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: 1 August 2024 | Viewed by 1159

Special Issue Editors


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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

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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
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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

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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

Published Papers (2 papers)

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Research

18 pages, 478 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
Viewed by 217
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
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 651
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
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