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Symmetry
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Functional Analysis, Fractional Operators and Symmetry/Asymmetry
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Functional Analysis, Fractional Operators and Symmetry/Asymmetry

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

Deadline for manuscript submissions: closed (31 July 2023) | Viewed by 14513

Special Issue Editors

Dr. J. Vanterler Da C. Sousa

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Guest Editor
Department of Mathematics, Aerospace Engineering, PPGEA-UEMA, DEMATI-UEMA, São Luís 65054, MA, Brazil
Interests: fractional differential equations; functional analysis; variational approach; frac-tional calculus; analysis mathematics
Special Issues, Collections and Topics in MDPI journals
Dr. Jiabin Zuo

E-Mail Website
Guest Editor
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
Interests: fractional laplacian equations; partial differential equations
Special Issues, Collections and Topics in MDPI journals
Dr. Cesar E. Torres Ledesma

E-Mail Website
Guest Editor
Departamento de Matemáticas, Universidad Nacional de Trujillo, Trujillo, Peru
Interests: fractional laplacian; fractional calculus; fractional elliptic equations; nonlocal equations

Special Issue Information

Dear Colleagues,

As we all know, the role and consequences of the notion of symmetry in mathematics and related sciences are very important. On many occasions, symmetries have appeared in mathematical formulations that have become essential for solving problems or delving further into research. In this Special Issue, we aim to establish some theorical results and applications in the fields of functional analysis and fractional operators, in which the concept of symmetry plays an essential role. Among others, papers on these topics are welcome.

Submit your paper and select the Journal “Symmetry” and the Special Issue “Functional Analysis, Fractional Operators and Symmetry/Asymmetry” via: MDPI submission system. Our papers will be published on a rolling basis and we will be pleased to receive your submission once you have finished it.

Dr. J. Vanterler Da C. Sousa
Dr. Jiabin Zuo
Dr. Cesar E. Torres Ledesma
Guest Editors

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

  • Dynamical systems
  • Partial differential equations
  • Mathematical physics
  • Symmetry operators
  • Fractional operators
  • Applied mathematics
  • Discrete mathematics and graph theory
  • Mathematical analysis
  • Fractional differential equations
  • Extension of linear operators
  • Self-adjoint operators

Published Papers (12 papers)

