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Symmetry
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Symmetry in Mathematical Functional Equations
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Symmetry in Mathematical Functional Equations

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

Deadline for manuscript submissions: 30 April 2024 | Viewed by 7510

Special Issue Editor

Prof. Dr. Rabha W. Ibrahim

E-Mail Website
Guest Editor
Mathematics Research Center, Department of Mathematics, Near East University, Near East Boulevard, TRNC Mersin 10, Nicosia 99138, Turkey
Interests: mathematical analysis; functional analysis

Special Issue Information

Dear Colleagues,

Functional equations have a long and fascinating history in relation to mathematical physics, as well as having impacts in many other areas of mathematics. In this Special Issue, we will cover some of the ways they have developed in the context of both classical and quantum totally integrable systems. These equations are frequently referred to as differential-difference equations with retarded arguments, and have an amazingly broad and diverse range of applications. Control systems, biological growth behavior, and econometrics are the most common of these. To find solutions with specific qualities, it is possible to research the symmetry properties of the functions used to build mathematical formulas. Regarding these formulas, the study of special functions and special polynomials, taking into account their symmetry features, can yield some intriguing findings.

Prof. Dr. Rabha W. Ibrahim
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.

Keywords

  • symmetry in functional differential equations (real or complex)
  • symmetry in functional integral equations (real or complex) 
  • fractional functional differential equations (real or complex) 
  • fractional functional integral equation (real or complex) 
  • fractal formula 
  • evolution equations 
  • implicit functions 
  • bifurcation 
  • quantum calculus 
  • k-calculus 
  • applications

Published Papers (7 papers)

