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Fractal Fract
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Fractional Derivatives and Their Applications to Fractional Diffusion Problems
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Fractional Derivatives and Their Applications to Fractional Diffusion Problems

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: closed (1 June 2023) | Viewed by 7405

Special Issue Editors

Dr. Mohammed S. Abdo

E-Mail Website
Guest Editor
Department of Mathematics, College of Education, Hodeidah University, Al-Hudaydah 207416, Yemen
Interests: fractional calculus; theory of fractional differential equations; nonlinear dynamics; mathematical models
Dr. Khaled M. Saad

E-Mail Website
Guest Editor
1. Department of Mathematics, College of Arts and Sciences, Najran University, Najran P.O. Box 1988, Saudi Arabia
2. Department of Mathematics, Faculty of Applied Science, Taiz University, Taiz P.O. Box 6803, Yemen
Interests: special functions; numerical of fractional differential equations

Special Issue Information

Dear Colleagues,

Fractional calculus is an incredible tool for understanding the complex world. The significance of fractional calculus has been shown to be exceptionally compelling in various phenomena, such as diffusion processes, viscoelastics, long-range interactions, materials, and so on. It turns out that fractional calculus provides numerous advantageous features that offer interesting solutions to system modeling and control, optimization algorithm design, and machine learning. Fractional diffusion systems are fundamental mathematical systems for the evolution of probability densities.

The motivation behind this Special Issue is to introduce an assortment of articles showing recent developments and results that rely on the frame of fractional derivatives and their applications to fractional diffusion equations and other fractional differential equations. Topics that are invited for submission include (but are not limited to):

  • Fractional order systems;
  • Existence and Stability analysis of fractional order systems;
  • Numerical methods for fractional diffusion equations;
  • Analytical solutions of fractional diffusion equations;
  • Modeling in the frame of fractional derivatives.

Dr. Mohammed S. Abdo
Dr. Khaled M. Saad
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 derivatives
  • existence and stability analysis
  • fractional diffusion systems
  • analytical and numerical methods
  • modeling
  • applications

Published Papers (6 papers)

