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Fractal Fract
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Recent Developments on Linear and Nonlinear Fractional-Order Systems: Theory and...
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Recent Developments on Linear and Nonlinear Fractional-Order Systems: Theory and Application

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: closed (25 November 2023) | Viewed by 8511

Special Issue Editors

Dr. Mathiyalagan Kalidass

E-Mail Website
Guest Editor
Department of Mathematics, Bharathiar University, Coimbatore 641046, India
Interests: differential equations; control theory and its applications
Dr. Huiyan Zhang

E-Mail Website
Guest Editor
National Research Base of Intelligent Manufacturing Service, Chongqing Technology and Business University, Chongqing 400067, China
Interests: Markov jump systems; stochastic systems; event-triggered schemes; filtering design; controller design; cyber-attacks; time-delay; robust control
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The increasing interest in the class of fractional-order systems is motivated by their applications in various fields and has given rise to a type of equation that has not been covered in the standard mathematical literature. Fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. This is the main advantage of fractional derivatives in comparison to the classical integer-order models, in which such effects are neglected. Furthermore, the fractional differential equations are used to model complex phenomena and they play a crucial role in engineering, physics and applied mathematics. The advantages of fractional derivatives become apparent in modeling the mechanical and electrical properties of real materials. So, the effective general methods for solving systems involving fractional derivatives have a wide range of applications. Due to the fact that many real-world physical systems are well-characterized by the fractional-order state equations and real non-integer differentiators, some new theories and practical instruments are developed.

The objective of this Special Issue is to compile the recent developments in methodologies, techniques, and applications of fractional dynamical systems to explore the plethora of interesting applications of fractional-order systems in physics, chemistry, engineering, finance, and other sciences. The main aim of this Special Issue is to focus on novel results without reflecting many of the new results proposed recently in the theory of fractional-order systems. Potential topics include, but are not limited to:

  • Qualitative theory of nonlinear fractional-order systems;
  • Fractional-order control systems and applications;
  • Optimization techniques for fractional-order systems;
  • Time delays and uncertainty in fractional-order systems;
  • Fractional-order models in mathematical biology;
  • Fractional-order models in networked control systems;
  • Stochastic fractional dynamical systems.

Dr. Mathiyalagan Kalidass
Dr. Huiyan Zhang
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. Fractal and Fractional 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 2700 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-order models
  • qualitative theory
  • dynamical systems
  • optimization techniques
  • control design
  • stochastic systems

Published Papers (8 papers)

