Recent Advances in Special Functions and Applications

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 30 July 2024 | Viewed by 5030

Special Issue Editor


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Guest Editor
Department of Mathematics, Dankook University, Cheonan 31116, Republic of Korea
Interests: analytic Feynman integral; integral transform; special function; operator theory

Special Issue Information

Dear Colleagues,

With the development of scientific techniques, various transformations have appeared in diverse areas of the sciences, engineering and humanities, ranging from genomics and health sciences to economics, finance and machine learning. To handle physical problems related to various transformations, integral transform, Fourier–Feynman transform and several transformations in function space have become a hot topic in functional analysis.

Transform analysis poses many challenges, particularly for numerical analysis with non-polynomial dimensionality or high-dimensional structures. Therefore, new and innovative theories for transform need to be developed.

As one of hottest research topics in functional analysis, various transformations are of great attraction to researchers. We invite investigators to contribute original research articles as well as review articles that will stimulate the continuing efforts to develop efficient transformations and applications concerning the analysis of transformation.

Prof. Dr. Hyun Soo Chung
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.

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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.

Keywords

  • integral transform in function space
  • Fourier–Feynman transform
  • integral transform for physical models
  • high-dimensional transform with applications
  • dimension reduction of high-dimensional transform
  • transform and differential equations

Published Papers (6 papers)

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Research

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17 pages, 303 KiB  
Article
A Comprehensive Study of Generalized Lambert, Generalized Stieltjes, and Stieltjes–Poisson Transforms
by Jeetendrasingh Maan and E. R. Negrín
Axioms 2024, 13(5), 283; https://doi.org/10.3390/axioms13050283 - 23 Apr 2024
Viewed by 583
Abstract
In this paper, we explore the properties of the generalized Lambert transform, the L-transform, the generalized Stieltjes transform, and the Stieltjes–Poisson transform within the framework of Lebesgue spaces. We establish Parseval-type relations for each transform, providing a comprehensive analysis of their behaviour and [...] Read more.
In this paper, we explore the properties of the generalized Lambert transform, the L-transform, the generalized Stieltjes transform, and the Stieltjes–Poisson transform within the framework of Lebesgue spaces. We establish Parseval-type relations for each transform, providing a comprehensive analysis of their behaviour and mathematical characteristics. Full article
(This article belongs to the Special Issue Recent Advances in Special Functions and Applications)
11 pages, 248 KiB  
Article
A Series Approximation for the Analytic Fourier–Feynman Transform on Wiener Space
by Hyun Soo Chung
Axioms 2024, 13(4), 237; https://doi.org/10.3390/axioms13040237 - 3 Apr 2024
Viewed by 647
Abstract
In this paper, we first establish an evaluation formula to calculate Wiener integrals of functionals on Wiener space. We then apply our evaluation formula to carry out easy an calculation for the analytic Fourier–Feynman transform of the functionals. Some examples are furnished to [...] Read more.
In this paper, we first establish an evaluation formula to calculate Wiener integrals of functionals on Wiener space. We then apply our evaluation formula to carry out easy an calculation for the analytic Fourier–Feynman transform of the functionals. Some examples are furnished to illustrate the usefulness of the evaluation formula. Finally, using the evaluation formula, we establish the series approximation for the analytic Fourier–Feynman transform. Full article
(This article belongs to the Special Issue Recent Advances in Special Functions and Applications)
10 pages, 232 KiB  
Article
Transformation Properties of a Class of Variable Coefficient Boiti–Leon–Manna–Pempinelli Equations
by Christodoulos Sophocleous
Axioms 2024, 13(2), 82; https://doi.org/10.3390/axioms13020082 - 25 Jan 2024
Viewed by 680
Abstract
We derive the enhanced Lie group classification for a general class of variable coefficient Boiti–Leon–Manna–Pempinelli equations. This task is achieved with the use of the equivalence group admitted by the class. Using the admitted equivalence group, we transform the general class into a [...] Read more.
We derive the enhanced Lie group classification for a general class of variable coefficient Boiti–Leon–Manna–Pempinelli equations. This task is achieved with the use of the equivalence group admitted by the class. Using the admitted equivalence group, we transform the general class into a much simpler class of equations. Additionally, examples of non-Lie reduction operators are presented. Full article
(This article belongs to the Special Issue Recent Advances in Special Functions and Applications)
11 pages, 295 KiB  
Article
On a Generalization of the Kummer’s Quadratic Transformation and a Resolution of an Isolated Case
by Mohamed Jalel Atia and Arjun Kumar Rathie
Axioms 2023, 12(9), 821; https://doi.org/10.3390/axioms12090821 - 27 Aug 2023
Viewed by 735
Abstract
The resolution of isolated cases of the well-known quadratic transformation due to Kummer was thoroughly examined in two recent publications by Atia as well as Atia and Al-Mohaimeed. The objective of this paper is twofold. We establish generalizations of the quadratic transformation due [...] Read more.
The resolution of isolated cases of the well-known quadratic transformation due to Kummer was thoroughly examined in two recent publications by Atia as well as Atia and Al-Mohaimeed. The objective of this paper is twofold. We establish generalizations of the quadratic transformation due to Kummer in the most general case in the first section, and in the second section, an effort is made to discuss the resolution of an isolated case of a quadratic transformation contiguous to that of Kummer established by Choi and Rathie. Full article
(This article belongs to the Special Issue Recent Advances in Special Functions and Applications)
15 pages, 285 KiB  
Article
Application of Double Sumudu-Generalized Laplace Decomposition Method for Solving 2+1-Pseudoparabolic Equation
by Hassan Eltayeb
Axioms 2023, 12(8), 799; https://doi.org/10.3390/axioms12080799 - 19 Aug 2023
Cited by 4 | Viewed by 745
Abstract
The main purpose of this research paper is to discuss the solution of the singular two-dimensional pseudoparabolic equation by employing the double Sumudu-generalized Laplace transform decomposition method (DSGLTDM). We establish two theorems related to the partial derivatives. Furthermore, to investigate the relevance of [...] Read more.
The main purpose of this research paper is to discuss the solution of the singular two-dimensional pseudoparabolic equation by employing the double Sumudu-generalized Laplace transform decomposition method (DSGLTDM). We establish two theorems related to the partial derivatives. Furthermore, to investigate the relevance of the proposed method to solving singular two-dimensional pseudo parabolic equations, three examples are provided. Full article
(This article belongs to the Special Issue Recent Advances in Special Functions and Applications)

