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: 25 July 2025 | Viewed by 6471

Special Issue Editors


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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
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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

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
Guest Editor

Dr. Leila Gholizadeh Zivlaei
Dr. Mohammad Dehghan
Co-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

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Related Special Issue

Published Papers (6 papers)

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Research

13 pages, 282 KiB  
Article
New Results on a Nonlocal Sturm–Liouville Eigenvalue Problem with Fractional Integrals and Fractional Derivatives
by Yunyang Zhang, Shaojie Chen and Jing Li
Fractal Fract. 2025, 9(2), 70; https://doi.org/10.3390/fractalfract9020070 - 23 Jan 2025
Viewed by 546
Abstract
In this paper, we investigate the eigenvalue properties of a nonlocal Sturm–Liouville equation involving fractional integrals and fractional derivatives under different boundary conditions. Based on these properties, we obtained the geometric multiplicity of eigenvalues for the nonlocal Sturm–Liouville problem with a non-Dirichlet boundary [...] Read more.
In this paper, we investigate the eigenvalue properties of a nonlocal Sturm–Liouville equation involving fractional integrals and fractional derivatives under different boundary conditions. Based on these properties, we obtained the geometric multiplicity of eigenvalues for the nonlocal Sturm–Liouville problem with a non-Dirichlet boundary condition. Furthermore, we discussed the continuous dependence of the eigenvalues on the potential function for a nonlocal Sturm–Liouville equation under a Dirichlet boundary condition. Full article
23 pages, 476 KiB  
Article
Positive Solution Pairs for Coupled p-Laplacian Hadamard Fractional Differential Model with Singular Source Item on Time Variable
by Cheng Li and Limin Guo
Fractal Fract. 2024, 8(12), 682; https://doi.org/10.3390/fractalfract8120682 - 21 Nov 2024
Viewed by 516
Abstract
The mathematical theories and methods of fractional calculus are relatively mature, which have been widely used in signal processing, control systems, nonlinear dynamics, financial models, etc. The studies of some basic theories of fractional differential equations can provide more understanding of mechanisms for [...] Read more.
The mathematical theories and methods of fractional calculus are relatively mature, which have been widely used in signal processing, control systems, nonlinear dynamics, financial models, etc. The studies of some basic theories of fractional differential equations can provide more understanding of mechanisms for the applications. In this paper, the expression of the Green function as well as its special properties are acquired and presented through theoretical analyses. Subsequently, on the basis of these properties of the Green function, the existence and uniqueness of positive solutions are achieved for a singular p-Laplacian fractional-order differential equation with nonlocal integral and infinite-point boundary value systems by using the method of a nonlinear alternative of Leray–Schauder-type Guo–Krasnoselskii’s fixed point theorem in cone, and the Banach fixed point theorem, respectively. Some existence results are obtained for the case in which the nonlinearity is allowed to be singular with regard to the time variable. Several examples are correspondingly provided to show the correctness and applicability of the obtained results, where nonlinear terms are controlled by the integrable functions 1π(lnt)12(1lnt)12 and 1π(lnt)34(1lnt)34 in Example 1, and by the integrable functions θ,θ¯ and φ(v),ψ(u) in Example 2, respectively. The present work may contribute to the improvement and application of the coupled p-Laplacian Hadamard fractional differential model and further promote the development of fractional differential equations and fractional differential calculus. Full article
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22 pages, 4049 KiB  
Article
Fractal-Fractional-Order Modeling of Liver Fibrosis Disease and Its Mathematical Results with Subinterval Transitions
by Amjad E. Hamza, Osman Osman, Arshad Ali, Amer Alsulami, Khaled Aldwoah, Alaa Mustafa and Hicham Saber
Fractal Fract. 2024, 8(11), 638; https://doi.org/10.3390/fractalfract8110638 - 29 Oct 2024
Cited by 2 | Viewed by 1176
Abstract
In this paper, we study human liver disease with a different approach of interval-based investigation by introducing subintervals. This investigation may be referred to as a short memory investigation. Such concepts are useful in problems where a transition is observed when transitioning from [...] Read more.
In this paper, we study human liver disease with a different approach of interval-based investigation by introducing subintervals. This investigation may be referred to as a short memory investigation. Such concepts are useful in problems where a transition is observed when transitioning from one subinterval to the other one. We use the classical and fractal-fractional-order derivative in each subinterval. We study the existence of solutions by using Banach’s and Krasnoselskii’s fixed-point theorems. Their stability is analyzed by adopting the Hyers–Ulam (H-U) stability approach. Also, using the extended Adams–Bashforth–Moulton (ABM) method, we simulate the results that visually present the numerical solutions for different fractal-fractional-order values. Full article
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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 1022
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
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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
Cited by 1 | Viewed by 782
Abstract
We consider fractional integral operators (IT)d,d(1,1) acting on functions g:ZνR,ν1, where T is the transition operator of a random [...] Read more.
We consider fractional integral operators (IT)d,d(1,1) acting on functions g:ZνR,ν1, 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),sZν of (IT)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 (IT)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
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 2 | Viewed by 1359
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
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