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Research

27 pages, 400 KiB  
Article
Parallel Subgradient-like Extragradient Approaches for Variational Inequality and Fixed-Point Problems with Bregman Relatively Asymptotical Nonexpansivity
by Lu-Chuan Ceng, Yun-Ling Cui, Sheng-Long Cao, Bing Li, Cong-Shan Wang and Hui-Ying Hu
Symmetry 2023, 15(9), 1749; https://doi.org/10.3390/sym15091749 - 12 Sep 2023
Viewed by 707
Abstract
In a uniformly smooth and p-uniformly convex Banach space, let the pair of variational inequality and fixed-point problems (VIFPPs) consist of two variational inequality problems (VIPs) involving two uniformly continuous and pseudomonotone mappings and two fixed-point problems implicating two uniformly continuous and [...] Read more.
In a uniformly smooth and p-uniformly convex Banach space, let the pair of variational inequality and fixed-point problems (VIFPPs) consist of two variational inequality problems (VIPs) involving two uniformly continuous and pseudomonotone mappings and two fixed-point problems implicating two uniformly continuous and Bregman relatively asymptotically nonexpansive mappings. This article designs two parallel subgradient-like extragradient algorithms with an inertial effect for solving this pair of VIFPPs, where each algorithm consists of two parts which are of a mutually symmetric structure. With the help of suitable registrations, it is proven that the sequences generated by the suggested algorithms converge weakly and strongly to a solution of this pair of VIFPPs, respectively. Lastly, an illustrative instance is presented to verify the implementability and applicability of the suggested approaches. Full article
(This article belongs to the Special Issue Functional Analysis, Fractional Operators and Symmetry/Asymmetry)
11 pages, 274 KiB  
Article
On Primary Decomposition of Hermite Projectors
by Boris Shekhtman, Lesław Skrzypek and Brian Tuesink
Symmetry 2023, 15(9), 1658; https://doi.org/10.3390/sym15091658 - 28 Aug 2023
Viewed by 550
Abstract
An ideal projector on the space of polynomials C[x]=C[x1,…,xd] is a projector whose kernel is an ideal in C[x]. Every ideal projector P can be [...] Read more.
An ideal projector on the space of polynomials C[x]=C[x1,…,xd] is a projector whose kernel is an ideal in C[x]. Every ideal projector P can be written as a sum of ideal projectors P(k) such that the intersection of their kernels kerP(k) is a primary decomposition of the ideal kerP. In this paper, we show that P is a limit of Lagrange projectors if and only if each P(k) is. As an application, we construct an ideal projector P whose kernel is a symmetric ideal, yet P is not a limit of Lagrange projectors. Full article
(This article belongs to the Special Issue Functional Analysis, Fractional Operators and Symmetry/Asymmetry)
17 pages, 358 KiB  
Article
A Mathematical Theoretical Study of a Coupled Fully Hybrid (k, Φ)-Fractional Order System of BVPs in Generalized Banach Spaces
by Abdellatif Boutiara, Sina Etemad, Sabri T. M. Thabet, Sotiris K. Ntouyas, Shahram Rezapour and Jessada Tariboon
Symmetry 2023, 15(5), 1041; https://doi.org/10.3390/sym15051041 - 08 May 2023
Cited by 4 | Viewed by 1333
Abstract
In this paper, we study a coupled fully hybrid system of (k,Φ)–Hilfer fractional differential equations equipped with non-symmetric (k,Φ)–Riemann-Liouville (RL) integral conditions. To prove the existence and uniqueness results, we use [...] Read more.
In this paper, we study a coupled fully hybrid system of (k,Φ)–Hilfer fractional differential equations equipped with non-symmetric (k,Φ)–Riemann-Liouville (RL) integral conditions. To prove the existence and uniqueness results, we use the Krasnoselskii and Perov fixed-point theorems with Lipschitzian matrix in the context of a generalized Banach space (GBS). Moreover, the Ulam–Hyers (UH) stability of the solutions is discussed by using the Urs’s method. Finally, an illustrated example is given to confirm the validity of our results. Full article
(This article belongs to the Special Issue Functional Analysis, Fractional Operators and Symmetry/Asymmetry)
23 pages, 407 KiB  
Article
Some New Fractional Hadamard and Pachpatte-Type Inequalities with Applications via Generalized Preinvexity
by Muhammad Tariq, Asif Ali Shaikh and Sotiris K. Ntouyas
Symmetry 2023, 15(5), 1033; https://doi.org/10.3390/sym15051033 - 07 May 2023
Viewed by 1099
Abstract
The term convexity associated with the theory of inequality in the sense of fractional analysis has a broad range of different and remarkable applications in the domain of applied sciences. The prime objective of this article is to investigate some new variants of [...] Read more.
