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Symmetry
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Fractional-Order Systems and Its Applications in Engineering
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Fractional-Order Systems and Its Applications in Engineering

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

Deadline for manuscript submissions: closed (31 December 2023) | Viewed by 16053

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editors

Dr. Hassen Fourati

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Guest Editor
Institute of Engineering, Université Grenoble Alpes, CNRS, Inria, Grenoble INP*, GIPSA-Lab, 38000 Grenoble, France
Interests: automatic control; data fusion; navigation; machine learning applied in navigation; estimation and filtering; sensors for robotics
Special Issues, Collections and Topics in MDPI journals
Dr. Abdellatif Ben Makhlouf

E-Mail Website
Guest Editor
Mathematics Department, College of Science, Jouf University, Sakaka P.O. Box 2014, Saudi Arabia
Interests: fractional-order systems; stability analysis; stabilization; stochastic differential equations; existence and uniqueness; numerical simulations; fractional derivative; differential equations; fixed point theory
Special Issues, Collections and Topics in MDPI journals
Dr. Omar Naifar

E-Mail Website
Guest Editor
Control and Energy Management Laboratory, National School of Engineering, Sfax University, Sfax 3038, Tunisia
Interests: mathematical models; control theory
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

As data systems become more effective, more and more mathematical approaches have been applied to real-world applications to achieve exceptional outcomes. Fractional approaches (such as fractional calculus, fractional Fourier analysis, and the linear canonical transform) are gaining importance in the area of mathematics and the community of applied mathematicians. The theory and method of fractional domain analysis may further define the dynamic process of signal translation from time domains to frequency domains, creating a new avenue for non-stationary signal analysis and treatment study. In technical domains such as radar, communications, and sonar, fractional approaches are preferable to traditional integral methods because they bring novel concepts, procedures, and ideas. Due to the unpredictability of the sent signal in actual engineering systems and the effect of different disturbances and noises on the transmission process, despite the numerous benefits of these new fractional approaches, a few critical issues still need to be resolved. Simultaneously, fraction theory is confronted with several practical limits in engineering, such as sampling and filtering in the sphere of multidimensional signals. This Special Issue focuses on the current successes and potential difficulties of fractional techniques in engineering theory and applications. Novel studies and literature reviews are encouraged.

Potential topics include, but are not limited to, the following: analysis of the stability of fractional-order systems; the observability and controllability of fractional-order systems; an estimation of the states of fractional-order systems; fractional-order system stabilization; the identification of fractional models in continuous time; and medical imaging, radar, navigation, communications, genetics, and optical system applications of fractional approaches.

Dr. Hassen Fourati
Dr. Abdellatif Ben Makhlouf
Dr. Omar Naifar
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

  • fractional differential equations
  • observability, controllability, stability of nonlinear systems
  • applications of fractional problems in engineering
  • stochastic fractional-order systems
  • variable-order differentiation and integration

Published Papers (13 papers)

