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Fractal Fract
Special Issues
Mathematical and Physical Analysis of Fractional Dynamical Systems
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Mathematical and Physical Analysis of Fractional Dynamical Systems

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

Deadline for manuscript submissions: 28 February 2025 | Viewed by 7417

Special Issue Editors

Prof. Dr. Dimplekumar Chalishajar

E-Mail Website
Guest Editor
Department of Applied Mathematics, Virginia Military Institute (VMI), Lexington, VA 24450, USA
Interests: control theory; dynamical system; fractional order systems; delay systems; stochastic system; partial differential equation
Dr. Kasinathan Ravikumar

E-Mail Website
Guest Editor
Department of Mathematics, PSG College of Arts & Science, Tamil Nadu 641014, India
Interests: control theory; stochastic process; probability and statistics; fractional order systems; applications

Special Issue Information

Dear Colleagues,

The fractional dynamic is a field of study in mathematics and physics investigating the behavior of objects and systems by using differentiations of fractional orders. This Special Issue, “Fractional Dynamical Systems: Recent Trends in Theory and Applications”, will be destined for the publication of high-quality mathematical papers in the area of theory and applications of fractional systems using fractional calculus. Emphasis is placed on developments in the theory of delay differential, integrodifferential, impulsive differential, and difference equations and their applications. Among possible topics of the papers, we can note, for example, the following: Various boundary value problems, the positivity/negativity of their solutions, Green’s functions and their properties, existence, and uniqueness solutions of nonlinear boundary value problems, optimization, and control theory, stability theory, oscillation and non-oscillation, variational problems, the use of functional differential equations in technology, economics, biology, and medicine. In this Special Topics issue select contributions on some recent developments in the theory and applications of fractional dynamical systems. Results on topics involving fundamental theory, qualitative theory, iterative methods, and numerous applications of fractional-order equations are welcome.

The scope includes (but is not limited to) original research works providing characterizations, explanations, predictions of systems, and phenomena supporting the emergence of potentially novel, useful applications which can even be at a very early stage of conception. Papers based on the advances in the theory of stochastic processes and stochastic models are also welcome. The papers devoted to local and nonlocal conditions and transference differential equations of heat and mass transference mathematical processes will be considered in continuous media with memory and in media with fractal structure. These papers shall investigate modified initial and mixed boundary value problems for generalized transfer differential equations of integral and fractional orders. The fields of qualitative and quantitative analysis of nonlinear evolution equations and their applications in image analysis. Both analytical studies, as well as simulation-based studies, will be considered. We will cover Mathematical Problems in Materials Science, Mathematical Approaches to Image Processing with Applications, Applications of Partial Differential Equations, Recent Advances in Delay Differential and Difference Equations, Nonlinear Optimization, Variational Inequalities, and Equilibrium Problems, Computational Methods in Analysis and Applications, and all applied mathematical fields.

Prof. Dr. Dimplekumar Chalishajar
Dr. Kasinathan Ravikumar
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

  • functional differential and integral systems with applications
  • fractional PDE with applications
  • fractional numerical analysis and numerical simulations
  • oscillation/nonoscillation
  • feedback control and various stability analysis
  • boundary value problems
  • Markov chain process
  • jump processes
  • coupled dynamics
  • fractional processes
  • space-time fractional equations
  • fractional Brownian motion and Rosenblatt process
  • long-range dependence
  • time–space fractional vibration equation
  • convergence
  • eigenvalue
  • green function

Published Papers (7 papers)