The term convexity associated with the theory of inequality in the sense of fractional analysis has a broad range of different and remarkable applications in the domain of applied sciences. The prime objective of this article is to investigate some new variants of Hermite–Hadamard and Pachpatte-type integral inequalities involving the idea of the preinvex function in the frame of a fractional integral operator, namely the Caputo–Fabrizio fractional operator. By employing our approach, a new fractional integral identity that correlates with preinvex functions for first-order differentiable mappings is presented. Moreover, we derive some refinements of the Hermite–Hadamard-type inequality for mappings, whose first-order derivatives are generalized preinvex functions in the Caputo–Fabrizio fractional sense. From an application viewpoint, to represent the usability of the concerning results, we presented several inequalities by using special means of real numbers. Integral inequalities in association with convexity in the frame of fractional calculus have a strong relationship with symmetry. Our investigation provides a better image of convex analysis in the frame of fractional calculus. Full article
(This article belongs to the Special Issue Functional Analysis, Fractional Operators and Symmetry/Asymmetry)
18 pages, 538 KiB  
Article
An Efficient Analytical Approach to Investigate Fractional Caudrey–Dodd–Gibbon Equations with Non-Singular Kernel Derivatives
by Dowlath Fathima, Reham A. Alahmadi, Adnan Khan, Afroza Akhter and Abdul Hamid Ganie
Symmetry 2023, 15(4), 850; https://doi.org/10.3390/sym15040850 - 02 Apr 2023
Cited by 12 | Viewed by 1110
Abstract
Fractional calculus is at this time an area where many models are still being developed, explored, and used in real-world applications in many branches of science and engineering where non-locality plays a key role. Although many wonderful discoveries have already been reported by [...] Read more.
Fractional calculus is at this time an area where many models are still being developed, explored, and used in real-world applications in many branches of science and engineering where non-locality plays a key role. Although many wonderful discoveries have already been reported by researchers in important monographs and review articles, there is still a great deal of non-local phenomena that have not been studied and are only waiting to be explored. As a result, we can continually learn about new applications and aspects of fractional modelling. In this study, a precise and analytical method with non-singular kernel derivatives is used to solve the Caudrey–Dodd–Gibbon (CDG) model, a modification of the fifth-order KdV equation (fKdV). The fractional derivative is taken into account by the Caputo–Fabrizio (CF) derivative and the Atangana–Baleanu derivative in the Caputo sense (ABC). This model illustrates the propagation of magneto-acoustic, shallow-water, and gravity–capillary waves in a plasma medium. The dynamic behaviour of the acquired solutions has been represented in a number of two- and three-dimensional figures. A number of simulations are also performed to demonstrate how the resulting solutions physically behave with respect to fractional order. The significance of the current research is that new solutions are obtained by using a strong analytical approach. Utilizing a fractional derivative operator to solve equivalent models is another benefit of this approach. The results of the present work have similar aspects to the symmetry of partial differential equations. Full article
(This article belongs to the Special Issue Functional Analysis, Fractional Operators and Symmetry/Asymmetry)
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13 pages, 1375 KiB  
Article
On the Solution of Fractional Biswas–Milovic Model via Analytical Method
by Pongsakorn Sunthrayuth, Muhammad Naeem, Nehad Ali Shah, Rasool Shah and Jae Dong Chung
Symmetry 2023, 15(1), 210; https://doi.org/10.3390/sym15010210 - 11 Jan 2023
Cited by 4 | Viewed by 1006
Abstract
Through the use of a unique approach, we study the fractional Biswas–Milovic model with Kerr and parabolic law nonlinearities in this paper. The Caputo approach is used to take the fractional derivative. The method employed here is the homotopy perturbation transform method (HPTM), [...] Read more.
Through the use of a unique approach, we study the fractional Biswas–Milovic model with Kerr and parabolic law nonlinearities in this paper. The Caputo approach is used to take the fractional derivative. The method employed here is the homotopy perturbation transform method (HPTM), which combines the homotopy perturbation method (HPM) and Yang transform (YT). The HPTM combines the homotopy perturbation method, He’s polynomials, and the Yang transform. He’s polynomial is a wonderful tool for dealing with nonlinear terms. To confirm the validity of each result, the technique was substituted into the equation. The described techniques can be used to find the solutions to these kinds of equations as infinite series, and when these series are in closed form, they give a precise solution. Graphs are used to show the derived numerical results. The maple software package is used to carry out the numerical simulation work. The results of this research are highly positive and demonstrate how effective the suggested method is for mathematical modeling of natural occurrences. Full article
(This article belongs to the Special Issue Functional Analysis, Fractional Operators and Symmetry/Asymmetry)
