Women’s Special Issue Series: Fractal and Fractional

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

Deadline for manuscript submissions: closed (26 November 2023) | Viewed by 14268

Special Issue Editors

Research Group on Dynamical Systems and Control (DYSC), Department of Electromechanical, Systems and Metal Engineering, Ghent University, B-9052 Ghent, Belgium
Interests: modelling and control; identification; anesthesia control; objective pain assessment; process control
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Special Issue Information

Dear colleagues,

This Special Issue is part of the excellent MDPI journal initiative to promote and support the contributions of women in research. The aim of the Guest Editors is to collect research articles as well as review articles that highlight the scientific achievements of women in the field of “Fractal and Fractional”. Recently introduced fractional and related modeling methodologies are also of great interest.

In recent years, fractional calculus has played a crucial role in modeling numerous real-world problems in studies in such areas as physics, thermodynamics, biophysics, aerodynamics, electrical circuits, electron-analytical chemistry, control theory, optimization, programming, associative memory, fitting of experimental data, etc. Significant results have been achieved as a result of the research based on fractals and fractional-order models.

We cordially invite researchers to submit their work on topics across all areas of “Fractal and Fractional”, including theoretical studies and practical applications. For this Special Issue, we welcome all research led by female scientists, where male scientists may offer support for the initiative as co-authors.

Dr. Ivanka Stamova
Dr. Carla M.A. Pinto
Dr. Dana Copot
Guest Editors

Women’s Special Issue Series

This Special Issue is part of Fractal and Fractional's Women’s Special Issue Series, hosted by women editors for women researchers. The Series advocates the advancement of women in science. We invite contributions to the Special Issue whose lead authors identify as women. The submission of articles with all-women authorship is especially encouraged. However, we do welcome articles from all authors, irrespective of gender.

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

  • Fractals and Fractional Calculus
  • Fractals and Fractional Calculus in Computing
  • Fractals and Fractional Calculus in Mathematical Physics
  • Fractals and Fractional Calculus in Biology and Neurocomputing
  • Fractals and Fractional Calculus in Engineering
  • Fractals and Fractional Calculus in Economics
  • Fractals and Fractional Calculus in Educational Technologies
  • Fractional Calculus and Control
  • Fractional Calculus and Optimization
  • Fractional Calculus and Stability
  • Fractional Models and Uncertainty
  • Related Fractional Modeling

Related Special Issue

Published Papers (8 papers)

