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Fractal Fract
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Mathematical and Physical Analysis of Fractional Dynamical Systems, Second Edition
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Mathematical and Physical Analysis of Fractional Dynamical Systems, Second Edition

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

Deadline for manuscript submissions: 30 April 2026 | Viewed by 1685

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
Special Issues, Collections and Topics in MDPI journals
Dr. Kasinathan Ravikumar

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

Special Issue Information

Dear Colleagues,

Fractional dynamics is a field of study in mathematics and physics that investigates the behavior of objects and systems by using differentiations of fractional orders. This Special Issue, “Mathematical and Physical Analysis of Fractional Dynamical Systems, Second Edition”, will present a selection high-quality mathematical papers covering some recent developments in the theory and applications of fractional systems using fractional calculus. Emphasis will be placed on developments in the theory of delay differential, integrodifferential, impulsive differential, and difference equations and their applications. The following is a list of possible topics welcome in this Special Issue: various boundary value problems and the positivity/negativity of their solutions; Green’s functions and their properties; the existing and unique 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; and economics, biology, and medicine.  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 and useful applications, even those at the very early stages of conception. Papers based on the advances in the theory of stochastic processes and stochastic models are also welcome. Papers on the differential equations of heat and mass transfer, both local and non-local, will be considered in media that have memory and in media with a fractional structure. These papers shall investigate modified initial and mixed-boundary value problems for generalized transfer differential equations of integral and fractional orders and the fields of the 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

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Further information on MDPI's Special Issue policies can be found here.

Related Special Issue

Published Papers (5 papers)

