Fractional Calculus: Methods and Modeling in Physics, Engineering and Applied Sciences — in Memory of Prof. J. A. Tenreiro Machado

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Functional Interpolation".

Deadline for manuscript submissions: closed (30 November 2022) | Viewed by 9750

Special Issue Editors

Department of mathematics, Bethune-Cookman University, Daytona Beach, FL 32114, USA
Interests: fractional differential equations; integral boundary conditions; Banach contraction principle; dynamical systems; fractional-order systems; delay differential equations; mathematical modelling; numerical methods; neural networks; optimization; control systems; mathematical medical; biology & environmental sciences

Special Issue Information

In Memory of Prof. J. A. Tenreiro Machado

On October 6, 2021 Prof. Tenreiro Machado passed away. We dedicate this Special Issue to his memory. We are not able to express our sadness for this loss. He was a pioneer in the field of fractional calculus. The breadth of Prof. Tenreiro Machado’s research, the global dissemination of his results, and his legacy belong to all scholars involved in any way in the study of fractional calculus. In particular, Professor Hammouch remembers that Prof. Tenreiro Machado was a deeply passionate scholar. His sense of humor was a gift to all those who knew him. We are proud that he was the Editor-in-Chief of Mathematics when this Special Issue was proposed.

Dear Colleagues,

During the last four decades, fractional calculus has been found to be remarkably popular and important, due mainly to their demonstrated applications in numerous seemingly diverse and widespread fields of mathematics, physics, engineering, etc. In particular, it extends the classical problems based on differential models. This allows us to generalize several mathematical models often described by derivatives of integer order (e.g., fractal media). Nevertheless, in current literature, there are several definitions of fractional derivative (Riemann–Liouville, Grünwald–Letnikov, etc.). Accordingly, many researchers work on overcoming this difficulty in order to get a unique definition of fractional derivative.

This would shed new light on the fractional calculus. Moreover, fractional calculus provides several tools for solving differential, integral, integro-differential equations and nonlinear models in mathematical physics. In the last ten years, considerable attention has been paid to fractional calculus in the complex plane. Several publications have appeared documenting the aforementioned interest, especially about the class of holomorphic functions.

Consequently, the fractional calculus of special functions represents one of the most interesting research topics in contemporary mathematics. As a result, the link between fractional calculus and other mathematical fields may provide new results and applications by opening up new frontiers in research. This is the case of the concept of symmetry, as shown by its recent applications in the concept of symmetry (e.g., Lie symmetry analysis of fractional PDEs, fractional supersymmetric quantum mechanics). Fractional calculus is nowadays applied in several fields of science, such as quantum mechanics, information theory, dynamical systems and health science. Moreover, many researchers have also found applications in non-scientific areas (e.g., life sciences, social sciences).

In this Special Issue, we invite and welcome review, expository and original papers dealing with recent advances in fractional calculus and from a more general point of view to all theoretical and practical studies in mathematics, physics and engineering focused in some way on this topic.

The main topics of this Special Issue include (but are not limited to):

  • Fractional equations;
  • Fractional modelling;
  • Fractional calculus of special functions;
  • Entropy concept and fractional calculus;
  • Fractional boundary value problems;
  • Fractional control theory;
  • Chaoticity and fractional calculus;
  • Fractal geometry and fractional calculus;
  • Fractional differential equations;
  • Integral boundary conditions;
  • Banach contraction principle;
  • Dynamical systems;
  • Fractional-order systems;
  • Delay differential equations;
  • Mathematical modelling;
  • Numerical methods;
  • Neural networks;
  • Optimization;
  • Control systems.

Dr. Emanuel Guariglia
Prof. Dr. Seenith Sivasundaram
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 derivative
  • fractional operator
  • symmetry
  • fractal media
  • chaoticity
  • special functions
  • control theory
  • dynamical systems

Published Papers (6 papers)

