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Fractal Fract
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Fractional Mathematical Modelling: Theory, Methods and Applications
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Fractional Mathematical Modelling: Theory, Methods and Applications

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Numerical and Computational Methods".

Deadline for manuscript submissions: 30 September 2024 | Viewed by 6608

Special Issue Editors

Dr. Faranak Rabiei

E-Mail
Guest Editor
Department of Mathematics, Texas A & M University Kingsville, 700 University Blvd, Kingsville, TX 78363, USA
Interests: numerical analysis; computational mathematics; fractional calculus; mathematical biological modeling; differential equations; stability analysis
Dr. Dongwook Kim

E-Mail Website
Guest Editor
Department of Mathematics, Texas A & M University Kingsville, 700 University Blvd, Kingsville, TX 78363, USA
Interests: differential equations; numerical analysis; mathematical biology modeling; nonlinear stability analysis; computational mathematics; fractional calculus
Dr. Zeeshan Ali

E-Mail Website
Guest Editor
School of Engineering, Monash University Malaysia, Selangor 47500, Malaysia
Interests: mathematical modeling; fractional calculus; mathematical biology; nonlinear analysis; computational mathematics; existence theory; stability analysis

Special Issue Information

Dear Colleagues,

The tools of fractional calculus serve as a new resource that is applicable in fields, including physics, fluid mechanics, hydrology, material science, signal processing, engineering, chemistry, biology, medicine, finance, and social sciences. This Special Issue aims to highlight the recent advancements in fractional calculus theory, innovative methodologies, and potential applications. We specifically invite authors to submit high-quality research that delves into the analysis of fractional differential/integral equations, the exploration of new definitions for fractional derivatives, the development of numerical methods to solve fractional equations, and the examination of applications in physical systems governed by fractional differential equations. The scope extends to include various other captivating research topics as well.

Dr. Faranak Rabiei
Dr. Dongwook Kim
Dr. Zeeshan Ali
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. Fractal and Fractional 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 2700 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 differential and integral equations
  • new fractional operators and their properties
  • existence and uniqueness of solutions
  • analytical and numerical methods
  • stability analysis
  • fractional calculus in physics
  • fractional dynamics in complex systems
  • applications to science and engineering

Published Papers (5 papers)