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Research

23 pages, 1107 KiB  
Article
Mathematical and Physical Analysis of the Fractional Dynamical Model
by Mohammed Ahmed Alomair and Haitham Qawaqneh
Fractal Fract. 2025, 9(7), 453; https://doi.org/10.3390/fractalfract9070453 - 11 Jul 2025
Viewed by 172
Abstract
This paper consists of various kinds of wave solitons to the mathematical model known as the truncated M-fractional FitzHugh–Nagumo model. This model explains the transmission of the electromechanical pulses in nerves. Through the application of the modified extended tanh function technique and the [...] Read more.
This paper consists of various kinds of wave solitons to the mathematical model known as the truncated M-fractional FitzHugh–Nagumo model. This model explains the transmission of the electromechanical pulses in nerves. Through the application of the modified extended tanh function technique and the modified (G′/G2)-expansion technique, we are able to achieve the series of exact solitons. The results differ from the current solutions because of the fractional derivative. These solutions could be helpful in the telecommunication and bioscience domains. Contour plots, in two and three dimensions, are used to describe the results. Stability analysis is used to check the stability of the obtained solutions. Moreover, the stationary solutions of the focusing equation are studied through modulation instability. Future research on the focused model in question will benefit from the findings. The techniques used are simple and effective. Full article
(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems, Second Edition)
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36 pages, 4192 KiB  
Article
Fractional Calculus for Neutrosophic-Valued Functions and Its Application in an Inventory Lot-Sizing Problem
by Rakibul Haque, Mostafijur Rahaman, Adel Fahad Alrasheedi, Dimplekumar Chalishajar and Sankar Prasad Mondal
Fractal Fract. 2025, 9(7), 433; https://doi.org/10.3390/fractalfract9070433 - 30 Jun 2025
Viewed by 293
Abstract
Past experiences and memory significantly contribute to self-learning and improved decision-making. These can assist decision-makers in refining their strategies for better outcomes. Fractional calculus is a tool that captures a system’s memory or past experience through its repeating patterns. In the realm of [...] Read more.
Past experiences and memory significantly contribute to self-learning and improved decision-making. These can assist decision-makers in refining their strategies for better outcomes. Fractional calculus is a tool that captures a system’s memory or past experience through its repeating patterns. In the realm of uncertainty, neutrosophic set theory demonstrates greater suitability, as it independently assesses membership, non-membership, and indeterminacy. In this article, we aim to extend the theory further by introducing fractional calculus for neutrosophic-valued functions. The proposed method is applied to an economic lot-sizing problem. Numerical simulations of the lot-sizing model suggest that strong memory employment with a memory index of 0.1 can lead to an increase in average profit in memory-independent phenomena with a memory index of 1 by approximately 44% to 49%. Additionally, the neutrosophic environment yields superior profitability results compared to both precise and imprecise settings. The synergy of fractional-order dynamics and neutrosophic uncertainty modeling paves the way for enhanced decision-making in complex, ambiguous environments. Full article
(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems, Second Edition)
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29 pages, 4033 KiB  
Article
A Virtual Element Method for a (2+1)-Dimensional Wave Equation with Time-Fractional Dissipation on Polygonal Meshes
by Zaffar Mehdi Dar, Chandru Muthusamy and Higinio Ramos
Fractal Fract. 2025, 9(7), 399; https://doi.org/10.3390/fractalfract9070399 - 20 Jun 2025
Viewed by 326
Abstract
We propose a novel space-time discretization method for a time-fractional dissipative wave equation. The approach employs a structured framework in which a fully discrete formulation is produced by combining virtual elements for spatial discretization and the Newmark predictor–corrector method for the temporal domain. [...] Read more.
We propose a novel space-time discretization method for a time-fractional dissipative wave equation. The approach employs a structured framework in which a fully discrete formulation is produced by combining virtual elements for spatial discretization and the Newmark predictor–corrector method for the temporal domain. The virtual element technique is regarded as a generalization of the finite element method for polygonal and polyhedral meshes within the Galerkin approximation framework. To discretize the time-fractional dissipation term, we utilize the Grünwald-Letnikov approximation in conjunction with the predictor–corrector scheme. The existence and uniqueness of the discrete solution are theoretically proved, together with the optimal convergence order achieved and an error analysis associated with the H1-seminorm and the L2-norm. Numerical experiments are presented to support the theoretical findings and demonstrate the effectiveness of the proposed method with both convex and non-convex polygonal meshes. Full article
(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems, Second Edition)
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25 pages, 2808 KiB  
Article
A Non-Standard Finite Difference Scheme for Time-Fractional Singularly Perturbed Convection–Diffusion Problems
by Pramod Chakravarthy Podila, Rahul Mishra and Higinio Ramos
Fractal Fract. 2025, 9(6), 333; https://doi.org/10.3390/fractalfract9060333 - 23 May 2025
Viewed by 371
Abstract
This paper introduces a stable non-standard finite difference (NSFD) method to solve time-fractional singularly perturbed convection–diffusion problems. The fractional derivative in time is defined in the Caputo sense. The proposed method shows high efficiency when applied using a uniform mesh and can be [...] Read more.
This paper introduces a stable non-standard finite difference (NSFD) method to solve time-fractional singularly perturbed convection–diffusion problems. The fractional derivative in time is defined in the Caputo sense. The proposed method shows high efficiency when applied using a uniform mesh and can be easily extended to a Shishkin mesh in the spatial domain. We discuss error estimates to demonstrate the convergence of the numerical scheme. Additionally, various numerical examples are presented to illustrate the behavior of the solution for different values of the perturbation parameter ϵ and the order of the fractional derivative. Full article
(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems, Second Edition)
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19 pages, 2581 KiB  
Article
Analytical and Dynamical Study of Solitary Waves in a Fractional Magneto-Electro-Elastic System
by Sait San, Beenish and Fehaid Salem Alshammari
Fractal Fract. 2025, 9(5), 309; https://doi.org/10.3390/fractalfract9050309 - 10 May 2025
Cited by 3 | Viewed by 342
Abstract
Magneto-electro-elastic materials, a novel class of smart materials, exhibit remarkable energy conversion properties, making them highly suitable for applications in nanotechnology. This study focuses on various aspects of the fractional nonlinear longitudinal wave equation (FNLWE) that models wave propagation in a magneto-electro-elastic circular [...] Read more.
Magneto-electro-elastic materials, a novel class of smart materials, exhibit remarkable energy conversion properties, making them highly suitable for applications in nanotechnology. This study focuses on various aspects of the fractional nonlinear longitudinal wave equation (FNLWE) that models wave propagation in a magneto-electro-elastic circular rod. Using the direct algebraic method, several new soliton solutions were derived under specific parameter constraints. In addition, Galilean transformation was employed to explore the system’s sensitivity and quasi-periodic dynamics. The study incorporates 2D, 3D, and time-series visualizations as effective tools for analyzing quasi-periodic behavior. The results contribute to a deeper understanding of the nonlinear dynamical features of such systems and demonstrate the robustness of the applied methodologies. This research not only extends existing knowledge of nonlinear wave equations but also introduces a substantial number of new solutions with broad applicability. Full article
(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems, Second Edition)
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