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Research

18 pages, 1280 KiB  
Article
Novel Computations of the Time-Fractional Coupled Korteweg–de Vries Equations via Non-Singular Kernel Operators in Terms of the Natural Transform
by Abdulrahman B. M. Alzahrani and Ghadah Alhawael
Symmetry 2023, 15(11), 2010; https://doi.org/10.3390/sym15112010 - 01 Nov 2023
Viewed by 671
Abstract
In the present research, we establish an effective method for determining the time-fractional coupled Korteweg–de Vries (KdV) equation’s approximate solution employing the fractional derivatives of Caputo–Fabrizio and Atangana–Baleanu. KdV models are crucial because they can accurately represent a variety of physical problems, including [...] Read more.
In the present research, we establish an effective method for determining the time-fractional coupled Korteweg–de Vries (KdV) equation’s approximate solution employing the fractional derivatives of Caputo–Fabrizio and Atangana–Baleanu. KdV models are crucial because they can accurately represent a variety of physical problems, including thin-film flows and waves on shallow water surfaces. Some theoretical physical features of quantum mechanics are also explained by the KdV model. Many investigations have been conducted on this precisely solvable model. Numerous academics have proposed new applications for the generation of acoustic waves in plasma from ions and crystal lattices. Adomian decomposition and natural transform decomposition techniques are combined in the natural decomposition method (NDM). We first apply the natural transform to examine the fractional order and obtain a recurrence relation. Second, we use the Adomian decomposition approach to the recurrence relation, and then, using successive iterations and the initial conditions, we can establish the series solution. We note that the proposed fractional model is highly accurate and valid when using this technique. The numerical outcomes demonstrate that only a small number of terms are required to arrive at an approximation that is exact, efficient, and trustworthy. Two examples are given to illustrate how the technique performs. Tables and 3D graphs display the best current numerical and analytical results. The suggested method provides a series form solution, which makes it quite easy to understand the behavior of the fractional models. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Functional Equations)
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17 pages, 333 KiB  
Article
Norm and Numerical Radius Inequalities Related to the Selberg Operator
by Najla Altwaijry, Cristian Conde, Silvestru Sever Dragomir and Kais Feki
Symmetry 2023, 15(10), 1860; https://doi.org/10.3390/sym15101860 - 03 Oct 2023
Viewed by 644
Abstract
The main focus of this paper is the study of the Selberg operator. It aims to establish appropriate bounds for the norm and numerical radius of the product of three bounded operators, with one of them being a Selberg operator. Moreover, it offers [...] Read more.
The main focus of this paper is the study of the Selberg operator. It aims to establish appropriate bounds for the norm and numerical radius of the product of three bounded operators, with one of them being a Selberg operator. Moreover, it offers several bounds involving the summation of operators, notably the Selberg operator. Through the examination of these properties and relationships, this study contributes to a better understanding of the Selberg operator and its influence on operator compositions. The paper also highlights the significance of symmetry in mathematics and its potential implications across various mathematical domains. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Functional Equations)
15 pages, 981 KiB  
Article
Complex-Variable Dynamic System of Layla and Majnun Model with Analytic Solutions
by Ibtisam Aldawish and Rabha W. Ibrahim
Symmetry 2023, 15(8), 1557; https://doi.org/10.3390/sym15081557 - 09 Aug 2023
Viewed by 733
Abstract
A complex Layla and Majnun model system (CLMMS) is suggested in this study for a complex variable in the open-unit disk. Analytic solutions are discovered by using a technique of bounded turning functions. The set of necessary conditions is illustrated involving some special [...] Read more.
A complex Layla and Majnun model system (CLMMS) is suggested in this study for a complex variable in the open-unit disk. Analytic solutions are discovered by using a technique of bounded turning functions. The set of necessary conditions is illustrated involving some special cases. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Functional Equations)
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11 pages, 393 KiB  
Article
Univalence and Starlikeness of Certain Classes of Analytic Functions
by Najla M. Alarifi and M. Obradović
Symmetry 2023, 15(5), 1014; https://doi.org/10.3390/sym15051014 - 02 May 2023
Cited by 2 | Viewed by 1044
Abstract
For the analytic functions ϕ(ζ)=ζ+∑k=n∞ϕkζk in the unit disk O, we calculate the values of n and α, where the condition [...] Read more.
For the analytic functions ϕ(ζ)=ζ+∑k=n∞ϕkζk in the unit disk O, we calculate the values of n and α, where the condition ℜ1+ζϕ″(ζ)/ϕ′(ζ)>−α or ℜ1+ζϕ″(ζ)/ϕ′(ζ)<1+α/2 yields univalence and starlikeness. Conditions imply ϕ in the class where all normalized analytic functions U, with ζ/ϕ(ζ)2ϕ′(ζ)−1<1 are obtained. Recent findings are gained, and unique cases are demonstrated. The generalization of the Jack lemma serves the proof of the main result and that our methodology is based on the idea of subordination. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Functional Equations)
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10 pages, 326 KiB  
Article
Applications of Gegenbauer Polynomials for Subfamilies of Bi-Univalent Functions Involving a Borel Distribution-Type Mittag-Leffler Function
by Abdullah Alatawi, Maslina Darus and Badriah Alamri
Symmetry 2023, 15(4), 785; https://doi.org/10.3390/sym15040785 - 23 Mar 2023
Cited by 5 | Viewed by 1066
Abstract
In this research, a novel linear operator involving the Borel distribution and Mittag-Leffler functions is introduced using Hadamard products or convolutions. This operator is utilized to develop new subfamilies of bi-univalent functions via the principle of subordination with Gegenbauer orthogonal polynomials. The investigation [...] Read more.
In this research, a novel linear operator involving the Borel distribution and Mittag-Leffler functions is introduced using Hadamard products or convolutions. This operator is utilized to develop new subfamilies of bi-univalent functions via the principle of subordination with Gegenbauer orthogonal polynomials. The investigation also focuses on the estimation of the coefficients |aℓ|(ℓ=2,3) and the Fekete–Szegö inequality for functions belonging to these subfamilies of bi-univalent functions. Several corollaries and implications of the findings are discussed. Overall, this study presents a new approach for constructing bi-univalent functions and provides valuable insights for further research in this area. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Functional Equations)
29 pages, 26411 KiB  
Article
Image Denoising Based on Quantum Calculus of Local Fractional Entropy
by Ala’a R. Al-Shamasneh and Rabha W. Ibrahim
Symmetry 2023, 15(2), 396; https://doi.org/10.3390/sym15020396 - 02 Feb 2023
Cited by 7 | Viewed by 1523
Abstract
Images are frequently disrupted by noise of all kinds, making image restoration very challenging. There have been many different image denoising models proposed over the last few decades. Some models preserve the image’s smooth region, while others preserve the texture margin. One of [...] Read more.
Images are frequently disrupted by noise of all kinds, making image restoration very challenging. There have been many different image denoising models proposed over the last few decades. Some models preserve the image’s smooth region, while others preserve the texture margin. One of these methods is by using quantum calculus. Quantum calculus is a branch of mathematics that deals with the manipulation of functions and operators in a quantum mechanical setting. It has been used in image processing to improve the speed and accuracy of image-processing algorithms. In quantum computing, entropy can be defined as a measure of the disorder or randomness of a quantum state. The concept of local fractional entropy has been used to study a wide range of quantum systems. In this study, an image denoising model is proposed based on the quantum calculus of local fractional entropy (QC-LFE) to remove a Gaussian noise. The local fractional entropy is used to estimate the image pixel probability, while the quantum calculus is used to estimate the convolution window mask for image denoising. A processing fractional mask with n x n elements was used in the suggested denoising algorithm. The proposed image denoising algorithm uses mask convolution to process each corrupted pixel one at a time. The proposed denoising algorithm’s effectiveness is assessed using peak signal-to-noise ratio and visual perception (PSNR). The experimental findings show that, compared to other similar fractional operators, the proposed method can better preserve texture details when denoising. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Functional Equations)
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18 pages, 649 KiB  
Article
On New Symmetric Schur Functions Associated with Integral and Integro-Differential Functional Expressions in a Complex Domain
by Samir B. Hadid and Rabha W. Ibrahim
Symmetry 2023, 15(1), 235; https://doi.org/10.3390/sym15010235 - 14 Jan 2023
Cited by 4 | Viewed by 931
Abstract
The symmetric Schur process has many different types of formals, such as the functional differential, functional integral, and special functional processes based on special functions. In this effort, the normalized symmetric Schur process (NSSP) is defined and then used to determine the geometric [...] Read more.
The symmetric Schur process has many different types of formals, such as the functional differential, functional integral, and special functional processes based on special functions. In this effort, the normalized symmetric Schur process (NSSP) is defined and then used to determine the geometric and symmetric interpretations of mathematical expressions in a complex symmetric domain (the open unit disk). To obtain more symmetric properties involving NSSP, we consider a symmetric differential operator. The outcome is a symmetric convoluted operator. Geometrically, studies are presented for the suggested operator. Our method is based on the theory of differential subordination. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Functional Equations)
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