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Mathematics
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Recent Research on Fractional Calculus: Theory and Applications
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Recent Research on Fractional Calculus: Theory and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: 28 February 2025 | Viewed by 5040

Special Issue Editors

Dr. Faïçal Ndaïrou

E-Mail Website
Guest Editor
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia 1113, Bulgaria
Interests: fractional calculus; mathematical biology; optimal control
Prof. Dr. Delfim F. M. Torres

grade E-Mail Website
Guest Editor
Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
Interests: fractional calculus; dynamics on time scales; mathematical biology; calculus of variations; optimal control
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional calculus is a theory that unifies differential and integral operators of arbitrary (integer and non-integer) orders to form differintegral operators. The theory is becoming more attractive due to its applicability to real-world problems, such as describing biological phenomena that are memory-dependent. The aim of this Special Issue is to contribute advancements in fractional calculus, both from theoretical and applicational points of view. We welcome papers on the following topics:

  • Properties of fractional operators defined by new kernels functions;
  • Static and dynamic optimization methods based on fractional operators;
  • Fractional nonlinear dynamical systems and control;
  • Mathematics of infectious diseases by fractional operators.

Dr. Faïçal Ndaïrou
Prof. Dr. Delfim F. M. Torres
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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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 operators
  • differintegral operators
  • memory-dependent phenomena
  • optimal control
  • mathematical modeling
  • bio-mathematics

Published Papers (7 papers)