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Research

10 pages, 4086 KiB  
Article
A Dynamic Behavior Analysis of a Rolling Mill’s Main Drive System with Fractional Derivative and Stochastic Disturbance
by Guobo Wang and Lifeng Ma
Symmetry 2023, 15(8), 1509; https://doi.org/10.3390/sym15081509 - 31 Jul 2023
Viewed by 614
Abstract
Taking the random factors into account, a fractional main drive system of a rolling mill with Gaussian white noise is developed. First, the potential deterministic bifurcation is investigated by a linearized stability analysis. The results indicate that the fractional order changes the system [...] Read more.
Taking the random factors into account, a fractional main drive system of a rolling mill with Gaussian white noise is developed. First, the potential deterministic bifurcation is investigated by a linearized stability analysis. The results indicate that the fractional order changes the system from a stable point to a limit cycle with symmetric phase trajectories. Then, the stochastic response is obtained with the aid of the equivalent transformation of the fractional derivative and stochastic averaging methods. It is found that the joint stationary probability density function appears to have symmetric distribution. Finally, the influence of the fractional order and noise intensity on system dynamics behavior is discussed. The study is beneficial to understand the intrinsic mechanisms of vibration abatement. Full article
(This article belongs to the Special Issue Fractional-Order Systems and Its Applications in Engineering)
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15 pages, 1638 KiB  
Article
Unknown Input Observer Scheme for a Class of Nonlinear Generalized Proportional Fractional Order Systems
by Ali Omar M. Alsharif, Assaad Jmal, Omar Naifar, Abdellatif Ben Makhlouf, Mohamed Rhaima and Lassaad Mchiri
Symmetry 2023, 15(6), 1233; https://doi.org/10.3390/sym15061233 - 09 Jun 2023
Viewed by 949
Abstract
In this study, an unknown input observer is proposed for a class of nonlinear GPFOSs. For this class of systems, both full-order and reduced-order observers have been established. The investigated system satisfies the one-sided Lipschitz nonlinear condition, which is an improvement of the [...] Read more.
In this study, an unknown input observer is proposed for a class of nonlinear GPFOSs. For this class of systems, both full-order and reduced-order observers have been established. The investigated system satisfies the one-sided Lipschitz nonlinear condition, which is an improvement of the classic Lipschitz condition. Sufficient conditions have been proposed to ensure the error dynamics’ Mittag–Leffler stability. The value of this work lies in the fact that, to the best of the authors’ knowledge, this is the first research work that investigates the issue of Observer Design (OD) for GPFOSs. To exemplify the usefulness of the suggested observers, an illustrative numerical example is suggested. Full article
(This article belongs to the Special Issue Fractional-Order Systems and Its Applications in Engineering)
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26 pages, 3056 KiB  
Article
Numerical Approach for Solving a Fractional-Order Norovirus Epidemic Model with Vaccination and Asymptomatic Carriers
by Aeshah A. Raezah, Rahat Zarin and Zehba Raizah
Symmetry 2023, 15(6), 1208; https://doi.org/10.3390/sym15061208 - 05 Jun 2023
Cited by 1 | Viewed by 961
Abstract
This paper explored the impact of population symmetry on the spread and control of a norovirus epidemic. The study proposed a mathematical model for the norovirus epidemic that takes into account asymptomatic infected individuals and vaccination effects using a non-singular fractional operator of [...] Read more.
This paper explored the impact of population symmetry on the spread and control of a norovirus epidemic. The study proposed a mathematical model for the norovirus epidemic that takes into account asymptomatic infected individuals and vaccination effects using a non-singular fractional operator of Atanganaa–Baleanu Caputo (ABC). Fixed point theory, specifically Schauder and Banach’s fixed point theory, was used to investigate the existence and uniqueness of solutions for the proposed model. The study employed MATLAB software to generate simulation results and demonstrate the effectiveness of the fractional order q. A general numerical algorithm based on Adams–Bashforth and Newton’s Polynomial method was developed to approximate the solution. Furthermore, the stability of the proposed model was analyzed using Ulam–Hyers stability techniques. The basic reproductive number was calculated with the help of next-generation matrix techniques. The sensitivity analysis of the model parameters was performed to test which parameter is the most sensitive for the epidemic. The values of the parameters were estimated with the help of least square curve fitting tools. The results of the study provide valuable insights into the behavior of the proposed model and demonstrate the potential applications of fractional calculus in solving complex problems related to disease transmission. Full article
(This article belongs to the Special Issue Fractional-Order Systems and Its Applications in Engineering)
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10 pages, 1085 KiB  
