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Fractal Fract
Special Issues
Advances in Fractional Differential Operators and Their Applications, 2nd Edition
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Advances in Fractional Differential Operators and Their Applications, 2nd 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: 30 November 2024 | Viewed by 2019

Special Issue Editors

Prof. Dr. Angelo B. Mingarelli

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Carleton University, Ottawa, ON K1S 5B6, Canada
Interests: celestial mechanics; spectral theory of differential operators; fuzzy cellular automata; irrationality questions in number theory
Special Issues, Collections and Topics in MDPI journals
Dr. Leila Gholizadeh Zivlaei

E-Mail Website
Co-Guest Editor
School of Mathematics and Statistics, Carleton University, Ottawa, ON K1S 5B6, Canada
Interests: functional analysis; mathematical analysis; real analysis; measure theory; differential equations
Dr. Mohammad Dehghan

E-Mail Website
Co-Guest Editor
School of Mathematics and Statistics, Carleton University, Ottawa, ON K1S 5B6, Canada
Interests: numerical methods; differential equations

Special Issue Information

Dear Colleagues,

This Special Issue, published by MDPI, is dedicated to the theory and practice of fractional differential operators and corresponding equations. Although the field of fractional derivatives is, in general, relatively old, there remain numerous unsolved problems and wide scope for further research. Thus, we welcome the submission of papers in the area of hybrid fractional equations (e.g., mixed Riemann–Liouville/Caputo derivatives and other combinations of such derivatives). The scope of this Special Issue includes, but is not limited to, the following topics:

  • Generalized and fractional derivatives and integrals;
  • Riemann–Liouville derivatives and integrals;
  • Caputo derivatives and integrals;
  • Spectral and asymptotic theory;
  • Qualitative theory;
  • Variational principles;
  • Applications of fractional derivatives to any area of science or the humanities.

We invite experts in this field to contribute their significant research to this Special Issue so that it can be employed to lay the groundwork for future research in the specified areas. We encourage authors to address open questions within their submissions in order to garner more attention regarding specific problems considered of importance.

Feel free to read and download all our published articles in the 1st volume: https://www.mdpi.com/journal/fractalfract/special_issues/fract_diff_operator and the book: https://www.mdpi.com/books/book/8012.

Prof. Dr. Angelo B. Mingarelli
Dr. Leila Gholizadeh Zivlaei
Dr. Mohammad Dehghan
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

  • generalized derivatives
  • riemann–liouville
  • caputo derivatives
  • spectral theory
  • asymptotic theory
  • qualitative theory
  • variational theorems
  • applications

Related Special Issue

Published Papers (3 papers)

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Research

17 pages, 4991 KiB  
Article
Application of Fractional-Order Multi-Wing Chaotic System to Weak Signal Detection
by Hongcun Mao, Yuling Feng, Xiaoqian Wang, Chao Gao and Zhihai Yao
Fractal Fract. 2024, 8(7), 417; https://doi.org/10.3390/fractalfract8070417 - 16 Jul 2024
Viewed by 238
Abstract
This work investigates a fractional-order multi-wing chaotic system for detecting weak signals. The influence of the order of fractional calculus on chaotic systems’ dynamical behavior is examined using phase diagrams, bifurcation diagrams, and SE complexity diagrams. Then, the principles and methods for determining [...] Read more.
This work investigates a fractional-order multi-wing chaotic system for detecting weak signals. The influence of the order of fractional calculus on chaotic systems’ dynamical behavior is examined using phase diagrams, bifurcation diagrams, and SE complexity diagrams. Then, the principles and methods for determining the frequencies and amplitudes of weak signals are examined utilizing fractional-order multi-wing chaotic systems. The findings indicate that the lowest order at which this kind of fractional-order multi-wing chaotic system appears chaotic is 2.625 at a=4, b=8, and c=1, and that this value decreases as the driving force increases. The four-wing and double-wing change dynamics phenomenon will manifest in a fractional-order chaotic system when the order exceeds the lowest order. This phenomenon can be utilized to detect weak signal amplitudes and frequencies because the system parameters control it. A detection array is built to determine the amplitude using the noise-resistant properties of both four-wing and double-wing chaotic states. Deep learning images are then used to identify the change in the array’s wing count, which can be used to determine the test signal’s amplitude. When frequencies detection is required, the MUSIC method estimates the frequencies using chaotic synchronization to transform the weak signal’s frequencies to the synchronization error’s frequencies. This solution adds to the contact between fractional-order calculus and chaos theory. It offers suggestions for practically implementing the chaotic weak signal detection theory in conjunction with deep learning. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications, 2nd Edition)
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20 pages, 412 KiB  
Article
Fractional Operators and Fractionally Integrated Random Fields on Zν
by Vytautė Pilipauskaitė and Donatas Surgailis
Fractal Fract. 2024, 8(6), 353; https://doi.org/10.3390/fractalfract8060353 - 13 Jun 2024
Viewed by 355
Abstract
We consider fractional integral operators (I−T)d,d∈(−1,1) acting on functions g:Zν→R,ν≥1, where T is the transition operator of a random [...] Read more.
We consider fractional integral operators (I−T)d,d∈(−1,1) acting on functions g:Zν→R,ν≥1, where T is the transition operator of a random walk on Zν. We obtain the sufficient and necessary conditions for the existence, invertibility, and square summability of kernels τ(s;d),s∈Zν of (I−T)d. The asymptotic behavior of τ(s;d) as |s|→∞ is identified following the local limit theorem for random walks. A class of fractionally integrated random fields X on Zν solving the difference equation (I−T)dX=ε with white noise on the right-hand side is discussed and their scaling limits. Several examples, including fractional lattice Laplace and heat operators, are studied in detail. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications, 2nd Edition)
18 pages, 330 KiB  
Article
Existence and Uniqueness of Some Unconventional Fractional Sturm–Liouville Equations
by Leila Gholizadeh Zivlaei and Angelo B. Mingarelli
Fractal Fract. 2024, 8(3), 148; https://doi.org/10.3390/fractalfract8030148 - 3 Mar 2024
Cited by 1 | Viewed by 968
Abstract
In this paper, we provide existence and uniqueness results for the initial value problems associated with mixed Riemann–Liouville/Caputo differential equations in the real domain. We show that, under appropriate conditions in a fractional order, solutions are always square-integrable on the finite interval under [...] Read more.
In this paper, we provide existence and uniqueness results for the initial value problems associated with mixed Riemann–Liouville/Caputo differential equations in the real domain. We show that, under appropriate conditions in a fractional order, solutions are always square-integrable on the finite interval under consideration. The results are valid for equations that have sign-indefinite leading terms and measurable coefficients. Existence and uniqueness theorem results are also provided for two-point boundary value problems in a closed interval. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications, 2nd Edition)
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