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Research

21 pages, 1494 KiB  
Article
On the Analysis of a Fractional Tuberculosis Model with the Effect of an Imperfect Vaccine and Exogenous Factors under the Mittag–Leffler Kernel
by Saeed Ahmad, Sedat Pak, Mati ur Rahman and Afrah Al-Bossly
Fractal Fract. 2023, 7(7), 526; https://doi.org/10.3390/fractalfract7070526 - 03 Jul 2023
Cited by 6 | Viewed by 1098
Abstract
This research study aims to investigate the effects of vaccination on reducing disease burden by analyzing a complex nonlinear ordinary differential equation system. The study focuses on five distinct sub-classes within the system to comprehensively explore the impact of vaccination. Specifically, the mathematical [...] Read more.
This research study aims to investigate the effects of vaccination on reducing disease burden by analyzing a complex nonlinear ordinary differential equation system. The study focuses on five distinct sub-classes within the system to comprehensively explore the impact of vaccination. Specifically, the mathematical model employed in this investigation is a fractional representation of tuberculosis, utilizing the Atangana–Baleanu fractional derivative in the Caputo sense. The validity of the proposed model is established through a rigorous qualitative analysis. The existence and uniqueness of the solution are rigorously determined by applying the fundamental theorems of the fixed point approach. The stability analysis of the model is conducted using the Ulam–Hyers approach. Additionally, the study employs the widely recognized iterative Adams–Bashforth technique to obtain an approximate solution for the suggested model. The numerical simulation of the tuberculosis model is comprehensively discussed, with a particular focus on the assumptions made regarding vaccination. The model assumes that only a limited portion of the population is vaccinated at a steady rate, and the efficacy of the vaccine is a critical factor in reducing disease burden. The findings of the study indicate that the proposed model can effectively assess the impact of vaccination on mitigating the spread of tuberculosis. Furthermore, the numerical simulation underscores the significance of vaccination as an effective control measure against tuberculosis. Full article
(This article belongs to the Special Issue Fractional Derivatives and Their Applications to Fractional Diffusion Problems)
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20 pages, 393 KiB  
Article
Existence of Positive Solutions for a Coupled System of p-Laplacian Semipositone Hadmard Fractional BVP
by Sabbavarapu Nageswara Rao, Manoj Singh, Ahmed Hussein Msmali and Abdullah Ali H. Ahmadini
Fractal Fract. 2023, 7(7), 499; https://doi.org/10.3390/fractalfract7070499 - 23 Jun 2023
Viewed by 611
Abstract
The existence of a positive solution to a system of nonlinear semipositone Hadamard fractional BVP with the p-Laplacian operator is examined in this research. The boundary value problem’s associated Green’s function and some of its properties are first obtained. Additionally, the existence [...] Read more.
The existence of a positive solution to a system of nonlinear semipositone Hadamard fractional BVP with the p-Laplacian operator is examined in this research. The boundary value problem’s associated Green’s function and some of its properties are first obtained. Additionally, the existence results are established using the nonlinear alternative of the Leray–Schauder theorem and the Guo–Krasnosel’skii fixed-point theorem. Full article
(This article belongs to the Special Issue Fractional Derivatives and Their Applications to Fractional Diffusion Problems)
18 pages, 496 KiB  
Article
Block-Centered Finite-Difference Methods for Time-Fractional Fourth-Order Parabolic Equations
by Taixiu Zhang, Zhe Yin and Ailing Zhu
Fractal Fract. 2023, 7(6), 471; https://doi.org/10.3390/fractalfract7060471 - 14 Jun 2023
Viewed by 1026
Abstract
The block-centered finite-difference method has many advantages, and the time-fractional fourth-order equation is widely used in physics and engineering science. In this paper, we consider variable-coefficient fourth-order parabolic equations of fractional-order time derivatives with Neumann boundary conditions. The fractional-order time derivatives are approximated [...] Read more.
The block-centered finite-difference method has many advantages, and the time-fractional fourth-order equation is widely used in physics and engineering science. In this paper, we consider variable-coefficient fourth-order parabolic equations of fractional-order time derivatives with Neumann boundary conditions. The fractional-order time derivatives are approximated by L1 interpolation. We propose the block-centered finite-difference scheme for fourth-order parabolic equations with fractional-order time derivatives. We prove the stability of the block-centered finite-difference scheme and the second-order convergence of the discrete L2 norms of the approximate solution and its derivatives of every order. Numerical examples are provided to verify the effectiveness of the block-centered finite-difference scheme. Full article
(This article belongs to the Special Issue Fractional Derivatives and Their Applications to Fractional Diffusion Problems)
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21 pages, 475 KiB  
Article
A Cotangent Fractional Derivative with the Application
by Lakhlifa Sadek
Fractal Fract. 2023, 7(6), 444; https://doi.org/10.3390/fractalfract7060444 - 30 May 2023
Cited by 9 | Viewed by 1210
Abstract
In this work, we present a new type of fractional derivatives (FD) involving exponential cotangent function in their kernels called Riemann–Liouville Dσ,γ and Caputo cotangent fractional derivatives CDσ,γ, respectively, and their corresponding integral [...] Read more.
In this work, we present a new type of fractional derivatives (FD) involving exponential cotangent function in their kernels called Riemann–Liouville Dσ,γ and Caputo cotangent fractional derivatives CDσ,γ, respectively, and their corresponding integral Iσ,γ. The advantage of the new fractional derivatives is that they achieve a semi-group property, and we have special cases; if γ=1 we obtain the Riemann–Liouville FD (RL-FD), Caputo FD (C-FD), and Riemann–Liouville fractional integral (RL-FI). We give some theorems and lemmas, and we give solutions to linear cotangent fractional differential equations using the Laplace transform of the Dσ,γ, CDσ,γ and Iσ,γ. Finally, we give the application of this new type on the SIR model. This new type of fractional calculus can help other researchers who still work on the actual subject. Full article
(This article belongs to the Special Issue Fractional Derivatives and Their Applications to Fractional Diffusion Problems)
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17 pages, 358 KiB  
Article
Nonlinear Piecewise Caputo Fractional Pantograph System with Respect to Another Function
by Mohammed S. Abdo, Wafa Shammakh, Hadeel Z. Alzumi, Najla Alghamd and M. Daher Albalwi
Fractal Fract. 2023, 7(2), 162; https://doi.org/10.3390/fractalfract7020162 - 06 Feb 2023
Cited by 6 | Viewed by 1043
Abstract
The existence, uniqueness, and various forms of Ulam–Hyers (UH)-type stability results for nonlocal pantograph equations are developed and extended in this study within the frame of novel psi-piecewise Caputo fractional derivatives, which generalize the piecewise operators recently presented in the [...] Read more.
The existence, uniqueness, and various forms of Ulam–Hyers (UH)-type stability results for nonlocal pantograph equations are developed and extended in this study within the frame of novel psi-piecewise Caputo fractional derivatives, which generalize the piecewise operators recently presented in the literature. The required results are proven using Banach’s contraction mapping and Krasnoselskii’s fixed-point theorem. Additionally, results pertaining to UH stability are obtained using traditional procedures of nonlinear functional analysis. Additionally, in light of our current findings, a more general challenge for the pantograph system is presented that includes problems similar to the one considered. We provide a pertinent example as an application to support the theoretical findings. Full article
(This article belongs to the Special Issue Fractional Derivatives and Their Applications to Fractional Diffusion Problems)
14 pages, 302 KiB  
Article
Topological Structure of Solution Sets of Fractional Control Delay Problem
by Ahmed A. Al Ghafli, Ramsha Shafqat, Azmat Ullah Khan Niazi, Kinda Abuasbeh and Muath Awadalla
Fractal Fract. 2023, 7(1), 59; https://doi.org/10.3390/fractalfract7010059 - 03 Jan 2023
Cited by 3 | Viewed by 1068
Abstract
This paper is concerned with the existence of a mild solution for the fractional delay control system. Firstly, we will study the control problem. Then, we will deal with the topological structure of the solution set consisting of the compactness and Rσ [...] Read more.
This paper is concerned with the existence of a mild solution for the fractional delay control system. Firstly, we will study the control problem. Then, we will deal with the topological structure of the solution set consisting of the compactness and Rσ property. We will derive a mild solution to the above delay control problem by using the Laplace transform method. Full article
(This article belongs to the Special Issue Fractional Derivatives and Their Applications to Fractional Diffusion Problems)
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