Article
State Feedback Controller Design for a Class of Generalized Proportional Fractional Order Nonlinear Systems
by Ali Omar M. Alsharif, Assaad Jmal, Omar Naifar, Abdellatif Ben Makhlouf, Mohamed Rhaima and Lassaad Mchiri
Symmetry 2023, 15(6), 1168; https://doi.org/10.3390/sym15061168 - 29 May 2023
Viewed by 905
Abstract
The state feedback controller design for a class of Generalized Proportional Fractional Order (GPFO) Nonlinear Systems is presented in this paper. The design is based on the combination of the One-Sided Lipschitz (OSL) system class with GPFO modeling. The main contribution of this [...] Read more.
The state feedback controller design for a class of Generalized Proportional Fractional Order (GPFO) Nonlinear Systems is presented in this paper. The design is based on the combination of the One-Sided Lipschitz (OSL) system class with GPFO modeling. The main contribution of this study is that, to the best of the authors’ knowledge, this work presents the first state feedback control design for GPFO systems. The suggested state feedback controller is intended to ensure the system’s generalized Mittag Leffler (GML) stability and to deliver optimal performance. The findings of this paper show that the proposed strategy is effective in stabilizing Generalized Proportional Fractional Order Nonlinear Systems. A numerical example is presented to demonstrate the usefulness of the stated theoretical conclusions. Full article
(This article belongs to the Special Issue Fractional-Order Systems and Its Applications in Engineering)
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31 pages, 2498 KiB  
Article
A Fractional-Order Density-Dependent Mathematical Model to Find the Better Strain of Wolbachia
by Dianavinnarasi Joseph, Raja Ramachandran, Jehad Alzabut, Sayooj Aby Jose and Hasib Khan
Symmetry 2023, 15(4), 845; https://doi.org/10.3390/sym15040845 - 01 Apr 2023
Cited by 7 | Viewed by 1419
Abstract
The primary objective of the current study was to create a mathematical model utilizing fractional-order calculus for the purpose of analyzing the symmetrical characteristics of Wolbachia dissemination among Aedesaegypti mosquitoes. We investigated various strains of Wolbachia to determine the most sustainable one through [...] Read more.
The primary objective of the current study was to create a mathematical model utilizing fractional-order calculus for the purpose of analyzing the symmetrical characteristics of Wolbachia dissemination among Aedesaegypti mosquitoes. We investigated various strains of Wolbachia to determine the most sustainable one through predicting their dynamics. Wolbachia is an effective tool for controlling mosquito-borne diseases, and several strains have been tested in laboratories and released into outbreak locations. This study aimed to determine the symmetrical features of the most efficient strain from a mathematical perspective. This was accomplished by integrating a density-dependent death rate and the rate of cytoplasmic incompatibility (CI) into the model to examine the spread of Wolbachia and non-Wolbachia mosquitoes. The fractional-order mathematical model developed here is physically meaningful and was assessed for equilibrium points in the presence and absence of disease. Eight equilibrium points were determined, and their local and global stability were determined using the Routh–Hurwitz criterion and linear matrix inequality theory. The basic reproduction number was calculated using the next-generation matrix method. The research also involved conducting numerical simulations to evaluate the behavior of the basic reproduction number for different equilibrium points and identify the optimal CI value for reducing disease spread. Full article
(This article belongs to the Special Issue Fractional-Order Systems and Its Applications in Engineering)
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13 pages, 293 KiB  
Article
On s-Convexity of Dual Simpson Type Integral Inequalities
by Tarek Chiheb, Hamid Boulares, Moheddine Imsatfia, Badreddine Meftah and Abdelkader Moumen
Symmetry 2023, 15(3), 733; https://doi.org/10.3390/sym15030733 - 15 Mar 2023
Cited by 2 | Viewed by 764
Abstract
Integral inequalities are a powerful tool for estimating errors of quadrature formulas. In this study, some symmetric dual Simpson type integral inequalities for the classes of s-convex, bounded and Lipschitzian functions are proposed. The obtained results are based on a new identity [...] Read more.
Integral inequalities are a powerful tool for estimating errors of quadrature formulas. In this study, some symmetric dual Simpson type integral inequalities for the classes of s-convex, bounded and Lipschitzian functions are proposed. The obtained results are based on a new identity and the use of some standard techniques such as Hölder as well as power mean inequalities. We give at the end some applications to the estimation of quadrature rules and to particular means. Full article
(This article belongs to the Special Issue Fractional-Order Systems and Its Applications in Engineering)
15 pages, 306 KiB  
Article
Symmetrical Solutions for Non-Local Fractional Integro-Differential Equations via Caputo–Katugampola Derivatives
by Khalil S. Al-Ghafri, Awad T. Alabdala, Saleh S. Redhwan, Omar Bazighifan, Ali Hasan Ali and Loredana Florentina Iambor