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14 pages, 474 KiB  
Article
Numerical Simulations of the Fractional Systems of Volterra Integral Equations within the Chebyshev Pseudo-Spectral Method
by Pongsakorn Sunthrayuth, Muhammad Naeem, Nehad Ali Shah, Rasool Shah and Jae Dong Chung
Symmetry 2022, 14(12), 2575; https://doi.org/10.3390/sym14122575 - 06 Dec 2022
Viewed by 978
Abstract
In this article, we find the solutions to fractional Volterra-type integral equation nonlinear systems through a Chebyshev pseudo-spectral method (CPM). The fractional derivative is described in the Caputo manner. The suggested method’s accuracy and reliability are confirmed by the results. The proposed method [...] Read more.
In this article, we find the solutions to fractional Volterra-type integral equation nonlinear systems through a Chebyshev pseudo-spectral method (CPM). The fractional derivative is described in the Caputo manner. The suggested method’s accuracy and reliability are confirmed by the results. The proposed method is implemented for solving various nonlinear systems; the results we obtained were compared with the exact solution and other method solutions. The graphical representation and tables show that our method’s error quickly converges as compared to other methods. By comparing the proposed method’s solution with the actual solution and other methods, we can confirm that CPM is more accurate and closer to the exact solution. We display the pointwise solution in the tables, which verifies the proposed method’s accuracy at each point and aids in a better comprehension of the suggested approach. Moreover, the results of using the suggested method at different fractional orders are examined, showing that when a value moves from a fractional order to an integer order, the result is closer to the precise solution. Furthermore, the proposed technique for handling fractional-order linear and non-linear physical problems in science and engineering is straightforward to implement. Full article
(This article belongs to the Special Issue Functional Analysis, Fractional Operators and Symmetry/Asymmetry)
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20 pages, 3217 KiB  
Article
Fractional Series Solution Construction for Nonlinear Fractional Reaction-Diffusion Brusselator Model Utilizing Laplace Residual Power Series
by Aisha Abdullah Alderremy, Rasool Shah, Naveed Iqbal, Shaban Aly and Kamsing Nonlaopon
Symmetry 2022, 14(9), 1944; https://doi.org/10.3390/sym14091944 - 18 Sep 2022
Cited by 30 | Viewed by 1413
Abstract
This article investigates different nonlinear systems of fractional partial differential equations analytically using an attractive modified method known as the Laplace residual power series technique. Based on a combination of the Laplace transformation and the residual power series technique, we achieve analytic and [...] Read more.
This article investigates different nonlinear systems of fractional partial differential equations analytically using an attractive modified method known as the Laplace residual power series technique. Based on a combination of the Laplace transformation and the residual power series technique, we achieve analytic and approximation results in rapid convergent series form by employing the notion of the limit, with less time and effort than the residual power series method. Three challenges are evaluated and simulated to validate the suggested method’s practicability, efficiency, and simplicity. The analysis of the acquired findings demonstrates that the method mentioned above is simple, accurate, and appropriate for investigating the solutions to nonlinear applied sciences models. Full article
(This article belongs to the Special Issue Functional Analysis, Fractional Operators and Symmetry/Asymmetry)
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10 pages, 284 KiB  
Article
A Class of Fourth-Order Symmetrical Kirchhoff Type Systems
by Yong Wu, Said Taarabti, Zakaria El Allali, Khalil Ben Hadddouch and Jiabin Zuo
Symmetry 2022, 14(8), 1630; https://doi.org/10.3390/sym14081630 - 08 Aug 2022
Cited by 1 | Viewed by 1109
Abstract
This paper deals with the existence and multiplicity of solutions for a perturbed nonlocal fourth-order class of p(·)&q(·)-Kirchhoff elliptic systems under Navier boundary conditions. By using the variational method and Ricceri’s critical point theorem, [...] Read more.
This paper deals with the existence and multiplicity of solutions for a perturbed nonlocal fourth-order class of p(·)&q(·)-Kirchhoff elliptic systems under Navier boundary conditions. By using the variational method and Ricceri’s critical point theorem, we can find a proper conditions to ensure that the perturbed fourth-order of (p(x),q(x))-Kirchhoff systems has at least three weak solutions. We have extended and improved some recent results. The complexity of the combination of variable exponent theory and fourth-order Kirchhoff systems makes the results of this work novel and new contribution. Finally, a very concrete example is given to illustrate the applications of our results. Full article