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Research

19 pages, 346 KiB  
Article
Controllability of Mild Solution to Hilfer Fuzzy Fractional Differential Inclusion with Infinite Continuous Delay
by Aeshah Abdullah Muhammad Al-Dosari
Fractal Fract. 2024, 8(4), 235; https://doi.org/10.3390/fractalfract8040235 - 17 Apr 2024
Viewed by 482
Abstract
This work investigates the solvability of the generalized Hilfer fractional inclusion associated with the solution set of a controlled system of minty type–fuzzy mixed quasi-hemivariational inequality (FMQHI). We explore the assumed inclusion via the infinite delay and the semi-group arguments in the area [...] Read more.
This work investigates the solvability of the generalized Hilfer fractional inclusion associated with the solution set of a controlled system of minty type–fuzzy mixed quasi-hemivariational inequality (FMQHI). We explore the assumed inclusion via the infinite delay and the semi-group arguments in the area of solid continuity that sculpts the compactness area. The conformable Hilfer fractional time derivative, the theory of fuzzy sets, and the infinite delay arguments support the solution set’s controllability. We explain the existence due to the convergence properties of Mittage–Leffler functions (Eα,β), that is, hatching the existing arguments according to FMQHI and the continuity of infinite delay, which has not been presented before. To prove the main results, we apply the Leray–Schauder nonlinear alternative thereom in the interpolation of Banach spaces. This problem seems to draw new extents on the controllability field of stochastic dynamic models. Full article
(This article belongs to the Special Issue Fractional Mathematical Modelling: Theory, Methods and Applications)
19 pages, 4410 KiB  
Article
Modeling and Analysis of Caputo–Fabrizio Definition-Based Fractional-Order Boost Converter with Inductive Loads
by Donghui Yu, Xiaozhong Liao and Yong Wang
Fractal Fract. 2024, 8(2), 81; https://doi.org/10.3390/fractalfract8020081 - 26 Jan 2024
Viewed by 1002
Abstract
This paper proposes a modeling and analysis method for a Caputo–Fabrizio (C-F) definition-based fractional-order Boost converter with fractional-order inductive loads. The proposed method analyzes the system characteristics of a fractional-order circuit with three state variables. Firstly, this paper constructs a large signal model [...] Read more.
This paper proposes a modeling and analysis method for a Caputo–Fabrizio (C-F) definition-based fractional-order Boost converter with fractional-order inductive loads. The proposed method analyzes the system characteristics of a fractional-order circuit with three state variables. Firstly, this paper constructs a large signal model of a fractional-order Boost converter by taking advantage of the state space averaging method, providing accurate analytical solutions for the quiescent operating point and the ripple parameters of the circuit with three state variables. Secondly, this paper constructs a small signal model of the C-F definition-based fractional-order Boost converter by small signal linearization, providing the transfer function of the fractional-order system with three state variables. Finally, this paper conducts circuit-oriented simulation experiments where the steady-state parameters and the transfer function of the circuit are obtained, and then the effect of the order of capacitor, induced inductor, and load inductor on the quiescent operating point and ripple parameters is analyzed. The experimental results show that the simulation results are consistent with those obtained by the proposed mathematical model and that the three fractional orders in the fractional model with three state variables have a significant impact on the DC component and steady-state characteristics of the fractional-order Boost converter. In conclusion, the proposed mathematical model can more comprehensively analyze the system characteristics of the C-F definition-based fractional-order Boost converter with fractional-order inductive loads, benefiting the circuit design of Boost converters. Full article
(This article belongs to the Special Issue Fractional Mathematical Modelling: Theory, Methods and Applications)
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12 pages, 2379 KiB  
Article
Fractional Photoconduction and Nonlinear Optical Behavior in ZnO Micro and Nanostructures
by Victor Manuel Garcia-de-los-Rios, Jose Alberto Arano-Martínez, Martin Trejo-Valdez, Martha Leticia Hernández-Pichardo, Mónica Araceli Vidales-Hurtado and Carlos Torres-Torres
Fractal Fract. 2023, 7(12), 885; https://doi.org/10.3390/fractalfract7120885 - 15 Dec 2023
Viewed by 1149
Abstract
A fractional description for the optically induced mechanisms responsible for conductivity and multiphotonic effects in ZnO nanomaterials is studied here. Photoconductive, electrical, and nonlinear optical phenomena exhibited by pure micro and nanostructured ZnO samples were analyzed. A hydrothermal approach was used to synthetize [...] Read more.
A fractional description for the optically induced mechanisms responsible for conductivity and multiphotonic effects in ZnO nanomaterials is studied here. Photoconductive, electrical, and nonlinear optical phenomena exhibited by pure micro and nanostructured ZnO samples were analyzed. A hydrothermal approach was used to synthetize ZnO micro-sized crystals, while a spray pyrolysis technique was employed to prepare ZnO nanostructures. A contrast in the fractional electrical behavior and photoconductivity was identified for the samples studied. A positive nonlinear refractive index was measured on the nanoscale sample using the z-scan technique, which endows it with a dominant real part for the third-order optical nonlinearity. The absence of nonlinear optical absorption, along with a strong optical Kerr effect in the ZnO nanostructures, shows favorable perspectives for their potential use in the development of all-optical switching devices. Fractional models for predicting electronic and nonlinear interactions in nanosystems could pave the way for the development of optoelectronic circuits and ultrafast functions controlled by ZnO photo technology. Full article