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Research

21 pages, 1940 KiB  
Article
Oscillation Criteria for Nonlinear Third-Order Delay Dynamic Equations on Time Scales Involving a Super-Linear Neutral Term
by Qinghua Feng and Bin Zheng
Fractal Fract. 2024, 8(2), 115; https://doi.org/10.3390/fractalfract8020115 - 14 Feb 2024
Viewed by 886
Abstract
In the sense of an arbitrary time scale, some new sufficient conditions on oscillation are presented in this paper for a class of nonlinear third-order delay dynamic equations involving a local fractional derivative with a super-linear neutral term. The established oscillation results include [...] Read more.
In the sense of an arbitrary time scale, some new sufficient conditions on oscillation are presented in this paper for a class of nonlinear third-order delay dynamic equations involving a local fractional derivative with a super-linear neutral term. The established oscillation results include known Kamenev and Philos-type oscillation criteria and are new oscillation results so far in the literature. Some inequalities, the Riccati transformation, the integral technique, and the theory of time scale are used in the establishment of these oscillation criteria. The proposed results unify continuous and discrete analysis, and the process of deduction is further extended to another class of nonlinear third-order delay dynamic equations involving a local fractional derivative with a super-linear neutral term and a damping term. As applications for the established oscillation criteria, some examples are given. Full article
(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems)
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21 pages, 481 KiB  
Article
Processing the Controllability of Control Systems with Distinct Fractional Derivatives via Kalman Filter and Gramian Matrix
by Muath Awadalla, Abir Chaouk, Maher Jneid, Kinda Abuasbeh and Jihan Alahmadi
Fractal Fract. 2024, 8(1), 52; https://doi.org/10.3390/fractalfract8010052 - 13 Jan 2024
Cited by 1 | Viewed by 886
Abstract
In this paper, we investigate the controllability conditions of linear control systems involving distinct local fractional derivatives. Sufficient conditions for controllability using Kalman rank conditions and the Gramian matrix are presented. We show that the controllability of the local fractional system can be [...] Read more.
In this paper, we investigate the controllability conditions of linear control systems involving distinct local fractional derivatives. Sufficient conditions for controllability using Kalman rank conditions and the Gramian matrix are presented. We show that the controllability of the local fractional system can be determined by the invertibility of the Gramian matrix and the full rank of the Kalman matrix. We also show that the local fractional system involving distinct orders is controllable if and only if the Gramian matrix is invertible. Illustrative examples and an application are provided to demonstrate the validity of the theoretical findings. Full article
(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems)
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16 pages, 329 KiB  
Article
Finite-Approximate Controllability of ν-Caputo Fractional Systems
by Muath Awadalla, Nazim I. Mahmudov and Jihan Alahmadi
Fractal Fract. 2024, 8(1), 21; https://doi.org/10.3390/fractalfract8010021 - 26 Dec 2023
Viewed by 943
Abstract
This paper introduces a methodology for examining finite-approximate controllability in Hilbert spaces for linear/semilinear ν-Caputo fractional evolution equations. A novel criterion for achieving finite-approximate controllability in linear ν-Caputo fractional evolution equations is established, utilizing resolvent-like operators. Additionally, we identify a control [...] Read more.
This paper introduces a methodology for examining finite-approximate controllability in Hilbert spaces for linear/semilinear ν-Caputo fractional evolution equations. A novel criterion for achieving finite-approximate controllability in linear ν-Caputo fractional evolution equations is established, utilizing resolvent-like operators. Additionally, we identify a control strategy that not only satisfies the approximative controllability property but also ensures exact finite-dimensional controllability. Leveraging the approximative controllability of the corresponding linear ν-Caputo fractional evolution system, we establish sufficient conditions for achieving finite-approximative controllability in the semilinear ν-Caputo fractional evolution equation. These findings extend and build upon recent advancements in this field. The paper also explores applications to ν-Caputo fractional heat equations. Full article
(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems)
18 pages, 1148 KiB  
Article
The Generalized Discrete Proportional Derivative and Its Applications
by Rajiniganth Pandurangan, Saravanan Shanmugam, Mohamed Rhaima and Hamza Ghoudi
Fractal Fract. 2023, 7(12), 838; https://doi.org/10.3390/fractalfract7120838 - 27 Nov 2023
Cited by 2 | Viewed by 855
Abstract
The aim of this paper is to define the generalized discrete proportional derivative (GDPD) and illustrate the application of the Leibniz theorem, the binomial expansion, and Montmort’s formulas in the context of the generalized discrete proportional case. Furthermore, we introduce the generalized discrete [...] Read more.