(This article belongs to the Special Issue Functional Analysis, Fractional Operators and Symmetry/Asymmetry)
13 pages, 273 KiB  
Article
2D Discrete Hodge–Dirac Operator on the Torus
by Volodymyr Sushch
Symmetry 2022, 14(8), 1556; https://doi.org/10.3390/sym14081556 - 28 Jul 2022
Viewed by 1060
Abstract
We discuss a discretization of the de Rham–Hodge theory in the two-dimensional case based on a discrete exterior calculus framework. We present discrete analogues of the Hodge–Dirac and Laplace operators in which key geometric aspects of the continuum counterpart are captured. We provide [...] Read more.
We discuss a discretization of the de Rham–Hodge theory in the two-dimensional case based on a discrete exterior calculus framework. We present discrete analogues of the Hodge–Dirac and Laplace operators in which key geometric aspects of the continuum counterpart are captured. We provide and prove a discrete version of the Hodge decomposition theorem. The goal of this work is to develop a satisfactory discrete model of the de Rham–Hodge theory on manifolds that are homeomorphic to the torus. Special attention has been paid to discrete models on a combinatorial torus. In this particular case, we also define and calculate the cohomology groups. Full article
(This article belongs to the Special Issue Functional Analysis, Fractional Operators and Symmetry/Asymmetry)
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17 pages, 324 KiB  
Article
An Efficient Method for Solving Second-Order Fuzzy Order Fuzzy Initial Value Problems
by Qamar Dallashi and Muhammed I. Syam
Symmetry 2022, 14(6), 1218; https://doi.org/10.3390/sym14061218 - 13 Jun 2022
Cited by 2 | Viewed by 1117
Abstract
In this paper, we present an accurate numerical approach based on the reproducing kernel method (RKM) for solving second-order fuzzy initial value problems (FIVP) with symmetry coefficients such as symmetric triangles and symmetric trapezoids. Finding the exact solution of FIVP is not an [...] Read more.
In this paper, we present an accurate numerical approach based on the reproducing kernel method (RKM) for solving second-order fuzzy initial value problems (FIVP) with symmetry coefficients such as symmetric triangles and symmetric trapezoids. Finding the exact solution of FIVP is not an easy task since the definition will produce a complicated optimization problem. To overcome this difficulty, a numerical method is developed to solve this type of problems. We start by introducing the necessary definitions and theorems about the fuzzy logic. Then, we derived the kernels for two Hilbert spaces. The RKM is derived for the second-order IVP in the Boolean sense, and then we generalize it for the fuzzy sense. Numerical and theoretical results will be given to obtain the accuracy of the developed technique. We solved four linear and non-linear fuzzy IVPs numerically using the proposed method, and we compute the error in each case to show the efficiency of the method. The absolute error was very small in the four examples. Full article
(This article belongs to the Special Issue Functional Analysis, Fractional Operators and Symmetry/Asymmetry)
26 pages, 356 KiB  
Article
Study of a Coupled System with Sub-Strip and Multi-Valued Boundary Conditions via Topological Degree Theory on an Infinite Domain
by Sahibzada Waseem Ahmad, Muhammed Sarwar, Kamal Shah, Eiman and Thabet Abdeljawad
Symmetry 2022, 14(5), 841; https://doi.org/10.3390/sym14050841 - 19 Apr 2022
Cited by 8 | Viewed by 1205
Abstract
The existence and uniqueness of solutions for a coupled system of Liouville–Caputo type fractional integro-differential equations with multi-point and sub-strip boundary conditions are investigated in this study. The fractional integro-differential equations contain a finite number of Riemann–Liouville fractional integral and non-integral type nonlinearities, [...] Read more.
The existence and uniqueness of solutions for a coupled system of Liouville–Caputo type fractional integro-differential equations with multi-point and sub-strip boundary conditions are investigated in this study. The fractional integro-differential equations contain a finite number of Riemann–Liouville fractional integral and non-integral type nonlinearities, as well as Caputo differential operators of various orders subject to fractional boundary conditions on an infinite interval. At the boundary conditions, we use sub-strip and multi-point contribution. There are various techniques to solve such type of differential equations and one of the most common is known as symmetry analysis. The symmetry analysis has widely been used in problems involving differential equations, although determining the symmetries can be computationally intensive compared to other methods. Therefore, we employ the degree theory due to the Mawhin involving measure of a non-compactness technique to arrive at our desired findings. An interesting pertinent problem has also been provided to demonstrate the applicability of our results. Full article
(This article belongs to the Special Issue Functional Analysis, Fractional Operators and Symmetry/Asymmetry)
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