed by 808
Abstract
The main aim of this study is to implement the general integral transform technique to determine Ulam-type stability and Ulam–Hyers–Mittag–Leffer stability. We are given suitable examples to validate and support the theoretical results. As an application, the general integral transform is used to [...] Read more.
The main aim of this study is to implement the general integral transform technique to determine Ulam-type stability and Ulam–Hyers–Mittag–Leffer stability. We are given suitable examples to validate and support the theoretical results. As an application, the general integral transform is used to find Ulam stability of differential equations arising in Thevenin equivalent electrical circuit system. The results are graphically represented, which provides a clear and thorough explanation of the suggested method. Full article
(This article belongs to the Special Issue Theory and Applications of Special Functions II)
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13 pages, 712 KiB  
Article
A Note on the Lambert W Function: Bernstein and Stieltjes Properties for a Creep Model in Linear Viscoelasticity
by Francesco Mainardi, Enrico Masina and Juan Luis González-Santander
Symmetry 2023, 15(9), 1654; https://doi.org/10.3390/sym15091654 - 26 Aug 2023
Viewed by 798
Abstract
The purpose of this note is to propose an application of the Lambert W function in linear viscoelasticity based on the Bernstein and Stieltjes properties of this function. In particular, we recognize the role of its main branch, W0(t) [...] Read more.
The purpose of this note is to propose an application of the Lambert W function in linear viscoelasticity based on the Bernstein and Stieltjes properties of this function. In particular, we recognize the role of its main branch, W0(t), in a peculiar model of creep with two spectral functions in frequency that completely characterize the creep model. In order to calculate these spectral functions, it turns out that the conjugate symmetry property of the Lambert W function along its branch cut on the negative real axis is essential. We supplement our analysis by computing the corresponding relaxation function and providing the plots of all computed functions. Full article
(This article belongs to the Special Issue Theory and Applications of Special Functions II)
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12 pages, 364 KiB  
Article
Monomiality and a New Family of Hermite Polynomials
by Giuseppe Dattoli and Silvia Licciardi
Symmetry 2023, 15(6), 1254; https://doi.org/10.3390/sym15061254 - 13 Jun 2023
Cited by 3 | Viewed by 781
Abstract
The monomiality principle is based on an abstract definition of the concept of derivative and multiplicative operators. This allows to treat different families of special polynomials as ordinary monomials. The procedure underlines a generalization of the Heisenberg–Weyl group, along with the relevant technicalities [...] Read more.
The monomiality principle is based on an abstract definition of the concept of derivative and multiplicative operators. This allows to treat different families of special polynomials as ordinary monomials. The procedure underlines a generalization of the Heisenberg–Weyl group, along with the relevant technicalities and symmetry properties. In this article, we go deeply into the formulation and meaning of the monomiality principle and employ it to study the properties of a set of polynomials, which, asymptotically, reduce to the ordinary two-variable Kampè dè Fèrièt family. We derive the relevant differential equations and discuss the associated orthogonality properties, along with the relevant generalized forms. Full article
(This article belongs to the Special Issue Theory and Applications of Special Functions II)
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13 pages, 287 KiB  
Article
Poly-Cauchy Numbers with Higher Level
by Takao Komatsu and Víctor F. Sirvent
Symmetry 2023, 15(2), 354; https://doi.org/10.3390/sym15020354 - 28 Jan 2023
Cited by 1 | Viewed by 694
Abstract
In this article, mainly from the analytical aspect, we introduce poly-Cauchy numbers with higher levels (level s) as a kind of extensions of poly-Cauchy numbers with level 2 and the original poly-Cauchy numbers and investigate their properties. Such poly-Cauchy numbers with higher [...] Read more.