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Research

13 pages, 318 KiB  
Article
Controllability Analysis of Impulsive Multi-Term Fractional-Order Stochastic Systems Involving State-Dependent Delay
by G. Arthi, M. Vaanmathi and Yong-Ki Ma
Fractal Fract. 2023, 7(10), 727; https://doi.org/10.3390/fractalfract7100727 - 30 Sep 2023
Viewed by 826
Abstract
This study deals with the controllability of multi-term fractional-order stochastic systems with impulsive effects and state-dependent delay that exhibit damping behavior. Based on fractional calculus theory, the Caputo fractional derivative is utilized to analyze the controllability of fractional-order systems. Mittag–Leffler functions and Laplace [...] Read more.
This study deals with the controllability of multi-term fractional-order stochastic systems with impulsive effects and state-dependent delay that exhibit damping behavior. Based on fractional calculus theory, the Caputo fractional derivative is utilized to analyze the controllability of fractional-order systems. Mittag–Leffler functions and Laplace transform are used to derive the solution set of the problem. Sufficient conditions for the controllability of nonlinear systems are achieved using fixed-point techniques and stochastic theory. Finally, the results stated in the paper are validated using examples. Full article
(This article belongs to the Special Issue Recent Developments on Linear and Nonlinear Fractional-Order Systems: Theory and Application)
20 pages, 1706 KiB  
Article
Adaptive Control Design for Euler–Lagrange Systems Using Fixed-Time Fractional Integral Sliding Mode Scheme
by Saim Ahmed, Ahmad Taher Azar, Mohamed Tounsi and Ibraheem Kasim Ibraheem
Fractal Fract. 2023, 7(10), 712; https://doi.org/10.3390/fractalfract7100712 - 27 Sep 2023
Cited by 5 | Viewed by 881
Abstract
This paper presents an adaptive fixed-time fractional integral control for externally disturbed Euler–Lagrange systems. In the first step of the control design, the approach of fractional-order fixed-time integral nonsingular terminal sliding mode control (FoIFxTSM) is introduced. This scheme combines the benefits of fractional [...] Read more.
This paper presents an adaptive fixed-time fractional integral control for externally disturbed Euler–Lagrange systems. In the first step of the control design, the approach of fractional-order fixed-time integral nonsingular terminal sliding mode control (FoIFxTSM) is introduced. This scheme combines the benefits of fractional calculus with integral sliding mode control, resulting in fast convergence, smooth nonsingular control inputs, and fixed-time stability. By integrating an adaptive scheme, the proposed method is used to control the dynamical system in the presence of uncertainty and external perturbations. The findings of the fixed-time stability using Lyapunov analyses are provided for the closed-loop system. The simulation results are compared with the adaptive fractional-order sliding mode control scheme, and they show the better tracking and convergence performance of the proposed method. Full article
(This article belongs to the Special Issue Recent Developments on Linear and Nonlinear Fractional-Order Systems: Theory and Application)
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18 pages, 1285 KiB  
Article
Numerical Analysis of Direct and Inverse Problems for a Fractional Parabolic Integro-Differential Equation
by Miglena N. Koleva and Lubin G. Vulkov
Fractal Fract. 2023, 7(8), 601; https://doi.org/10.3390/fractalfract7080601 - 04 Aug 2023
Viewed by 755
Abstract
A mathematical model consisting of weakly coupled time fractional one parabolic PDE and one ODE equations describing dynamical processes in porous media is our physical motivation. As is often performed, by solving analytically the ODE equation, such a system is reduced to an [...] Read more.
A mathematical model consisting of weakly coupled time fractional one parabolic PDE and one ODE equations describing dynamical processes in porous media is our physical motivation. As is often performed, by solving analytically the ODE equation, such a system is reduced to an integro-parabolic equation. We focus on the numerical reconstruction of a diffusion coefficient at finite number space-points measurements. The well-posedness of the direct problem is investigated and energy estimates of their solutions are derived. The second order in time and space finite difference approximation of the direct problem is analyzed. The approach of Lagrangian multiplier adjoint equations is utilized to compute the Fréchet derivative of the least-square cost functional. A numerical solution based on the conjugate gradient method (CGM) of the inverse problem is studied. A number of computational examples are discussed. Full article
(This article belongs to the Special Issue Recent Developments on Linear and Nonlinear Fractional-Order Systems: Theory and Application)
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13 pages, 6474 KiB  
Article
Finite-Time Stabilization of Unstable Orbits in the Fractional Difference Logistic Map
by Ernestas Uzdila, Inga Telksniene, Tadas Telksnys and Minvydas Ragulskis
Fractal Fract. 2023, 7(8), 570; https://doi.org/10.3390/fractalfract7080570 - 25 Jul 2023
Viewed by 627
Abstract
A control scheme for finite-time stabilization of unstable orbits of the fractional difference logistic map is proposed in this paper. The presented technique is based on isolated perturbation impulses used to correct the evolution of the map’s trajectory after it deviates too far [...] Read more.
A control scheme for finite-time stabilization of unstable orbits of the fractional difference logistic map is proposed in this paper. The presented technique is based on isolated perturbation impulses used to correct the evolution of the map’s trajectory after it deviates too far from the neighborhood of the unstable orbit, and does not require any feedback control loops. The magnitude of the control impulses is determined by means of H-rank algorithm, which helps to reveal the pseudo-manifold of non-asymptotic convergence of the fractional difference logistic map. Numerical experiments are used to illustrate the effectiveness and the feasibility of the proposed approach, which is applicable beyond the studied fractional difference logistic map. Full article
(This article belongs to the Special Issue Recent Developments on Linear and Nonlinear Fractional-Order Systems: Theory and Application)
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16 pages, 13398 KiB  
Article
Solitary and Periodic Wave Solutions of the Space-Time Fractional Extended Kawahara Equation