Symmetry 2023, 15(3), 662; https://doi.org/10.3390/sym15030662 - 06 Mar 2023
Cited by 7 | Viewed by 1105
Abstract
Fractional calculus, which deals with the concept of fractional derivatives and integrals, has become an important area of research, due to its ability to capture memory effects and non-local behavior in the modeling of real-world phenomena. In this work, we study a new [...] Read more.
Fractional calculus, which deals with the concept of fractional derivatives and integrals, has become an important area of research, due to its ability to capture memory effects and non-local behavior in the modeling of real-world phenomena. In this work, we study a new class of fractional Volterra–Fredholm integro-differential equations, involving the Caputo–Katugampola fractional derivative. By applying the Krasnoselskii and Banach fixed-point theorems, we prove the existence and uniqueness of solutions to this problem. The modified Adomian decomposition method is used, to solve the resulting fractional differential equations. This technique rapidly provides convergent successive approximations of the exact solution to the given problem; therefore, we investigate the convergence of approximate solutions, using the modified Adomian decomposition method. Finally, we provide an example, to demonstrate our results. Our findings contribute to the current understanding of fractional integro-differential equations and their solutions, and have the potential to inform future research in this area. Full article
(This article belongs to the Special Issue Fractional-Order Systems and Its Applications in Engineering)
21 pages, 1442 KiB  
Article
A Method for Solving Time-Fractional Initial Boundary Value Problems of Variable Order
by Kinda Abuasbeh, Asia Kanwal, Ramsha Shafqat, Bilal Taufeeq, Muna A. Almulla and Muath Awadalla
Symmetry 2023, 15(2), 519; https://doi.org/10.3390/sym15020519 - 15 Feb 2023
Cited by 5 | Viewed by 1426
Abstract
Various scholars have lately employed a wide range of strategies to resolve specific types of symmetrical fractional differential equations. This paper introduces a new implicit finite difference method with variable-order time-fractional Caputo derivative to solve semi-linear initial boundary value problems. Despite its extensive [...] Read more.
Various scholars have lately employed a wide range of strategies to resolve specific types of symmetrical fractional differential equations. This paper introduces a new implicit finite difference method with variable-order time-fractional Caputo derivative to solve semi-linear initial boundary value problems. Despite its extensive use in other areas, fractional calculus has only recently been applied to physics. This paper aims to find a solution for the fractional diffusion equation using an implicit finite difference scheme, and the results are displayed graphically using MATLAB and the Fourier technique to assess stability. The findings show the unconditional stability of the implicit time-fractional finite difference method. This method employs a variable-order fractional derivative of time, enabling greater flexibility and the ability to tackle more complicated problems. Full article
(This article belongs to the Special Issue Fractional-Order Systems and Its Applications in Engineering)
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13 pages, 289 KiB  
Article
Multiplicatively Simpson Type Inequalities via Fractional Integral
by Abdelkader Moumen, Hamid Boulares, Badreddine Meftah, Ramsha Shafqat, Tariq Alraqad, Ekram E. Ali and Zennir Khaled
Symmetry 2023, 15(2), 460; https://doi.org/10.3390/sym15020460 - 09 Feb 2023
Cited by 11 | Viewed by 1127
Abstract
Multiplicative calculus, also called non-Newtonian calculus, represents an alternative approach to the usual calculus of Newton (1643–1727) and Leibniz (1646–1716). This type of calculus was first introduced by Grossman and Katz and it provides a defined calculation, from the start, for positive real [...] Read more.
Multiplicative calculus, also called non-Newtonian calculus, represents an alternative approach to the usual calculus of Newton (1643–1727) and Leibniz (1646–1716). This type of calculus was first introduced by Grossman and Katz and it provides a defined calculation, from the start, for positive real numbers only. In this investigation, we propose to study symmetrical fractional multiplicative inequalities of the Simpson type. For this, we first establish a new fractional identity for multiplicatively differentiable functions. Based on that identity, we derive new Simpson-type inequalities for multiplicatively convex functions via fractional integral operators. We finish the study by providing some applications to analytic inequalities. Full article
(This article belongs to the Special Issue Fractional-Order Systems and Its Applications in Engineering)
12 pages, 281 KiB  
Article
Fractional Multiplicative Bullen-Type Inequalities for Multiplicative Differentiable Functions
by Hamid Boulares, Badreddine Meftah, Abdelkader Moumen, Ramsha Shafqat, Hicham Saber, Tariq Alraqad and Ekram E. Ali
Symmetry 2023, 15(2), 451; https://doi.org/10.3390/sym15020451 - 08 Feb 2023
Cited by 12 | Viewed by 1004
Abstract
