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Fractional Calculus—Theory and Applications, 3rd Edition
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Fractional Calculus—Theory and Applications, 3rd Edition

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 31 December 2025 | Viewed by 3141

Special Issue Editor

Prof. Dr. Jorge E. Macías Díaz

E-Mail Website
Guest Editor
1. Department of Mathematics, School of Digital Technologies, Tallinn University, 10120 Tallinn, Estonia
2. Department of Mathematics and Physics, Autonomous University of Aguascalientes, Aguascalientes 20131, Mexico
Interests: fractional calculus; fractional analysis; numerical methods for fractional differential equations; nonlinear fractional analysis; simulation of fractional systems; nonlinear systems
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In recent years, fractional calculus has witnessed tremendous progress in various areas of sciences and mathematics. On the one hand, new definitions of fractional derivatives and integrals have appeared in recent years, extending the classical definitions in some sense or another. Moreover, the rigorous analysis of the functional properties of those new definitions has been an active area of research in mathematical analysis. Systems considering differential equations with fractional-order operators have been investigated rigorously from the analytical and numerical points of view, and potential applications have been proposed in science and in technology. The purpose of this Special Issue is to serve as a specialized forum for the dissemination of recent progress in the theory of fractional calculus and its potential applications. We invite authors to submit high-quality reports on the analysis of fractional-order differential/integral equations, the analysis of new definitions of fractional derivatives, numerical methods for fractional-order equations, and applications to physical systems governed by fractional differential equations, among other interesting topics of research.

Prof. Dr. Jorge E. Macías Díaz
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. Axioms 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 2400 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-order differential/integral equations
  • existence and regularity of solutions
  • numerical methods for fractional equations
  • analysis of convergence and stability
  • applications to science and technology

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Related Special Issue

Published Papers (5 papers)