(This article belongs to the Special Issue Fractional Mathematical Modelling: Theory, Methods and Applications)
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11 pages, 533 KiB  
Article
Application of Riemann–Liouville Derivatives on Second-Order Fractional Differential Equations: The Exact Solution
by Abdulrahman B. Albidah
Fractal Fract. 2023, 7(12), 843; https://doi.org/10.3390/fractalfract7120843 - 28 Nov 2023
Viewed by 938
Abstract
This paper applies two different types of Riemann–Liouville derivatives to solve fractional differential equations of second order. Basically, the properties of the Riemann–Liouville fractional derivative depend mainly on the lower bound of the integral involved in the Riemann–Liouville fractional definition. The Riemann–Liouville fractional [...] Read more.
This paper applies two different types of Riemann–Liouville derivatives to solve fractional differential equations of second order. Basically, the properties of the Riemann–Liouville fractional derivative depend mainly on the lower bound of the integral involved in the Riemann–Liouville fractional definition. The Riemann–Liouville fractional derivative of first type considers the lower bound as a zero while the second type applies negative infinity as a lower bound. Due to the differences in properties of the two operators, two different solutions are obtained for the present two classes of fractional differential equations under appropriate initial conditions. It is shown that the zeroth lower bound implies implicit solutions in terms of the Mittag–Leffler functions while explicit solutions are derived when negative infinity is taken as a lower bound. Such explicit solutions are obtained for the current two classes in terms of trigonometric and hyperbolic functions. Some theoretical results are introduced to facilitate the solutions procedures. Moreover, the characteristics of the obtained solutions are discussed and interpreted. Full article
(This article belongs to the Special Issue Fractional Mathematical Modelling: Theory, Methods and Applications)
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23 pages, 1284 KiB  
Article
Fractional-Order Zener Model with Temperature-Order Equivalence for Viscoelastic Dampers
by Kang Xu, Liping Chen, António M. Lopes, Mingwu Wang, Ranchao Wu and Min Zhu
Fractal Fract. 2023, 7(10), 714; https://doi.org/10.3390/fractalfract7100714 - 28 Sep 2023
Viewed by 802
Abstract
Viscoelastic (VE) dampers show good performance in dissipating energy, being widely used for reducing vibration in engineering structures caused by earthquakes and winds. Experimental studies have shown that ambient temperature has great influence on the mechanical behavior of VE dampers. Therefore, it is [...] Read more.
Viscoelastic (VE) dampers show good performance in dissipating energy, being widely used for reducing vibration in engineering structures caused by earthquakes and winds. Experimental studies have shown that ambient temperature has great influence on the mechanical behavior of VE dampers. Therefore, it is important to accurately model VE dampers considering the effect of temperature. In this paper, a new fractional-order Zener (AEF-Zener) model of VE dampers is proposed. Firstly, the important influence of fractional orders on the energy dissipation ability of materials is analyzed. Secondly, an equivalent AEF-Zener model is developed that incorporates the ambient temperature and fractional-order equivalence principle. Finally, the chaotic fractional-order particle swarm optimization (CFOPSO) algorithm is used to determine the model’s parameters. The accuracy of the AEF-Zener model is verified by comparing model simulations with experimental results. This study is helpful for designing and analyzing vibration reduction techniques for civil structures with VE dampers under the influence of temperature. Full article
(This article belongs to the Special Issue Fractional Mathematical Modelling: Theory, Methods and Applications)
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Planned Papers

The below list represents only planned manuscripts. Some of these manuscripts have not been received by the Editorial Office yet. Papers submitted to MDPI journals are subject to peer-review.

Title: Fractal-Fractional Third-Order Adams-Bashforth Method

Author: Zeeshan Ali, Faranak Rabiei, Dongwook Kim et al.

Abstract
Fractal-fractional order differential and integral operators have recently been proposed and proven to be an effective mathematical tool for modelling complex real-world problems that could not be modelled with classical and nonlocal differential and integral operators of a single order. Therefore, numerous researchers are trying to propose a numerical technique to solve the mentioned above problems. This paper proposes a new numerical method called the fractal-fractional third-order Adams-Bashforth (AB3) technique to solve the differential equations of fractal-fractional order. Error analysis and convergence of proposed method are discussed and proved. Several test problems, including linear and nonlinear systems of the fractal-fractional order differential equations are tested with various numerical methods from other studies to validate the method reliability and efficiency.
Keywords: Fractal-fractional derivatives; Fractal-fractional order Adams-Bashforth method, Convergence analysis

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