Various scholars have lately employed a wide range of strategies to resolve specific types of symmetrical fractional differential equations. In this paper, we propose a new fractional identity for multiplicatively differentiable functions; based on this identity, we establish some new fractional multiplicative Bullen-type [...] Read more.
Various scholars have lately employed a wide range of strategies to resolve specific types of symmetrical fractional differential equations. In this paper, we propose a new fractional identity for multiplicatively differentiable functions; based on this identity, we establish some new fractional multiplicative Bullen-type inequalities for multiplicative differentiable convex functions. Some applications of the obtained results are given. Full article
(This article belongs to the Special Issue Fractional-Order Systems and Its Applications in Engineering)
15 pages, 1897 KiB  
Article
A Novel Three-Step Numerical Solver for Physical Models under Fractal Behavior
by Muath Awadalla, Sania Qureshi, Amanullah Soomro and Kinda Abuasbeh
Symmetry 2023, 15(2), 330; https://doi.org/10.3390/sym15020330 - 24 Jan 2023
Cited by 6 | Viewed by 1055
Abstract
In this paper, we suggest an iterative method for solving nonlinear equations that can be used in the physical sciences. This response is broken down into three parts. Our methodology is inspired by both the standard Taylor’s method and an earlier Halley’s method. [...] Read more.
In this paper, we suggest an iterative method for solving nonlinear equations that can be used in the physical sciences. This response is broken down into three parts. Our methodology is inspired by both the standard Taylor’s method and an earlier Halley’s method. Three evaluations of the given function and two evaluations of its first derivative are all that are needed for each iteration with this method. Because of this, the unique methodology can complete its goal far more quickly than many of the other methods currently in use. We looked at several additional practical research models, including population growth, blood rheology, and neurophysiology. Polynomiographs can be used to show the convergence zones of certain polynomials with complex values. Polynomiographs are produced as a byproduct, and these end up having an appealing look and being artistically engaging. The twisting of polynomiographs is symmetric when the parameters are all real and asymmetric when some of the parameters are imaginary. Full article
(This article belongs to the Special Issue Fractional-Order Systems and Its Applications in Engineering)
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21 pages, 1200 KiB  
Article
Analysis of the Mathematical Modelling of COVID-19 by Using Mild Solution with Delay Caputo Operator
by Kinda Abuasbeh, Ramsha Shafqat, Ammar Alsinai and Muath Awadalla
Symmetry 2023, 15(2), 286; https://doi.org/10.3390/sym15020286 - 19 Jan 2023
Cited by 8 | Viewed by 1340
Abstract
This work investigates a mathematical fractional-order model that depicts the Caputo growth of a new coronavirus (COVID-19). We studied the existence and uniqueness of the linked solution using the fixed point theory method. Using the Laplace Adomian decomposition method (LADM), we explored the [...] Read more.
This work investigates a mathematical fractional-order model that depicts the Caputo growth of a new coronavirus (COVID-19). We studied the existence and uniqueness of the linked solution using the fixed point theory method. Using the Laplace Adomian decomposition method (LADM), we explored the precise solution of our model and obtained results that are stated in terms of infinite series. Numerical data were then used to demonstrate the use of the new derivative and the symmetric structure that we created. When compared to the traditional order derivatives, our results under the new hypothesis show that the innovative coronavirus model performs better. Full article
(This article belongs to the Special Issue Fractional-Order Systems and Its Applications in Engineering)
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20 pages, 348 KiB  
Article
Existence and Ulam–Hyers Stability Results for a System of Coupled Generalized Liouville–Caputo Fractional Langevin Equations with Multipoint Boundary Conditions
by Muath Awadalla, Muthaiah Subramanian and Kinda Abuasbeh
Symmetry 2023, 15(1), 198; https://doi.org/10.3390/sym15010198 - 09 Jan 2023
Cited by 8 | Viewed by 901
Abstract
We study the existence and uniqueness of solutions for coupled Langevin differential equations of fractional order with multipoint boundary conditions involving generalized Liouville–Caputo fractional derivatives. Furthermore, we discuss Ulam–Hyers stability in the context of the problem at hand. The results are shown with [...] Read more.
We study the existence and uniqueness of solutions for coupled Langevin differential equations of fractional order with multipoint boundary conditions involving generalized Liouville–Caputo fractional derivatives. Furthermore, we discuss Ulam–Hyers stability in the context of the problem at hand. The results are shown with examples. Results are asymmetric when a generalized Liouville–Caputo fractional derivative (ρ) parameter is changed. Full article
(This article belongs to the Special Issue Fractional-Order Systems and Its Applications in Engineering)
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