Review

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17 pages, 1227 KiB  
Review
Topological Comparison of Some Dimension Reduction Methods Using Persistent Homology on EEG Data
by Eddy Kwessi
Axioms 2023, 12(7), 699; https://doi.org/10.3390/axioms12070699 - 18 Jul 2023
Cited by 1 | Viewed by 998
Abstract
In this paper, we explore how to use topological tools to compare dimension reduction methods. We first make a brief overview of some of the methods often used in dimension reduction such as isometric feature mapping, Laplacian Eigenmaps, fast independent component analysis, kernel [...] Read more.
In this paper, we explore how to use topological tools to compare dimension reduction methods. We first make a brief overview of some of the methods often used in dimension reduction such as isometric feature mapping, Laplacian Eigenmaps, fast independent component analysis, kernel ridge regression, and t-distributed stochastic neighbor embedding. We then give a brief overview of some of the topological notions used in topological data analysis, such as barcodes, persistent homology, and Wasserstein distance. Theoretically, when these methods are applied on a data set, they can be interpreted differently. From EEG data embedded into a manifold of high dimension, we discuss these methods and we compare them across persistent homologies of dimensions 0, 1, and 2, that is, across connected components, tunnels and holes, shells around voids, or cavities. We find that from three dimension clouds of points, it is not clear how distinct from each other the methods are, but Wasserstein and Bottleneck distances, topological tests of hypothesis, and various methods show that the methods qualitatively and significantly differ across homologies. We can infer from this analysis that topological persistent homologies do change dramatically at seizure, a finding already obtained in previous analyses. This suggests that looking at changes in homology landscapes could be a predictor of seizure. Full article
(This article belongs to the Special Issue Recent Advances in Special Functions and Applications)
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