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Fractal Fract
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Recent Developments in Multidimensional Fractional Differential Equations and Fractional Difference...
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Recent Developments in Multidimensional Fractional Differential Equations and Fractional Difference Equations

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

Deadline for manuscript submissions: 18 July 2026 | Viewed by 1854

Special Issue Editors

Prof. Dr. Marko Kostić

E-Mail Website
Guest Editor
Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovića 6, 21125 Novi Sad, Serbia
Interests: functional analysis; fractional calculus; fractional integro-differential equations; fractional integro-difference equations
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Wei-Shih Du

E-Mail Website
Guest Editor
Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 82444, Taiwan
Interests: nonlinear analysis; fixed point theory and its applications; variational principles and inequalities; optimization theory; fractional calculus theory.
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The theory of abstract fractional differential equations is an active field of research for many authors. Fractional calculus and fractional differential equations, which have emerged as the most important field of applied mathematics in the last century, can be viewed as a special aspect of this theory.

On the other hand, fractional calculus and fractional differential equations have received significant attention in recent years. Muti-dimensional fractional calculus is fundamentally important in the modeling of various phenomena concerning complex dynamic systems, frequency response analysis, image processing, interval–valued systems and neural networks.

The main aim of this Special Issue is to present recent developments in the theory and application of fractional calculus, fractional integro–differential-difference equations and multidimensional fractional calculus. We strongly encourage the submission of papers concerning fractional partial differential–difference equations that depend on several variables.

We would like to thank Professor Vladimir E. Fedorov for assisting us with this Special Issue.

Prof. Dr. Marko Kostić
Prof. Dr. Wei-Shih Du
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

  • fractional calculus
  • fractional differential equations
  • fractional difference equations
  • multidimensional fractional calculus
  • multidimensional fractional differential equations
  • multidimensional partial fractional differential equations
  • theory and applications

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Published Papers (3 papers)

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23 pages, 21905 KB  
Article
Fractional Calculus in Nuclear Multistep Decay: Analytical Solutions, Existence and Uniqueness Analysis of the Actinium Series
by Mohammed Shqair, Areej Almuneef, Emad Jaradat, Rahat Zarin and Ahmed Hagag
Fractal Fract. 2025, 9(9), 601; https://doi.org/10.3390/fractalfract9090601 - 16 Sep 2025
Viewed by 172
Abstract
This paper provides a thorough examination of the Actinium radioactive decay series, which converts Uranium-235 into the stable Lead-207 isotope via a succession of alpha, beta, and gamma decays. For the first time, the series is modeled using fractional calculus, employing two innovative [...] Read more.
This paper provides a thorough examination of the Actinium radioactive decay series, which converts Uranium-235 into the stable Lead-207 isotope via a succession of alpha, beta, and gamma decays. For the first time, the series is modeled using fractional calculus, employing two innovative analytical methods: the Sumudu Residual Power Series Method (SRPSM) and the Temimi Ansari Method (TAM). The study discusses the well-posedness of the fractional-order model in the Caputo sense within a Banach space setting. These fractional models capture complex, non-ideal decay behaviors more accurately than traditional exponential models. Mathematica is used to do numerical computations for four different Actinium series scenarios. The results are tabulated and visually depicted to show how radionuclide concentrations change over time. The findings demonstrate that SRPSM and TAM effectively simplify the complex differential equations governing nuclear decay, offering enhanced precision and flexibility. This work provides a robust framework for modeling the Actinium series, with potential applications in nuclear physics, radiometric dating, and radiation safety studies. Full article
(This article belongs to the Special Issue Recent Developments in Multidimensional Fractional Differential Equations and Fractional Difference Equations)
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29 pages, 3058 KB  
Article
Existence, Uniqueness, and Stability of Weighted Fuzzy Fractional Volterra–Fredholm Integro-Differential Equation
by Sahar Abbas, Abdul Ahad Abro, Syed Muhammad Daniyal, Hanaa A. Abdallah, Sadique Ahmad, Abdelhamied Ashraf Ateya and Noman Bin Zahid
Fractal Fract. 2025, 9(8), 540; https://doi.org/10.3390/fractalfract9080540 - 16 Aug 2025
Viewed by 439
Abstract
This paper investigates a novel class of weighted fuzzy fractional Volterra–Fredholm integro-differential equations (FWFVFIDEs) subject to integral boundary conditions. The analysis is conducted within the framework of Caputo-weighted fractional calculus. Employing Banach’s and Krasnoselskii’s fixed-point theorems, we establish the existence and uniqueness of [...] Read more.
This paper investigates a novel class of weighted fuzzy fractional Volterra–Fredholm integro-differential equations (FWFVFIDEs) subject to integral boundary conditions. The analysis is conducted within the framework of Caputo-weighted fractional calculus. Employing Banach’s and Krasnoselskii’s fixed-point theorems, we establish the existence and uniqueness of solutions. Stability is analyzed in the Ulam–Hyers (UHS), generalized Ulam–Hyers (GUHS), and Ulam–Hyers–Rassias (UHRS) senses. A modified Adomian decomposition method (MADM) is introduced to derive explicit solutions without linearization, preserving the problem’s original structure. The first numerical example validates the theoretical findings on existence, uniqueness, and stability, supplemented by graphical results obtained via the MADM. Further examples illustrate fuzzy solutions by varying the uncertainty level (r), the variable (x), and both parameters simultaneously. The numerical results align with the theoretical analysis, demonstrating the efficacy and applicability of the proposed method. Full article
(This article belongs to the Special Issue Recent Developments in Multidimensional Fractional Differential Equations and Fractional Difference Equations)
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16 pages, 1058 KB  
Article
Ulam–Hyers Stability of Fractional Difference Equations with Hilfer Derivatives
by Marko Kostić, Halis Can Koyuncuoğlu and Jagan Mohan Jonnalagadda
Fractal Fract. 2025, 9(7), 417; https://doi.org/10.3390/fractalfract9070417 - 26 Jun 2025
Viewed by 548
Abstract
This paper investigates the Ulam–Hyers stability of both linear and nonlinear delayed neutral Hilfer fractional difference equations. We utilize the nabla Laplace transform, known as the N-transform, along with a generalized discrete Gronwall inequality to derive sufficient conditions for stability. For the [...] Read more.
This paper investigates the Ulam–Hyers stability of both linear and nonlinear delayed neutral Hilfer fractional difference equations. We utilize the nabla Laplace transform, known as the N-transform, along with a generalized discrete Gronwall inequality to derive sufficient conditions for stability. For the linear case, we provide an explicit solution formula involving discrete Mittag-Leffler functions and establish its stability properties. In the nonlinear case, we concentrate on delayed neutral Hilfer fractional difference equations, a class of systems that appears to be unexplored in the existing literature with respect to Ulam–Hyers stability. In particular, for the linear case, the absolute difference between the solution of the linear Hilfer fractional difference equation and the solution of the corresponding perturbed equation is bounded by the function of ε when the perturbed term is bounded by ε. In the case of the neutral fractional delayed Hilfer difference equation, the absolute difference is bounded by a constant multiple of ε. Our results fill this gap by offering novel stability criteria. We support our theoretical findings with illustrative numerical examples and simulations, which visually confirm the predicted stability behavior and demonstrate the applicability of the results in discrete fractional dynamic systems. Full article
(This article belongs to the Special Issue Recent Developments in Multidimensional Fractional Differential Equations and Fractional Difference Equations)
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