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Axioms
Special Issues
Fractional Calculus and Its Applications: Historical and Recent Developments
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Fractional Calculus and Its Applications: Historical and Recent Developments

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

Deadline for manuscript submissions: 30 April 2025 | Viewed by 4571

Special Issue Editors

Dr. Anabela S. Silva

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Guest Editor
Center for Research and Development in Mathematics and Applications (CIDMA), University of Aveiro, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal
Interests: fractional calculus; fractional boundary or initial value problems; existence and uniqueness of solutions; stability of solutions; convolution operators
Dr. Cristiana J. Silva

E-Mail Website
Guest Editor
Department of Mathematics, ISTA, Instituto Universitário de Lisboa (ISCTE-IUL), Av. das Forças Armadas, 1649-026 Lisbon, Portugal
Interests: optimal control theory; mathematical modeling; optimization methods; applications to biology and epidemiology
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional calculus is a field of mathematical analysis that generalizes the concept of differentiation and integration to non-integer orders. The history of fractional calculus dates back to the 17th century, with developments by mathematicians like Leibniz and L'Hôpital. However, the interest in fractional calculus faded for several centuries, and it was not until the 19th century that the topic regained attention. Nowadays, fractional calculus provides a powerful mathematical tool for describing systems with memory effects or systems involving fractal geometry. Its applications range from areas such as signal processing, viscoelasticity, and control theory to the modeling of complex phenomena.

This Special Issue will accept high-quality articles on the theory and applications of fractional calculus, showing the latest developments in the area and its evolution over the years. The aim is to bring together researchers in mathematics, physics, engineering, or other fields, to showcase their most recent work, contextualizing it with existing results and highlighting the importance of this area in mathematics.

Dr. Anabela S. Silva
Dr. Cristiana J. Silva
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. 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 integrals and derivatives
  • history of fractional calculus
  • fractional ordinary and partial differential equations
  • properties of solutions
  • mathematical modelling involving fractional ODEs and PDEs calculus
  • numerical methods
  • applications

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Published Papers (5 papers)

