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Fractal Fract
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Advances in Fractional Order Derivatives and Their Applications, 3rd Edition
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Advances in Fractional Order Derivatives and Their Applications, 3rd Edition

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

Deadline for manuscript submissions: closed (20 March 2025) | Viewed by 3704

Special Issue Editor

Prof. Dr. Sameerah Jamal

E-Mail Website
Guest Editor
School of Mathematics, University of the Witwatersrand, Johannesburg 2001, South Africa
Interests: differential equations; symmetries; conservation laws; exact solutions; cosmology
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional order derivatives have had a revolutionary impact on the scientific community. Its study has grown in leaps and bounds, from analytical methods to numerical techniques. Consequently, applications of fractional order differential equations are now widespread across every possible area of research. 

The focus of this Special Issue is on the advancement of research on fractional order derivatives and their multi-faceted applications. Topics that are invited for submission include (but are not limited to) the following:

  • Mathematical modeling with fractional order derivatives;
  • Symmetry analysis of fractional order equations;
  • Conserved quantities related to fractional order models;
  • The various techniques for solving fractional order equations;
  • Special functions that are linked to the solution of fractional order equations;
  • Software to aid computations and analysis for fractional order derivatives and equations.

Also, please feel free to read and download all the published articles in our first volume:

https://www.mdpi.com/journal/fractalfract/special_issues/fractional_derivative

Our second volume is also available：

https://www.mdpi.com/journal/fractalfract/special_issues/fractional_derivative_ii

Prof. Dr. Sameerah Jamal
Guest Editor

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 derivatives
  • fractional calculus
  • Caputo derivatives
  • Riemann–Liouville derivatives
  • numerical analysis
  • modeling
  • application

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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