by Dilek Varol
Fractal Fract. 2023, 7(7), 539; https://doi.org/10.3390/fractalfract7070539 - 12 Jul 2023
Cited by 6 | Viewed by 708
Abstract
The extended Kawahara (Gardner Kawahara) equation is the improved form of the Korteweg–de Vries (KdV) equation, which is one of the most significant nonlinear evolution equations in mathematical physics. In that research, the analytical solutions of the conformable fractional extended Kawahara equation were [...] Read more.
The extended Kawahara (Gardner Kawahara) equation is the improved form of the Korteweg–de Vries (KdV) equation, which is one of the most significant nonlinear evolution equations in mathematical physics. In that research, the analytical solutions of the conformable fractional extended Kawahara equation were acquired by utilizing the Jacobi elliptic function expansion method. The given expansion method was applied to different fractional forms of the extended Kawahara equation, such as the fraction that occurs in time, space, or both time and space by suitably changing the variables. In addition, various types of fractional problems are exhibited to expose the realistic application of the given method, and some of the obtained solutions were illustrated in two- or three-dimensional graphics as proof of the visualization. Full article
(This article belongs to the Special Issue Recent Developments on Linear and Nonlinear Fractional-Order Systems: Theory and Application)
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13 pages, 368 KiB  
Article
On a System of Equations with General Fractional Derivatives Arising in Diffusion Theory
by Vesna Miskovic-Stankovic and Teodor M. Atanackovic
Fractal Fract. 2023, 7(7), 518; https://doi.org/10.3390/fractalfract7070518 - 30 Jun 2023
Cited by 2 | Viewed by 754
Abstract
A novel two-compartment model for drug release was formulated. The general fractional derivatives of a specific type and distributed order were used in the formulation. Earlier used models in pharmacokinetics with fractional derivatives follow as special cases of the model proposed here. As [...] Read more.
A novel two-compartment model for drug release was formulated. The general fractional derivatives of a specific type and distributed order were used in the formulation. Earlier used models in pharmacokinetics with fractional derivatives follow as special cases of the model proposed here. As a first application, we used this model to study the release of gentamicin from poly(vinyl alcohol)/chitosan/gentamicin (PVA/CHI/Gent) hydrogel aimed at wound dressing in the medical treatment of deep chronic wounds. As a second application, we studied the release of gentamicin from antibacterial biodynamic hydroxyapatite/poly(vinyl alcohol) /chitosan/gentamicin (HAP/PVA/CS/Gent) coating on a titanium substrate for bone tissue implants, which enables drug delivery directly to the infection site. In both cases. a good agreement is obtained between the measured data and the data calculated from the model proposed here. The form of the general fractional derivatives used here results in an additional parameter in the compartmental model used here. This, as a consequence, leads to a better approximation of the experimental data with only a slightly more complicated numerical procedure in obtaining the solution. Full article
(This article belongs to the Special Issue Recent Developments on Linear and Nonlinear Fractional-Order Systems: Theory and Application)
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20 pages, 421 KiB  
Article
Application of Fixed Points in Bipolar Controlled Metric Space to Solve Fractional Differential Equation
by Gunaseelan Mani, Rajagopalan Ramaswamy, Arul Joseph Gnanaprakasam, Amr Elsonbaty, Ola A. Ashour Abdelnaby and Stojan Radenović
Fractal Fract. 2023, 7(3), 242; https://doi.org/10.3390/fractalfract7030242 - 08 Mar 2023
Cited by 2 | Viewed by 1119
Abstract
Fixed point results and metric fixed point theory play a vital role to find the unique solution to differential and integral equations. Likewise, fractal calculus has vast physical applications. In this article, we introduce the concept of bipolar-controlled metric space and prove fixed [...] Read more.
Fixed point results and metric fixed point theory play a vital role to find the unique solution to differential and integral equations. Likewise, fractal calculus has vast physical applications. In this article, we introduce the concept of bipolar-controlled metric space and prove fixed point theorems. The derived results expand and extend certain well-known results from the research literature and are supported with a non-trivial example. We have applied the fixed point result to find the analytical solution to the integral equation and fractional differential equation. The analytical solution has been supplemented with numerical simulation. Full article
(This article belongs to the Special Issue Recent Developments on Linear and Nonlinear Fractional-Order Systems: Theory and Application)
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24 pages, 885 KiB  
Article
A New Modeling of Fractional-Order and Sensitivity Analysis for Hepatitis-B Disease with Real Data
by Mehmet Yavuz, Fatma Özköse, Muhittin Susam and Mathiyalagan Kalidass
Fractal Fract. 2023, 7(2), 165; https://doi.org/10.3390/fractalfract7020165 - 07 Feb 2023
Cited by 18 | Viewed by 1760
Abstract
In this study, we propose new illustrative and effective modeling to point out the behaviors of the Hepatitis-B virus (Hepatitis-B). Not only do we consider the mathematical modeling, equilibria, stabilities, and existence–uniqueness analysis of the model, but also, we make numerical simulations by [...] Read more.
In this study, we propose new illustrative and effective modeling to point out the behaviors of the Hepatitis-B virus (Hepatitis-B). Not only do we consider the mathematical modeling, equilibria, stabilities, and existence–uniqueness analysis of the model, but also, we make numerical simulations by using the Adams–Bashforth numerical scheme. However, we apply the parameter estimation method to determine our model parameters and find the curve that best fits the model. Additionally, in this study, the stability analysis of the aforementioned model is considered, and also the sensitivity analysis of R0 is examined. The results point out that the order of the fractional derivative has an essential effect on the dynamical process of the constructed model for Hepatitis-B. Full article
(This article belongs to the Special Issue Recent Developments on Linear and Nonlinear Fractional-Order Systems: Theory and Application)
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