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Research

15 pages, 1756 KiB  
Article
Hermite Wavelet Method for Nonlinear Fractional Differential Equations
Fractal Fract. 2023, 7(5), 346; https://doi.org/10.3390/fractalfract7050346 - 23 Apr 2023
Cited by 1 | Viewed by 1226
Abstract
Nonlinear fractional differential equations (FDEs) constitute the basis for many dynamical systems in various areas of engineering and applied science. Obtaining the numerical solutions to those nonlinear FDEs has quickly gained importance for the purposes of accurate modelling and fast prototyping among many [...] Read more.
Nonlinear fractional differential equations (FDEs) constitute the basis for many dynamical systems in various areas of engineering and applied science. Obtaining the numerical solutions to those nonlinear FDEs has quickly gained importance for the purposes of accurate modelling and fast prototyping among many others in recent years. In this study, we use Hermite wavelets to solve nonlinear FDEs. To this end, utilizing Hermite wavelets and block-pulse functions (BPF) for function approximation, we first derive the operational matrices for the fractional integration. The novel contribution provided by this method involves combining the orthogonal Hermite wavelets with their corresponding operational matrices of integrations to obtain sparser conversion matrices. Sparser conversion matrices require less computational load, and also converge rapidly. Using the generated approximate matrices, the original nonlinear FDE is converted into an algebraic equation in vector-matrix form. The obtained algebraic equation is then solved using the collocation points. The proposed method is used to find a number of nonlinear FDE solutions. Numerical results for several resolutions and comparisons are provided to demonstrate the value of the method. The convergence analysis is also provided for the proposed method. Full article
(This article belongs to the Special Issue Women’s Special Issue Series: Fractal and Fractional)
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10 pages, 3622 KiB  
Article
Spectra of Reduced Fractals and Their Applications in Biology
Fractal Fract. 2023, 7(1), 28; https://doi.org/10.3390/fractalfract7010028 - 28 Dec 2022
Cited by 1 | Viewed by 2015
Abstract
Fractals with different levels of self-similarity and magnification are defined as reduced fractals. It is shown that spectra of these reduced fractals can be constructed and used to describe levels of complexity of natural phenomena. Specific applications to biological systems, such as green [...] Read more.
Fractals with different levels of self-similarity and magnification are defined as reduced fractals. It is shown that spectra of these reduced fractals can be constructed and used to describe levels of complexity of natural phenomena. Specific applications to biological systems, such as green algae, are performed, and it is suggested that the obtained spectra can be used to classify the considered algae by identifying spectra associated with them. The ranges of these spectra for green algae are determined and their extension to other biological as well as other natural systems is proposed. Full article
(This article belongs to the Special Issue Women’s Special Issue Series: Fractal and Fractional)
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8 pages, 268 KiB  
Article
Some New Inequalities and Extremal Solutions of a Caputo–Fabrizio Fractional Bagley–Torvik Differential Equation
Fractal Fract. 2022, 6(9), 488; https://doi.org/10.3390/fractalfract6090488 - 31 Aug 2022
Cited by 4 | Viewed by 1016
Abstract
This paper studies the existence of extremal solutions for a nonlinear boundary value problem of Bagley–Torvik differential equations involving the Caputo–Fabrizio-type fractional differential operator with a non-singular kernel. With the help of a new inequality with a Caputo–Fabrizio fractional differential operator, the main [...] Read more.
This paper studies the existence of extremal solutions for a nonlinear boundary value problem of Bagley–Torvik differential equations involving the Caputo–Fabrizio-type fractional differential operator with a non-singular kernel. With the help of a new inequality with a Caputo–Fabrizio fractional differential operator, the main result is obtained by applying a monotone iterative technique coupled with upper and lower solutions. This paper concludes with an illustrative example. Full article
(This article belongs to the Special Issue Women’s Special Issue Series: Fractal and Fractional)
17 pages, 1337 KiB  
Article
Formal Verification of Fractional-Order PID Control Systems Using Higher-Order Logic
Fractal Fract. 2022, 6(9), 485; https://doi.org/10.3390/fractalfract6090485 - 30 Aug 2022
Cited by 5 | Viewed by 1270
Abstract
Fractional-order PID control is a landmark in the development of fractional-order control theory. It can improve the control precision and accuracy of systems and achieve more robust control results. As a theorem-proving formal verification method, it can be applied to an arbitrary system [...] Read more.
Fractional-order PID control is a landmark in the development of fractional-order control theory. It can improve the control precision and accuracy of systems and achieve more robust control results. As a theorem-proving formal verification method, it can be applied to an arbitrary system represented by a mathematical model. It is the ideal verification method because it is not subject to limits on state numbers. This paper presents the higher-order logic (HOL) formal verification and modeling of fractional-order PID controller systems. Firstly, a fractional-order PID controller was designed. The accuracy of fractional-order PID control can be supported by simulation, comparing integral-order PID controls. Secondly, the superior property of fractional-order PID control is validated via higher-order logic theorem proofs. An important basic property, the relationship between fractional-order differential calculus and integral-order differential calculus, was analyzed via a higher-order logic theorem proof. Then, the relations between the fractional-order PID controller and integral-order PID controller were verified based on the fractional-order Grünwald–Letnikov definition for higher-order logic theorem proofs. Formalization models of the fractional-order PID controller and the fractional-order closed-loop control system were established. Finally, the stability of the fractional-order control systems was verified based on established formal models and theorems. The results show that the fractional-order PID controllers can be conducive to the control performance of control systems, and the higher-order logic formal verification method can ensure the reliability and security of fractional-order control systems. Full article
(This article belongs to the Special Issue Women’s Special Issue Series: Fractal and Fractional)
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17 pages, 1897 KiB  
Article
A Mathematical Investigation of Sex Differences in Alzheimer’s Disease