14 pages, 1071 KiB  
Article
A Certain Numerical Algorithm for Solving a Fractional Partial Model with a Neumann Constraint in a Hilbert Space
by Rawya Al-Deiakeh, Shrideh Al-Omari, Amra Al kenany and Mohammed Al-Smadi
Fractal Fract. 2025, 9(4), 243; https://doi.org/10.3390/fractalfract9040243 - 11 Apr 2025
Viewed by 151
Abstract
This research examines a fractional partial advection–dispersion model, incorporating both mobile and immobile components, employing the Hilbert reproducing algorithm under an appropriate Neumann constraint condition. To effectively formulate the model while adhering to the specified constraints, two suitable Hilbert spaces are constructed, with [...] Read more.
This research examines a fractional partial advection–dispersion model, incorporating both mobile and immobile components, employing the Hilbert reproducing algorithm under an appropriate Neumann constraint condition. To effectively formulate the model while adhering to the specified constraints, two suitable Hilbert spaces are constructed, with the time-fractional Caputo derivative being utilized in the model’s formulation. Alongside the convergence analysis, a derived approximate solution formula is presented, and a systematic computational algorithm is developed to effectively implement the solution methodology. Numerical applications related to the proposed model are presented, complemented by tables and graphical illustrations. In conclusion, significant results are analyzed, and directions for future research are outlined. Full article
(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications, 3rd Edition)
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19 pages, 1606 KiB  
Article
Chaos in Fractional-Order Glucose–Insulin Models with Variable Derivatives: Insights from the Laplace–Adomian Decomposition Method and Generalized Euler Techniques
by Sayed Saber, Emad Solouma, Rasmiyah A. Alharb and Ahmad Alalyani
Fractal Fract. 2025, 9(3), 149; https://doi.org/10.3390/fractalfract9030149 - 27 Feb 2025
Cited by 2 | Viewed by 435
Abstract
This study investigates the complex dynamics and control mechanisms of fractional-order glucose–insulin regulatory systems, incorporating memory-dependent properties through fractional derivatives. Employing the Laplace–Adomian Decomposition Method (LADM) and the Generalized Euler Method (GEM), the research models glucose–insulin interactions with time-varying fractional orders to simulate [...] Read more.
This study investigates the complex dynamics and control mechanisms of fractional-order glucose–insulin regulatory systems, incorporating memory-dependent properties through fractional derivatives. Employing the Laplace–Adomian Decomposition Method (LADM) and the Generalized Euler Method (GEM), the research models glucose–insulin interactions with time-varying fractional orders to simulate long-term physiological processes. Key aspects include the derivation of Lyapunov exponents, bifurcation diagrams, and phase diagrams to explore system stability and chaotic behavior. A novel control strategy using simple linear controllers is introduced to stabilize chaotic oscillations. The effectiveness of this approach is validated through numerical simulations, where Lyapunov exponents are reduced from positive values (λ1=0.123) in the uncontrolled system to negative values (λ1=−0.045) post-control application, indicating successful stabilization. Additionally, bifurcation analysis demonstrates a shift from chaotic to periodic behavior when control is applied, and time-series plots confirm a significant reduction in glucose–insulin fluctuations. These findings underscore the importance of fractional calculus in accurately modeling nonlinear and memory-dependent glucose–insulin dynamics, paving the way for improved predictive models and therapeutic strategies. The proposed framework provides a foundation for personalized diabetes management, real-time glucose monitoring, and intelligent insulin delivery systems. Full article
(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications, 3rd Edition)
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30 pages, 1100 KiB  
Article
Probing Malware Propagation Model with Variable Infection Rates Under Integer, Fractional, and Fractal–Fractional Orders
by Nausheen Razi, Ambreen Bano, Umar Ishtiaq, Tayyab Kamran, Mubariz Garayev and Ioan-Lucian Popa
Fractal Fract. 2025, 9(2), 90; https://doi.org/10.3390/fractalfract9020090 - 1 Feb 2025
Viewed by 646
Abstract
Malware software has become a pervasive threat in computer and mobile technology attacks. Attackers use this software to obtain information about users of the digital world to obtain benefits by hijacking their data. Antivirus software has been developed to prevent the propagation of [...] Read more.
Malware software has become a pervasive threat in computer and mobile technology attacks. Attackers use this software to obtain information about users of the digital world to obtain benefits by hijacking their data. Antivirus software has been developed to prevent the propagation of malware, but this problem is not yet under control. To develop this software, we have to check the propagation of malware. In this paper, we explore an advanced malware propagation model with a time-delay factor and a variable infection rate. To better understand this model, we use fractal–fractional theory. We use an exponential decay kernel for this. For theoretical purposes (existence, uniqueness, and stability), we use the results from fixed-point theory, and, for numerical purposes, a Lagrange two-point interpolation polynomial is used to develop an algorithm. Matlab R2016a is used for simulation, and the physical significance is assessed. We examine the impact of different fractal and fractional orders for various parameters. Moreover, we compare four different mathematical models (classical, fractional, fractal, and fractal–fractional). Also, constant and variable fractional and fractal orders are compared using graphs. We investigate the idea that significant perturbation in infected nodes might be due to minor changes. This work may help with developing antivirus strategies in real life. Full article
(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications, 3rd Edition)
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16 pages, 2388 KiB  
Article
Variable Time Step Algorithm for Transient Response Analysis for Control and Optimization
by Igor Reznichenko, Primož Podržaj and Aljoša Peperko
Fractal Fract. 2024, 8(12), 710; https://doi.org/10.3390/fractalfract8120710 - 29 Nov 2024
Viewed by 914
Abstract
This work considers an optimization problem based on step response characteristics. We lay a foundation for it by designing a rapid transient response analysis algorithm with variable time steps. This method applies to linear ordinary differential equations with real order. Numerical tests of [...] Read more.
This work considers an optimization problem based on step response characteristics. We lay a foundation for it by designing a rapid transient response analysis algorithm with variable time steps. This method applies to linear ordinary differential equations with real order. Numerical tests of the algorithm in the integer case show significant improvement even for higher order systems. This suggests a new method for acquiring step response characteristics for the fractional order case for which we have constructed an explicit expression of the inverse Laplace transform. Full article
(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications, 3rd Edition)
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11 pages, 271 KiB  
Article
Advanced Differential Equations with Canonical Operators: New Criteria for the Oscillation
by Omar Bazighifan, Nawa Alshammari, Khalil S. Al-Ghafri and Loredana Florentina Iambor
Fractal Fract. 2024, 8(11), 670; https://doi.org/10.3390/fractalfract8110670 - 18 Nov 2024
Viewed by 776
Abstract
In this study, we use the integral averaging methodology, comparison with second-order differential equations, and the Riccati technique to determine the Philos-type and Hille–Nehari-type oscillation conditions of fourth-order advanced differential equations with canonical operators. In essence, these techniques supplement and generalize a wide [...] Read more.
In this study, we use the integral averaging methodology, comparison with second-order differential equations, and the Riccati technique to determine the Philos-type and Hille–Nehari-type oscillation conditions of fourth-order advanced differential equations with canonical operators. In essence, these techniques supplement and generalize a wide range of established oscillation conditions. Two instance cases demonstrate the importance of our outcomes and their significant improvement. Full article
(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications, 3rd Edition)
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