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Fractal Fract
Special Issues
Fractional Behaviors Analysis and Modelling
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Fractional Behaviors Analysis and Modelling

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

Deadline for manuscript submissions: closed (31 October 2023) | Viewed by 9116

Special Issue Editor

Prof. Dr. Jocelyn Sabatier

E-Mail Website
Guest Editor
Bordeaux Institute of Technology, Bordeaux University, 351, Cours de la Libération, 33400 Talence, France
Interests: fractional behaviors; modeling; fractals; fractional operators

Special Issue Information

Dear Colleagues,

An implicit link exists in the literature between fractional behaviors and fractional differentiation-based models. However, fractional behaviors and fractional-differentiation-based models are two distinct concepts. One designates a property or a particular behavior of a physical system, while the other designates a model class that can capture fractional behaviors.

Fractional behaviors appear in numerous domains (of physical, biological, thermal, etc. origin). They often result from stochastic physical phenomena (diffusion, diffusion reaction, adsorption, absorption, aggregation, fragmentation, etc.) that can operate on a fractal space of dimension d and that generate time kinetics (or fractional behaviors) in t^(ν/d). Fractional behaviors are ubiquitous, and faced with the drawbacks now associated with the fractional-differentiation-based models, new modeling tools must be found.

The goal of this Special Issue is to bring out new modeling tools for fractional behaviors (other than usual and strict fractional differentiation or integration-based operators), as well as to study their properties and their applications in engineering sciences. Considering fractional behaviors without being limited to fractional models opens up countless avenues of research in the field of model analysis and identification. To avoid unnecessary fractionalizations, this Special Issue also focuses on methods that allow characterizing the existence of fractional behavior in measured data and theoretical justifications for fractional behaviors.

Prof. Dr. Jocelyn Sabatier
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 behaviors
  • Modeling
  • Volterra equations
  • Non-singular kernels
  • Non-linear models
  • Time-varying models
  • Partial differential equations
  • Fractal

Published Papers (4 papers)

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Research

11 pages, 303 KiB  
Article
Nontrivial Solutions of a Class of Fourth-Order Elliptic Problems with Potentials
by Jiabin Zuo, Zakaria El Allali and Said Taarabti
Fractal Fract. 2022, 6(10), 568; https://doi.org/10.3390/fractalfract6100568 - 6 Oct 2022
Viewed by 918
Abstract
This paper deals with a fourth-order elliptic equation with Dirichlet boundary conditions. Using a variant form of the mountain pass theorem, we prove the existence of nontrivial solutions to this problem. Furthermore, we discuss the fundamental properties of the representation of the solution [...] Read more.
This paper deals with a fourth-order elliptic equation with Dirichlet boundary conditions. Using a variant form of the mountain pass theorem, we prove the existence of nontrivial solutions to this problem. Furthermore, we discuss the fundamental properties of the representation of the solution by considering two cases. Our results not only make previous results more general but also show new insights into fourth-order elliptic problems. Full article
(This article belongs to the Special Issue Fractional Behaviors Analysis and Modelling)
18 pages, 2727 KiB  
Article
Some Properties of Fractional Cumulative Residual Entropy and Fractional Conditional Cumulative Residual Entropy
by Keqiang Dong, Shushu Li and Dan Li
Fractal Fract. 2022, 6(7), 400; https://doi.org/10.3390/fractalfract6070400 - 21 Jul 2022
Cited by 1 | Viewed by 1749
Abstract
Fractional cumulative residual entropy is a powerful tool for the analysis of complex systems. In this paper, we first provide some properties of fractional cumulative residual entropy (FCRE). Secondly, we generate cumulative residual entropy (CRE) to the case of conditional entropy, named fractional [...] Read more.
Fractional cumulative residual entropy is a powerful tool for the analysis of complex systems. In this paper, we first provide some properties of fractional cumulative residual entropy (FCRE). Secondly, we generate cumulative residual entropy (CRE) to the case of conditional entropy, named fractional conditional cumulative residual entropy (FCCRE), and introduce some properties. Then, we verify the validity of these properties with randomly generated sequences that follow different distributions. Moreover, we give the definition of empirical fractional conditional accumulative residual entropy and prove that it can be used to approximate FCCRE. Finally, the empirical analysis of the aero-engine gas path data is carried out. The results show that FCRE and FCCRE can effectively capture complex information in the gas path system. Full article
(This article belongs to the Special Issue Fractional Behaviors Analysis and Modelling)
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11 pages, 1202 KiB  
Article
Fractal Operators and Fractional Dynamics with 1/2 Order in Biological Systems
by Yajun Yin, Jianqiao Guo, Gang Peng, Xiaobin Yu and Yiya Kong
Fractal Fract. 2022, 6(7), 378; https://doi.org/10.3390/fractalfract6070378 - 2 Jul 2022
Cited by 5 | Viewed by 1946
Abstract
This paper reports the new advances in biological fractal dynamics. The following contents are included: (1) physical (or functional) fractal spaces abstracted from biological materials, biological structures and biological motions; (2) fractal operators on fractal spaces; (3) 1/2-order fractional dynamics controlled by fractal [...] Read more.
This paper reports the new advances in biological fractal dynamics. The following contents are included: (1) physical (or functional) fractal spaces abstracted from biological materials, biological structures and biological motions; (2) fractal operators on fractal spaces; (3) 1/2-order fractional dynamics controlled by fractal operators; and (4) the origin of 1/2-order. Based on the new progress, we can make a judgment that all the two-bifurcation physical functional fractal motions in the living body can be attributed to the fractional dynamics with 1/2-order. Full article
(This article belongs to the Special Issue Fractional Behaviors Analysis and Modelling)
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14 pages, 707 KiB  
Article
Fractional Behaviours Modelling with Volterra Equations: Application to a Lithium-Ion Cell and Comparison with a Fractional Model
by Vincent Tartaglione, Christophe Farges and Jocelyn Sabatier
Fractal Fract. 2022, 6(3), 137; https://doi.org/10.3390/fractalfract6030137 - 1 Mar 2022
Cited by 4 | Viewed by 2960
Abstract
This paper proposes to model fractional behaviors using Volterra equations. As fractional differentiation-based models that are commonly used to model such behaviors exhibit several drawbacks and are particular cases of Volterra equations (in the kernel definition), it appears legitimate in a modeling approach [...] Read more.
This paper proposes to model fractional behaviors using Volterra equations. As fractional differentiation-based models that are commonly used to model such behaviors exhibit several drawbacks and are particular cases of Volterra equations (in the kernel definition), it appears legitimate in a modeling approach to work directly with Volterra equations. In this paper, a numerical method is thus developed to identify the kernel associated to a Volterra equation that describes the input–output behavior of a system. This method is used to model a lithium-ion cell using real data. The resulting model is compared to a fractional differentiation-based model with the same number of tunable parameters. Full article
(This article belongs to the Special Issue Fractional Behaviors Analysis and Modelling)
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