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Research

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14 pages, 362 KiB  
Article
A New Projection Method for a System of Fractional Cauchy Integro-Differential Equations via Vieta–Lucas Polynomials
by Abdelkader Moumen and Abdelaziz Mennouni
Mathematics 2023, 11(1), 32; https://0-doi-org.brum.beds.ac.uk/10.3390/math11010032 - 22 Dec 2022
Viewed by 1147
Abstract
This work presents a projection method based on Vieta–Lucas polynomials and an effective approach to solve a Cauchy-type fractional integro-differential equation system. The suggested established model overcomes two linear equation systems. We prove the existence of the problem’s approximate solution and conduct an [...] Read more.
This work presents a projection method based on Vieta–Lucas polynomials and an effective approach to solve a Cauchy-type fractional integro-differential equation system. The suggested established model overcomes two linear equation systems. We prove the existence of the problem’s approximate solution and conduct an error analysis in a weighted space. The theoretical results are numerically supported. Full article
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11 pages, 1731 KiB  
Article
Parameter Identification for Lithium-Ion Battery Based on Hybrid Genetic–Fractional Beetle Swarm Optimization Method
by Peng Guo, Xiaobo Wu, António M. Lopes, Anyu Cheng, Yang Xu and Liping Chen
Mathematics 2022, 10(17), 3056; https://0-doi-org.brum.beds.ac.uk/10.3390/math10173056 - 24 Aug 2022
Cited by 2 | Viewed by 1003
Abstract
This paper proposes a fractional order (FO) impedance model for lithium-ion batteries and a method for model parameter identification. The model is established based on electrochemical impedance spectroscopy (EIS). A new hybrid genetic–fractional beetle swarm optimization (HGA-FBSO) scheme is derived for parameter identification, [...] Read more.
This paper proposes a fractional order (FO) impedance model for lithium-ion batteries and a method for model parameter identification. The model is established based on electrochemical impedance spectroscopy (EIS). A new hybrid genetic–fractional beetle swarm optimization (HGA-FBSO) scheme is derived for parameter identification, which combines the advantages of genetic algorithms (GA) and beetle swarm optimization (BSO). The approach leads to an equivalent circuit model being able to describe accurately the dynamic behavior of the lithium-ion battery. Experimental results illustrate the effectiveness of the proposed method, yielding voltage estimation root-mean-squared error (RMSE) of 10.5 mV and mean absolute error (MAE) of 0.6058%. This corresponds to accuracy improvements of 32.26% and 7.89% for the RMSE, and 43.83% and 13.67% for the MAE, when comparing the results of the new approach to those obtained with the GA and the FBSO methods, respectively. Full article
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15 pages, 828 KiB  
Article
A Fractional-Order Compartmental Model of Vaccination for COVID-19 with the Fear Factor
by Amar Nath Chatterjee, Fahad Al Basir, Bashir Ahmad and Ahmed Alsaedi
Mathematics 2022, 10(9), 1451; https://0-doi-org.brum.beds.ac.uk/10.3390/math10091451 - 26 Apr 2022
Cited by 12 | Viewed by 1575
Abstract
During the past several years, the deadly COVID-19 pandemic has dramatically affected the world; the death toll exceeds 4.8 million across the world according to current statistics. Mathematical modeling is one of the critical tools being used to fight against this deadly infectious [...] Read more.
During the past several years, the deadly COVID-19 pandemic has dramatically affected the world; the death toll exceeds 4.8 million across the world according to current statistics. Mathematical modeling is one of the critical tools being used to fight against this deadly infectious disease. It has been observed that the transmission of COVID-19 follows a fading memory process. We have used the fractional order differential operator to identify this kind of disease transmission, considering both fear effects and vaccination in our proposed mathematical model. Our COVID-19 disease model was analyzed by considering the Caputo fractional operator. A brief description of this operator and a mathematical analysis of the proposed model involving this operator are presented. In addition, a numerical simulation of the proposed model is presented along with the resulting analytical findings. We show that fear effects play a pivotal role in reducing infections in the population as well as in encouraging the vaccination campaign. Furthermore, decreasing the fractional-order parameter α value minimizes the number of infected individuals. The analysis presented here reveals that the system switches its stability for the critical value of the basic reproduction number R0=1. Full article
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14 pages, 2694 KiB  
Article
New Type Modelling of the Circumscribed Self-Excited Spherical Attractor
by Mohammad Partohaghighi, Ali Akgül and Rubayyi T. Alqahtani
Mathematics 2022, 10(5), 732; https://0-doi-org.brum.beds.ac.uk/10.3390/math10050732 - 25 Feb 2022
Cited by 4 | Viewed by 989
Abstract
The fractal–fractional derivative with the Mittag–Leffler kernel is employed to design the fractional-order model of the new circumscribed self-excited spherical attractor, which is not investigated yet by fractional operators. Moreover, the theorems of Schauder’s fixed point and Banach fixed existence theory are used [...] Read more.
The fractal–fractional derivative with the Mittag–Leffler kernel is employed to design the fractional-order model of the new circumscribed self-excited spherical attractor, which is not investigated yet by fractional operators. Moreover, the theorems of Schauder’s fixed point and Banach fixed existence theory are used to guarantee that there are solutions to the model. Approximate solutions to the problem are presented by an effective method. To prove the efficiency of the given technique, different values of fractal and fractional orders as well as initial conditions are selected. Figures of the approximate solutions are provided for each case in different dimensions. Full article
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21 pages, 1000 KiB  
Article
Dynamical Analysis of Bio-Ethanol Production Model under Generalized Nonlocal Operator in Caputo Sense
by Rubayyi T. Alqahtani, Shabir Ahmad and Ali Akgül
Mathematics 2021, 9(19), 2370; https://0-doi-org.brum.beds.ac.uk/10.3390/math9192370 - 24 Sep 2021
Cited by 27 | Viewed by 1250
Abstract
The nonlinear fractional-order model of bioethanol production under a generalized nonlocal operator in the Caputo sense is investigated in this work. Theoretical and computational aspects of the considered model are discussed. We prove that the model has at least one solution and a [...] Read more.
The nonlinear fractional-order model of bioethanol production under a generalized nonlocal operator in the Caputo sense is investigated in this work. Theoretical and computational aspects of the considered model are discussed. We prove that the model has at least one solution and a unique solution using the Leray–Schauder and Banach contraction theorems. Using functional analysis, we investigate several types of Ulam–Hyres model stability. We use the predictor–corrector (P–C) method to construct a broad numerical scheme for the model’s solution. The proposed numerical method’s stability is demonstrated. Finally, we depict the numerical findings geometrically to demonstrate the model’s dynamics. Full article
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Review

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12 pages, 297 KiB  
Review
From Wavelet Analysis to Fractional Calculus: A Review
by Emanuel Guariglia, Rodrigo C. Guido and Gabriel J. P. Dalalana
Mathematics 2023, 11(7), 1606; https://0-doi-org.brum.beds.ac.uk/10.3390/math11071606 - 26 Mar 2023
Cited by 8 | Viewed by 1679
Abstract
In this note, we review some important results on wavelets, together with their main applications. Similarly, we present the main results on fractional calculus and their current applications in pure and applied science. We conclude the paper showing the close interconnection between wavelet [...] Read more.
In this note, we review some important results on wavelets, together with their main applications. Similarly, we present the main results on fractional calculus and their current applications in pure and applied science. We conclude the paper showing the close interconnection between wavelet analysis and fractional calculus. Full article
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