Fractal Fract. 2022, 6(8), 457; https://doi.org/10.3390/fractalfract6080457 - 21 Aug 2022
Cited by 1 | Viewed by 1341
Abstract
Alzheimer’s disease (AD) is an age-related degenerative disorder of the cerebral neuro-glial-vascular units. Not only are post-menopausal females, especially those who carry the APOE4 gene, at a higher risk of AD than males, but also AD in females appears to progress faster than [...] Read more.
Alzheimer’s disease (AD) is an age-related degenerative disorder of the cerebral neuro-glial-vascular units. Not only are post-menopausal females, especially those who carry the APOE4 gene, at a higher risk of AD than males, but also AD in females appears to progress faster than in aged-matched male patients. Currently, there is no cure for AD. Mathematical models can help us to understand mechanisms of AD onset, progression, and therapies. However, existing models of AD do not account for sex differences. In this paper a mathematical model of AD is proposed that uses variable-order fractional temporal derivatives to describe the temporal evolutions of some relevant cells’ populations and aggregation-prone amyloid-β fibrils. The approach generalizes the model of Puri and Li. The variable fractional order describes variable fading memory due to neuroprotection loss caused by AD progression with age which, in the case of post-menopausal females, is more aggressive because of fast estrogen decrease. Different expressions of the variable fractional order are used for the two sexes and a sharper decreasing function corresponds to the female’s neuroprotection decay. Numerical simulations show that the population of surviving neurons has decreased more in post-menopausal female patients than in males at the same stage of the disease. The results suggest that if a treatment that may include estrogen replacement therapy is applied to female patients, then the loss of neurons slows down at later times. Additionally, the sooner a treatment starts, the better the outcome is. Full article
(This article belongs to the Special Issue Women’s Special Issue Series: Fractal and Fractional)
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24 pages, 7902 KiB  
Article
Fractional-Order PI Controller Design Based on Reference–to–Disturbance Ratio
Fractal Fract. 2022, 6(4), 224; https://doi.org/10.3390/fractalfract6040224 - 15 Apr 2022
Cited by 2 | Viewed by 1567
Abstract
The presence of disturbances in practical control engineering applications is unavoidable. At the same time, they drive the closed-loop system’s response away from the desired behavior. For this reason, the attenuation of disturbance effects is a primary goal of the control loop. Fractional-order [...] Read more.
The presence of disturbances in practical control engineering applications is unavoidable. At the same time, they drive the closed-loop system’s response away from the desired behavior. For this reason, the attenuation of disturbance effects is a primary goal of the control loop. Fractional-order controllers have now been researched intensively in terms of improving the closed-loop results and robustness of the control system, compared to the standard integer-order controllers. In this study, a novel tuning method for fractional-order controllers is developed. The tuning is based on improving the disturbance attenuation of periodic disturbances with an estimated frequency. For this, the reference–to–disturbance ratio is used as a quantitative measure of the control system’s ability to reject disturbances. Numerical examples are included to justify the approach, quantify the advantages and demonstrate the robustness. The simulation results provide for a validation of the proposed tuning method. Full article
(This article belongs to the Special Issue Women’s Special Issue Series: Fractal and Fractional)
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11 pages, 40159 KiB  
Article
Synchronization in a Multiplex Network of Nonidentical Fractional-Order Neurons
Fractal Fract. 2022, 6(3), 169; https://doi.org/10.3390/fractalfract6030169 - 18 Mar 2022
Cited by 10 | Viewed by 1848
Abstract
Fractional-order neuronal models that include memory effects can describe the rich dynamics of the firing of the neurons. This paper studies synchronization problems in a multiple network of Caputo–Fabrizio type fractional order neurons in which the orders of the derivatives in the layers [...] Read more.
Fractional-order neuronal models that include memory effects can describe the rich dynamics of the firing of the neurons. This paper studies synchronization problems in a multiple network of Caputo–Fabrizio type fractional order neurons in which the orders of the derivatives in the layers are different. It is observed that the intralayer synchronization state occurs in weaker intralayer couplings when using nonidentical fractional-order derivatives rather than integer-order or identical fractional orders. Furthermore, the needed interlayer coupling strength for interlayer near synchronization decreases for lower fractional orders. The dynamics of the neurons in nonidentical layers are also considered. It is shown that in lower fractional orders, the neurons’ dynamics change to periodic when the near synchronization state occurs. Moreover, decreasing the derivative order leads to incrementing the frequency of the bursts in the synchronization manifold, which is in contrast to the behavior of the single neuron. Full article
(This article belongs to the Special Issue Women’s Special Issue Series: Fractal and Fractional)
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12 pages, 1191 KiB  
Article
The Influence of Noise on the Solutions of Fractional Stochastic Bogoyavlenskii Equation
Fractal Fract. 2022, 6(3), 156; https://doi.org/10.3390/fractalfract6030156 - 13 Mar 2022
Cited by 11 | Viewed by 1597
Abstract
We look at the stochastic fractional-space Bogoyavlenskii equation in the Stratonovich sense, which is driven by multiplicative noise. Our aim is to acquire analytical fractional stochastic solutions to this stochastic fractional-space Bogoyavlenskii equation via two different methods such as the [...] Read more.
We look at the stochastic fractional-space Bogoyavlenskii equation in the Stratonovich sense, which is driven by multiplicative noise. Our aim is to acquire analytical fractional stochastic solutions to this stochastic fractional-space Bogoyavlenskii equation via two different methods such as the exp(Φ(η))-expansion method and sine–cosine method. Since this equation is used to explain the hydrodynamic model of shallow-water waves, the wave of leading fluid flow, and plasma physics, scientists will be able to characterize a wide variety of fascinating physical phenomena with these solutions. Furthermore, we evaluate the influence of noise on the behavior of the acquired solutions using 2D and 3D graphical representations. Full article
(This article belongs to the Special Issue Women’s Special Issue Series: Fractal and Fractional)
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