In this article, mainly from the analytical aspect, we introduce poly-Cauchy numbers with higher levels (level s) as a kind of extensions of poly-Cauchy numbers with level 2 and the original poly-Cauchy numbers and investigate their properties. Such poly-Cauchy numbers with higher levels are yielded from the inverse relationship with an s-step function of the exponential function. We show such a function with recurrence relations and give the expressions of poly-Cauchy numbers with higher levels. Poly-Cauchy numbers with higher levels can be also expressed in terms of iterated integrals and a combinatorial summation. Poly-Cauchy numbers with higher levels for negative indices have a double summation formula. In addition, Cauchy numbers with higher levels can be also expressed in terms of determinants. Full article
(This article belongs to the Special Issue Theory and Applications of Special Functions II)
15 pages, 8858 KiB  
Article
Bell’s Polynomials and Laplace Transform of Higher Order Nested Functions
by Diego Caratelli and Paolo Emilio Ricci
Symmetry 2022, 14(10), 2139; https://doi.org/10.3390/sym14102139 - 13 Oct 2022
Cited by 1 | Viewed by 1083
Abstract
Using Bell’s polynomials it is possible to approximate the Laplace Transform of composite functions. The same methodology can be adopted for the evaluation of the Laplace Transform of higher-order nested functions. In this case, a suitable extension of Bell’s polynomials, as previously introduced [...] Read more.
Using Bell’s polynomials it is possible to approximate the Laplace Transform of composite functions. The same methodology can be adopted for the evaluation of the Laplace Transform of higher-order nested functions. In this case, a suitable extension of Bell’s polynomials, as previously introduced in the scientific literature, is used, namely higher order Bell’s polynomials used in the representation of the derivatives of multiple nested functions. Some worked examples are shown, and some of the polynomials used are reported in the Appendices. Full article
(This article belongs to the Special Issue Theory and Applications of Special Functions II)
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18 pages, 470 KiB  
Article
Implementation of Two-Mode Gaussian States Whose Covariance Matrix Has the Standard Form
by Gianfranco Cariolaro and Roberto Corvaja
Symmetry 2022, 14(7), 1485; https://doi.org/10.3390/sym14071485 - 20 Jul 2022
Cited by 1 | Viewed by 1918
Abstract
This paper deals with the covariance matrix (CM) of two-mode Gaussian states, which, together with the mean vector, fully describes these states. In the two-mode states, the (ordinary) CM is a real symmetric matrix of order 4; therefore, it depends on 10 real [...] Read more.
This paper deals with the covariance matrix (CM) of two-mode Gaussian states, which, together with the mean vector, fully describes these states. In the two-mode states, the (ordinary) CM is a real symmetric matrix of order 4; therefore, it depends on 10 real variables. However, there is a very efficient representation of the CM called the standard form (SF) that reduces the degrees of freedom to four real variables, while preserving all the relevant information on the state. The SF can be easily evaluated using a set of symplectic invariants. The paper starts from the SF, introducing an architecture that implements with primitive components the given two-mode Gaussian state having the CM with the SF. The architecture consists of a beam splitter, followed by the parallel set of two single–mode real squeezers, followed by another beam splitter. The advantage of this architecture is that it gives a precise non-redundant physical meaning of the generation of the Gaussian state. Essentially, all the relevant information is contained in this simple architecture. Full article
(This article belongs to the Special Issue Theory and Applications of Special Functions II)
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16 pages, 407 KiB  
Article
Gaussian States: Evaluation of the Covariance Matrix from the Implementation with Primitive Component
by Gianfranco Cariolaro, Roberto Corvaja and Filippo Miatto
Symmetry 2022, 14(7), 1286; https://doi.org/10.3390/sym14071286 - 21 Jun 2022
Cited by 1 | Viewed by 1370
Abstract
Quantum Gaussian states play a fundamental role in quantum communications and in quantum information. This paper deals with the implementation of multimode, and particularly of two-mode Gaussian unitaries and Gaussian states with primitive components (phase shifters, single-mode real squeezers, displacements, and beam splitters). [...] Read more.
Quantum Gaussian states play a fundamental role in quantum communications and in quantum information. This paper deals with the implementation of multimode, and particularly of two-mode Gaussian unitaries and Gaussian states with primitive components (phase shifters, single-mode real squeezers, displacements, and beam splitters). The architecture thus obtained allows one to obtain an insight into the physical meaning of each variable involved. Moreover, following the implementation architecture, it is possible to formulate an easy algebra (radical free) for the main operations and transformations of Gaussian states. Full article
(This article belongs to the Special Issue Theory and Applications of Special Functions II)
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