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Research

18 pages, 338 KiB  
Article
Existence of Solutions for Caputo-Type Fractional (p,q)-Difference Equations Under Robin Boundary Conditions
by Hailong Ma and Hongyu Li
Axioms 2025, 14(4), 318; https://doi.org/10.3390/axioms14040318 - 21 Apr 2025
Viewed by 113
Abstract
In this paper, we investigate the existence results of solutions for Caputo-type fractional (p,q)-difference equations. Using Banach’s fixed-point theorem, we obtain the existence and uniqueness results. Meanwhile, by applying Krasnoselskii’s fixed-point theorem and Leray-Schauder’s nonlinear alternative, we also [...] Read more.
In this paper, we investigate the existence results of solutions for Caputo-type fractional (p,q)-difference equations. Using Banach’s fixed-point theorem, we obtain the existence and uniqueness results. Meanwhile, by applying Krasnoselskii’s fixed-point theorem and Leray-Schauder’s nonlinear alternative, we also obtain the existence results of non-trivial solutions. Finally, we provide examples to verify the correctness of the given results. Moreover, relevant applications are presented through specific examples. Full article
(This article belongs to the Special Issue Fractional Calculus—Theory and Applications, 3rd Edition)
20 pages, 307 KiB  
Article
Existence and Asymptotic Estimates of the Maximal and Minimal Solutions for a Coupled Tempered Fractional Differential System with Different Orders
by Peng Chen, Xinguang Zhang, Lishuang Li, Yongsheng Jiang and Yonghong Wu
Axioms 2025, 14(2), 92; https://doi.org/10.3390/axioms14020092 - 26 Jan 2025
Viewed by 527
Abstract
In this paper, we focus on the existence and asymptotic estimates of the maximal and minimal solutions for a coupled tempered fractional differential system with different orders. By introducing an order reduction technique and some new growth conditions, we establish some new results [...] Read more.
In this paper, we focus on the existence and asymptotic estimates of the maximal and minimal solutions for a coupled tempered fractional differential system with different orders. By introducing an order reduction technique and some new growth conditions, we establish some new results on the existence of positive extremal solutions for the tempered fractional differential system, meanwhile, we also obtain the asymptotic estimate of the positive extreme solution by an iterative technique, which possesses a sharp asymptotic estimate. In particular, the iterative sequences converging to maximal and minimal solutions starting from two known initial values are easy to compute. Moreover, the weight function ℏi is allowed to have an infinite number of singular points in [0,1]. Full article
(This article belongs to the Special Issue Fractional Calculus—Theory and Applications, 3rd Edition)
13 pages, 258 KiB  
Article
Analyzing Uniqueness of Solutions in Nonlinear Fractional Differential Equations with Discontinuities Using Lebesgue Spaces
by Farva Hafeez, Mdi Begum Jeelani and Nouf Abdulrahman Alqahtani
Axioms 2025, 14(1), 26; https://doi.org/10.3390/axioms14010026 - 31 Dec 2024
Viewed by 576
Abstract
We explore the existence and uniqueness of solutions to nonlinear fractional differential equations (FDEs), defined in the sense of RL-fractional derivatives of order η∈(1,2). The nonlinear term is assumed to have a discontinuity at zero. By [...] Read more.
We explore the existence and uniqueness of solutions to nonlinear fractional differential equations (FDEs), defined in the sense of RL-fractional derivatives of order η∈(1,2). The nonlinear term is assumed to have a discontinuity at zero. By employing techniques from Lebesgue spaces, including Holder’s inequality, we establish uniqueness theorems for this problem, analogous to Nagumo, Krasnoselskii–Krein, and Osgood-type results. These findings provide a fundamental framework for understanding the properties of solutions to nonlinear FDEs with discontinuous nonlinearities. Full article
(This article belongs to the Special Issue Fractional Calculus—Theory and Applications, 3rd Edition)
22 pages, 2198 KiB  
Article
A Fractional Gompertz Model with Generalized Conformable Operators to Forecast the Dynamics of Mexico’s Hotel Demand and Tourist Area Life Cycle
by Fidel Meléndez-Vázquez, Josué N. Gutiérrez-Corona, Luis A. Quezada-Téllez, Guillermo Fernández-Anaya and Jorge E. Macías-Díaz
Axioms 2024, 13(12), 876; https://doi.org/10.3390/axioms13120876 - 17 Dec 2024
Viewed by 652
Abstract
This study explores the application of generalized conformable derivatives in modeling hotel demand dynamics in Mexico, using the Gompertz-type model. The research focuses on customizing conformable functions to fit the unique characteristics of the Mexican hotel industry, considering the Tourist Area Life Cycle [...] Read more.
This study explores the application of generalized conformable derivatives in modeling hotel demand dynamics in Mexico, using the Gompertz-type model. The research focuses on customizing conformable functions to fit the unique characteristics of the Mexican hotel industry, considering the Tourist Area Life Cycle (TALC) model and aiming to enhance forecasting accuracy. The parameter adjustment in all cases was made by designing a convex function, which represents the difference between the theoretical model and real data. Results demonstrate the effectiveness of the generalized conformable derivative approach in predicting hotel demand trends, showcasing its potential for improving decision-making processes in the Mexican hospitality sector. The comparison between the logistic and Gompertz models, in both integer and fractional versions, provides insights into the suitability of these modeling techniques for capturing the dynamics of hotel demand in the studied regions. Full article
(This article belongs to the Special Issue Fractional Calculus—Theory and Applications, 3rd Edition)
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12 pages, 257 KiB  
Article
The Orthogonal Riesz Fractional Derivative
by Fethi Bouzeffour
Axioms 2024, 13(10), 715; https://doi.org/10.3390/axioms13100715 - 16 Oct 2024
Viewed by 872
Abstract
The aim of this paper is to extend the concept of the orthogonal derivative to provide a new integral representation of the fractional Riesz derivative. Specifically, we investigate the orthogonal derivative associated with Gegenbauer polynomials Cn(ν)(x) [...] Read more.
The aim of this paper is to extend the concept of the orthogonal derivative to provide a new integral representation of the fractional Riesz derivative. Specifically, we investigate the orthogonal derivative associated with Gegenbauer polynomials Cn(ν)(x), where ν>−12. Building on the work of Diekema and Koornwinder, the n-th derivative is obtained as the limit of an integral involving Gegenbauer polynomials as the kernel. When this limit is omitted, it results in the approximate Gegenbauer orthogonal derivative, which serves as an effective approximation of the n-th order derivative. Using this operator, we introduce a novel extension of the fractional Riesz derivative, denoted as Dαx, providing an alternative framework for fractional calculus. Full article
(This article belongs to the Special Issue Fractional Calculus—Theory and Applications, 3rd Edition)
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