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Research

24 pages, 709 KiB  
Article
The Stability of Solutions of the Variable-Order Fractional Optimal Control Model for the COVID-19 Epidemic in Discrete Time
by Meriem Boukhobza, Amar Debbouche, Lingeshwaran Shangerganesh and Juan J. Nieto
Mathematics 2024, 12(8), 1236; https://doi.org/10.3390/math12081236 - 19 Apr 2024
Viewed by 396
Abstract
This article introduces a discrete-time fractional variable order over a SEIQR model, incorporated for COVID-19. Initially, we establish the well-possedness of solution. Further, the disease-free and the endemic equilibrium points are determined. Moreover, the local asymptotic stability of the model is analyzed. We [...] Read more.
This article introduces a discrete-time fractional variable order over a SEIQR model, incorporated for COVID-19. Initially, we establish the well-possedness of solution. Further, the disease-free and the endemic equilibrium points are determined. Moreover, the local asymptotic stability of the model is analyzed. We develop a novel discrete fractional optimal control problem tailored for COVID-19, utilizing a discrete mathematical model featuring a variable order fractional derivative. Finally, we validate the reliability of these analytical findings through numerical simulations and offer insights from a biological perspective. Full article
(This article belongs to the Special Issue Recent Research on Fractional Calculus: Theory and Applications)
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21 pages, 396 KiB  
Article
Existence, Uniqueness, and Averaging Principle of Fractional Neutral Stochastic Differential Equations in the Lp Space with the Framework of the Ψ-Caputo Derivative
by Abdelhamid Mohammed Djaouti, Zareen A. Khan, Muhammad Imran Liaqat and Ashraf Al-Quran
Mathematics 2024, 12(7), 1037; https://doi.org/10.3390/math12071037 - 30 Mar 2024
Viewed by 431
Abstract
In this research work, we use the concepts of contraction mapping to establish the existence and uniqueness results and also study the averaging principle in Lp space by using Jensen’s, Grönwall–Bellman’s, Hölder’s, and Burkholder–Davis–Gundy’s inequalities, and the interval translation technique for a [...] Read more.
In this research work, we use the concepts of contraction mapping to establish the existence and uniqueness results and also study the averaging principle in Lp space by using Jensen’s, Grönwall–Bellman’s, Hölder’s, and Burkholder–Davis–Gundy’s inequalities, and the interval translation technique for a class of fractional neutral stochastic differential equations. We establish the results within the framework of the Ψ-Caputo derivative. We generalize the two situations of p=2 and the Caputo derivative with the findings that we obtain. To help with the understanding of the theoretical results, we provide two applied examples at the end. Full article
(This article belongs to the Special Issue Recent Research on Fractional Calculus: Theory and Applications)
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14 pages, 268 KiB  
Article
Hyers–Ulam Stability of Caputo Fractional Stochastic Delay Differential Systems with Poisson Jumps
by Zhenyu Bai and Chuanzhi Bai
Mathematics 2024, 12(6), 804; https://doi.org/10.3390/math12060804 - 08 Mar 2024
Viewed by 430
Abstract
In this paper, we explore the stability of a new class of Caputo-type fractional stochastic delay differential systems with Poisson jumps. We prove the Hyers–Ulam stability of the solution by utilizing a version of fixed point theorem, fractional calculus, Cauchy–Schwartz inequality, Jensen inequality, [...] Read more.
In this paper, we explore the stability of a new class of Caputo-type fractional stochastic delay differential systems with Poisson jumps. We prove the Hyers–Ulam stability of the solution by utilizing a version of fixed point theorem, fractional calculus, Cauchy–Schwartz inequality, Jensen inequality, and some stochastic analysis techniques. Finally, an example is provided to illustrate the effectiveness of the results. Full article
(This article belongs to the Special Issue Recent Research on Fractional Calculus: Theory and Applications)
15 pages, 307 KiB  
Article
On the Fractional Derivative Duality in Some Transforms
by Manuel Duarte Ortigueira and Gabriel Bengochea
Mathematics 2023, 11(21), 4464; https://doi.org/10.3390/math11214464 - 27 Oct 2023
Viewed by 600
Abstract
Duality is one of the most interesting properties of the Laplace and Fourier transforms associated with the integer-order derivative. Here, we will generalize it for fractional derivatives and extend the results to the Mellin, Z and discrete-time Fourier transforms. The scale and nabla [...] Read more.
Duality is one of the most interesting properties of the Laplace and Fourier transforms associated with the integer-order derivative. Here, we will generalize it for fractional derivatives and extend the results to the Mellin, Z and discrete-time Fourier transforms. The scale and nabla derivatives are used. Some consequences are described. Full article
(This article belongs to the Special Issue Recent Research on Fractional Calculus: Theory and Applications)
12 pages, 286 KiB  
Article
Pontryagin Maximum Principle for Incommensurate Fractional-Orders Optimal Control Problems
by Faïçal Ndaïrou and Delfim F. M. Torres
Mathematics 2023, 11(19), 4218; https://doi.org/10.3390/math11194218 - 09 Oct 2023
Viewed by 1079
Abstract
We introduce a new optimal control problem where the controlled dynamical system depends on multi-order (incommensurate) fractional differential equations. The cost functional to be maximized is of Bolza type and depends on incommensurate Caputo fractional-orders derivatives. We establish continuity and differentiability of the [...] Read more.
We introduce a new optimal control problem where the controlled dynamical system depends on multi-order (incommensurate) fractional differential equations. The cost functional to be maximized is of Bolza type and depends on incommensurate Caputo fractional-orders derivatives. We establish continuity and differentiability of the state solutions with respect to perturbed trajectories. Then, we state and prove a Pontryagin maximum principle for incommensurate Caputo fractional optimal control problems. Finally, we give an example, illustrating the applicability of our Pontryagin maximum principle. Full article
(This article belongs to the Special Issue Recent Research on Fractional Calculus: Theory and Applications)
19 pages, 343 KiB  
Article
New Stability Results for Abstract Fractional Differential Equations with Delay and Non-Instantaneous Impulses
by Abdellatif Benchaib, Abdelkrim Salim, Saïd Abbas and Mouffak Benchohra
Mathematics 2023, 11(16), 3490; https://doi.org/10.3390/math11163490 - 12 Aug 2023
Cited by 1 | Viewed by 708
Abstract
This research delves into the field of fractional differential equations with both non-instantaneous impulses and delay within the framework of Banach spaces. Our objective is to establish adequate conditions that ensure the existence, uniqueness, and Ulam–Hyers–Rassias stability results for our problems. The studied [...] Read more.
This research delves into the field of fractional differential equations with both non-instantaneous impulses and delay within the framework of Banach spaces. Our objective is to establish adequate conditions that ensure the existence, uniqueness, and Ulam–Hyers–Rassias stability results for our problems. The studied problems encompass abstract impulsive fractional differential problems with finite delay, infinite delay, state-dependent finite delay, and state-dependent infinite delay. To provide clarity and depth, we augment our theoretical results with illustrative examples, illustrating the practical implications of our work. Full article
(This article belongs to the Special Issue Recent Research on Fractional Calculus: Theory and Applications)
16 pages, 309 KiB  
Article
Euler–Lagrange-Type Equations for Functionals Involving Fractional Operators and Antiderivatives
by Ricardo Almeida
Mathematics 2023, 11(14), 3208; https://doi.org/10.3390/math11143208 - 21 Jul 2023
Viewed by 579
Abstract
The goal of this paper is to present the necessary and sufficient conditions that every extremizer of a given class of functionals, defined on the set C1[a,b], must satisfy. The Lagrange function depends on a generalized [...] Read more.
The goal of this paper is to present the necessary and sufficient conditions that every extremizer of a given class of functionals, defined on the set C1[a,b], must satisfy. The Lagrange function depends on a generalized fractional derivative, on a generalized fractional integral, and on an antiderivative involving the previous fractional operators. We begin by obtaining the fractional Euler–Lagrange equation, which is a necessary condition to optimize a given functional. By imposing convexity conditions over the Lagrange function, we prove that it is also a sufficient condition for optimization. After this, we consider variational problems with additional constraints on the set of admissible functions, such as the isoperimetric and the holonomic problems. We end by considering a generalization of the fundamental problem, where the fractional order is not restricted to real values between 0 and 1, but may take any positive real value. We also present some examples to illustrate our results. Full article
(This article belongs to the Special Issue Recent Research on Fractional Calculus: Theory and Applications)
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