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23 pages, 543 KiB  
Article
Numerical Solutions for Nonlinear Ordinary and Fractional Duffing Equations Using Combined Fibonacci–Lucas Polynomials
by Waleed Mohamed Abd-Elhameed, Omar Mazen Alqubori, Amr Kamel Amin and Ahmed Gamal Atta
Axioms 2025, 14(4), 314; https://doi.org/10.3390/axioms14040314 - 19 Apr 2025
Viewed by 146
Abstract
Two nonlinear Duffing equations are numerically treated in this article. The nonlinear fractional-order Duffing equations and the second-order nonlinear Duffing equations are handled. Based on the collocation technique, we provide two numerical algorithms. To achieve this goal, a new family of basis functions [...] Read more.
Two nonlinear Duffing equations are numerically treated in this article. The nonlinear fractional-order Duffing equations and the second-order nonlinear Duffing equations are handled. Based on the collocation technique, we provide two numerical algorithms. To achieve this goal, a new family of basis functions is built by combining the sets of Fibonacci and Lucas polynomials. Several new formulae for these polynomials are developed. The operational matrices of integer and fractional derivatives of these polynomials, as well as some new theoretical results of these polynomials, are presented and used in conjunction with the collocation method to convert nonlinear Duffing equations into algebraic systems of equations by forcing the equation to hold at certain collocation points. To numerically handle the resultant nonlinear systems, one can use symbolic algebra solvers or Newton’s approach. Some particular inequalities are proved to investigate the convergence analysis. Some numerical examples show that our suggested strategy is effective and accurate. The numerical results demonstrate that the suggested collocation approach yields accurate solutions by utilizing Fibonacci–Lucas polynomials as basis functions. Full article
(This article belongs to the Special Issue Fractional Calculus and Its Applications: Historical and Recent Developments)
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12 pages, 258 KiB  
Article
Existence Result for a Class of Time-Fractional Nonstationary Incompressible Navier–Stokes–Voigt Equations
by Keji Xu and Biao Zeng
Axioms 2024, 13(8), 499; https://doi.org/10.3390/axioms13080499 - 25 Jul 2024
Viewed by 674
Abstract
We are devoted in this work to dealing with a class of time-fractional nonstationary incompressible Navier–Stokes–Voigt equation involving the Caputo fractional derivative. By exploiting the properties of the operators in the equation, we use the Rothe method to show the existence of weak [...] Read more.
We are devoted in this work to dealing with a class of time-fractional nonstationary incompressible Navier–Stokes–Voigt equation involving the Caputo fractional derivative. By exploiting the properties of the operators in the equation, we use the Rothe method to show the existence of weak solutions to the equation by verifying all the conditions of the surjectivity theorem for nonlinear weakly continuous operators. Full article
(This article belongs to the Special Issue Fractional Calculus and Its Applications: Historical and Recent Developments)
17 pages, 298 KiB  
Article
Fractional Sequential Coupled Systems of Hilfer and Caputo Integro-Differential Equations with Non-Separated Boundary Conditions
by Ayub Samadi, Sotiris K. Ntouyas and Jessada Tariboon
Axioms 2024, 13(7), 484; https://doi.org/10.3390/axioms13070484 - 18 Jul 2024
Cited by 1 | Viewed by 924
Abstract
In studying boundary value problems and coupled systems of fractional order in (1,2], involving Hilfer fractional derivative operators, a zero initial condition is necessary. The consequence of this fact is that boundary value problems and coupled systems of [...] Read more.
In studying boundary value problems and coupled systems of fractional order in (1,2], involving Hilfer fractional derivative operators, a zero initial condition is necessary. The consequence of this fact is that boundary value problems and coupled systems of fractional order with non-zero initial conditions cannot be studied. For example, such boundary value problems and coupled systems of fractional order are those including separated, non-separated, or periodic boundary conditions. In this paper, we propose a method for studying a coupled system of fractional order in (1,2], involving fractional derivative operators of Hilfer and Caputo with non-separated boundary conditions. More precisely, a sequential coupled system of fractional differential equations including Hilfer and Caputo fractional derivative operators and non-separated boundary conditions is studied in the present paper. As explained in the concluding section, the opposite combination of Caputo and Hilfer fractional derivative operators requires zero initial conditions. By using Banach’s fixed point theorem, the uniqueness of the solution is established, while by applying the Leray–Schauder alternative, the existence of solution is obtained. Numerical examples are constructed illustrating the main results. Full article
(This article belongs to the Special Issue Fractional Calculus and Its Applications: Historical and Recent Developments)
19 pages, 485 KiB  
Article
Weighted Convolution for Quaternion Linear Canonical Cosine Transform and Its Application
by Rongbo Wang and Qiang Feng
Axioms 2024, 13(6), 402; https://doi.org/10.3390/axioms13060402 - 14 Jun 2024
Cited by 2 | Viewed by 822
Abstract
Convolution plays a pivotal role in the domains of signal processing and optics. This paper primarily focuses on studying the weighted convolution for quaternion linear canonical cosine transform (QLCcT) and its application in multiplicative filter analysis. Firstly, we propose QLCcT by combining quaternion [...] Read more.
Convolution plays a pivotal role in the domains of signal processing and optics. This paper primarily focuses on studying the weighted convolution for quaternion linear canonical cosine transform (QLCcT) and its application in multiplicative filter analysis. Firstly, we propose QLCcT by combining quaternion algebra with linear canonical cosine transform (LCcT), which extends LCcT to Hamiltonian quaternion algebra. Secondly, we introduce weighted convolution and correlation operations for QLCcT, accompanied by their corresponding theorems. We also explore the properties of QLCcT. Thirdly, we utilize these proposed convolution structures to analyze multiplicative filter models that offer lower computational complexity compared to existing methods based on quaternion linear canonical transform (QLCT). Additionally, we discuss the rationale behind studying such transforms using quaternion functions as an illustrative example. Full article
(This article belongs to the Special Issue Fractional Calculus and Its Applications: Historical and Recent Developments)
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11 pages, 253 KiB  
Article
A Comprehensive Study of the Langevin Boundary Value Problems with Variable Order Fractional Derivatives
by John R. Graef, Kadda Maazouz and Moussa Daif Allah Zaak
Axioms 2024, 13(4), 277; https://doi.org/10.3390/axioms13040277 - 21 Apr 2024
Viewed by 1162
Abstract
The authors investigate Langevin boundary value problems containing a variable order Caputo fractional derivative. After presenting the background for the study, the authors provide the definitions, theorems, and lemmas that are required for comprehending the manuscript. The existence of solutions is proved using [...] Read more.
The authors investigate Langevin boundary value problems containing a variable order Caputo fractional derivative. After presenting the background for the study, the authors provide the definitions, theorems, and lemmas that are required for comprehending the manuscript. The existence of solutions is proved using Schauder’s fixed point theorem; the uniqueness of solutions is obtained by adding an additional hypothesis and applying Banach’s contraction principle. An example is provided to demonstrate the results. Full article
(This article belongs to the Special Issue Fractional Calculus and Its Applications: Historical and Recent Developments)
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