The aim of this paper is to define the generalized discrete proportional derivative (GDPD) and illustrate the application of the Leibniz theorem, the binomial expansion, and Montmort’s formulas in the context of the generalized discrete proportional case. Furthermore, we introduce the generalized discrete proportional Laplace transform and determine the GDPLT of various functions using the inverse operator. The results obtained are showcased through relevant examples and validated using MATLAB. Full article
(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems)
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25 pages, 1029 KiB  
Article
Exploring Propagating Soliton Solutions for the Fractional Kudryashov–Sinelshchikov Equation in a Mixture of Liquid–Gas Bubbles under the Consideration of Heat Transfer and Viscosity
by Rashid Ali, Ahmed S. Hendy, Mohamed R. Ali, Ahmed M. Hassan, Fuad A. Awwad and Emad A. A. Ismail
Fractal Fract. 2023, 7(11), 773; https://doi.org/10.3390/fractalfract7110773 - 24 Oct 2023
Cited by 3 | Viewed by 1443
Abstract
In this research work, we investigate the complex structure of soliton in the Fractional Kudryashov–Sinelshchikov Equation (FKSE) using conformable fractional derivatives. Our study involves the development of soliton solutions using the modified Extended Direct Algebraic Method (mEDAM). This approach involves a key variable [...] Read more.
In this research work, we investigate the complex structure of soliton in the Fractional Kudryashov–Sinelshchikov Equation (FKSE) using conformable fractional derivatives. Our study involves the development of soliton solutions using the modified Extended Direct Algebraic Method (mEDAM). This approach involves a key variable transformation, which successfully transforms the model into a Nonlinear Ordinary Differential Equation (NODE). Following that, by using a series form solution, the NODE is turned into a system of algebraic equations, allowing us to construct soliton solutions methodically. The FKSE is the governing equation, allowing for heat transmission and viscosity effects while capturing the behaviour of pressure waves in liquid–gas bubble mixtures. The solutions we discover include generalised trigonometric, hyperbolic, and rational functions with kinks, singular kinks, multi-kinks, lumps, shocks, and periodic waves. We depict two-dimensional, three-dimensional, and contour graphs to aid comprehension. These newly created soliton solutions have far-reaching ramifications not just in mathematical physics, but also in a wide range of subjects such as optical fibre research, plasma physics, and a variety of applied sciences. Full article
(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems)
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18 pages, 367 KiB  
Article
Approximate Controllability of Neutral Differential Systems with Fractional Deformable Derivatives
by Sreedharan Raju, Raja Balachandar Sevugan, Ramalingam Udhayakumar, Ghada AlNemer and Umamaheswaran Arunachalam
Fractal Fract. 2023, 7(10), 741; https://doi.org/10.3390/fractalfract7100741 - 08 Oct 2023
Viewed by 912
Abstract
This article deals with the existence and uniqueness of solutions, as well as the approximate controllability of fractional neutral differential equations (ACFNDEs) with deformable derivatives. The findings are achieved using Banach’s, Krasnoselskii’s, and Schauder’s fixed-point theorems and semigroup theory. Three numerical examples are [...] Read more.
This article deals with the existence and uniqueness of solutions, as well as the approximate controllability of fractional neutral differential equations (ACFNDEs) with deformable derivatives. The findings are achieved using Banach’s, Krasnoselskii’s, and Schauder’s fixed-point theorems and semigroup theory. Three numerical examples are used to illustrate the application of the theories discussed in the conclusion. Full article
(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems)
20 pages, 393 KiB  
Article
Trajectory Controllability of Clarke Subdifferential-Type Conformable Fractional Stochastic Differential Inclusions with Non-Instantaneous Impulsive Effects and Deviated Arguments
by Dimplekumar Chalishajar, Ramkumar Kasinathan, Ravikumar Kasinathan and Varshini Sandrasekaran
Fractal Fract. 2023, 7(7), 541; https://doi.org/10.3390/fractalfract7070541 - 13 Jul 2023
Viewed by 754
Abstract
In this study, the multivalued fixed point theorem, Clarke subdifferential properties, fractional calculus, and stochastic analysis are used to arrive at the system’s mild solution (1). Furthermore, the mean square moment for the aforementioned system (1) confirms the conditions for trajectory (T-)controllability. The [...] Read more.
In this study, the multivalued fixed point theorem, Clarke subdifferential properties, fractional calculus, and stochastic analysis are used to arrive at the system’s mild solution (1). Furthermore, the mean square moment for the aforementioned system (1) confirms the conditions for trajectory (T-)controllability. The last part of the paper uses two numerical applications to explain the novel theoretical results that were reached. Full article
(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems)
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