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Fractional Systems, Integrals and Derivatives: Theory and Application

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

Deadline for manuscript submissions: 30 June 2025 | Viewed by 11216

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Special Issue Editors

Prof. Dr. Andrey Zahariev

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Guest Editor

Faculty of Mathematics and Informatics, University of Plovdiv, Plovdiv 4000, Bulgaria
Interests: fractional differential equations; functional-differential equations; impulsive differential equations; differential equations in Banach spaces; integral equations; integral inequalities; stability analysis; real and functional analysis; applied mathematics
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Prof. Dr. Hristo Kiskinov

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Guest Editor

Faculty of Mathematics and Informatics, University of Plovdiv, Plovdiv 4000, Bulgaria
Interests: fractional differential equations; impulsive differential equations; functional-differential equations; differential equations in Banach spaces
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The present Special Issue is dedicated to new research in fractional calculus related to fractional integrals and derivatives. Our main interest is focused on works devoted to fractional differential equations or systems with different types of fractional orders: constant, variable or distributed types; as well as fractional equation or systems without delay; or those with delayed argument (retarded or neutral types). Articles devoted to partial fractional and stochastic differential equations with fractional derivatives are welcome too. The works can contain results concerning the existence and/or uniqueness of the solutions, different kinds of integral representations of these solutions, and their continuous dependence from the given data. Qualitative results concerning the asymptotic behaviour of solutions, and various types of stability properties, namely Lyapunov’s type, finite time stability, Mittag–Leffler stability, robust stability, Ulam–Hyers–Rassias stability, and so on, are also invited.

In addition, articles containing new applications, including fractional variants of well-known classical models in different scientific areas as economics, physics, engineering, and so on, are also welcome. We also encourage works that compare the advantages and disadvantages of fractional derivatives with integrable singular kernels, and those with regularized kernels calculating not only the mathematical (technical) convenience but also convenience from an applicability point of view, namely fractional models with which type kernels give a more adequate description of the dynamics of the studied physical or economical real-world phenomena.

Prof. Dr. Andrey Zahariev
Prof. Dr. Hristo Kiskinov
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 of constant order
distributed order fractional derivative
variable order fractional derivative
fractional differential equations/systems
functional-differential equations/systems
stability analysis of fractional-order systems

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Published Papers (12 papers)

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Research

26 pages, 1846 KiB

Open AccessArticle

Analytical Techniques for Studying Fractional-Order Jaulent–Miodek System Within Algebraic Context

by Yousuf Alkhezi and Ahmad Shafee

Fractal Fract. 2025, 9(1), 50; https://doi.org/10.3390/fractalfract9010050 - 17 Jan 2025

Viewed by 590

Abstract

The proposed study seeks to investigate various analytical and numerical techniques for solving fractional differential equations, with a particular focus on their applications in mathematical modeling and scientific research within the field of algebra. This study intends to investigate methods such as the Aboodh transform iteration method and the Aboodh residual power series method, specifically for addressing the Jaulent–Miodek system of partial differential equations. By analyzing the behavior of fractional-order differential equations and their solutions, this research seeks to contribute to a deeper understanding of complex mathematical phenomena. Furthermore, this study examines the role of the Caputo operator in fractional calculus, offering insights into its significance in modeling real-world systems within the algebraic context. Through this research, novel approaches for solving fractional differential equations are developed, offering essential tools for researchers in diverse fields of science and engineering, including algebraic applications. Full article

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)

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20 pages, 620 KiB

Open AccessArticle

Self-Convolution and Its Invariant Properties for the Kernel Function of the Aortic Fractal Operator

by Chaoqian Luo, Yajun Yin, Gang Peng, Tianyi Zhou, Xiaobin Yu and Dongan Li

Fractal Fract. 2024, 8(12), 726; https://doi.org/10.3390/fractalfract8120726 - 11 Dec 2024

Viewed by 507

Abstract

In this paper, we explore the self-convolution of the kernel function of the aortic fractal operator. Previous research has established a model named “physical fractal”, and confirmed that the hemodynamics of the aorta can be inscribed by a fractal operator and that the dominant component of the kernel function of the fractal operator is a weighted first-order Bessel function. These studies primarily focus on solving the fractal operator kernel function and examining the overall properties of the physical fractal. As we began to investigate the internal structure of physical fractals, we discovered that studying the powers of fractal operators is a necessary step. In this paper, we introduce the concept of kernel function self-convolution, establish its connection with the power of the fractal operator, and derive a series of invariant properties for the self-convolution of the aortic operator kernel function. These invariant properties, in turn, are deeply and intrinsically related to the invariant properties of the Bessel functions. The research findings of this paper enrich hemodynamics and biomechanics in physical fractal space and extend the scope of using fractal operators to characterize the dynamics of living organisms. Full article

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)

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26 pages, 478 KiB

Open AccessArticle

Modeling Ebola Dynamics with a Φ-Piecewise Hybrid Fractional Derivative Approach

by Tariq Alraqad, Mohammed A. Almalahi, Naglaa Mohammed, Ayman Alahmade, Khaled A. Aldwoah and Hicham Saber

Fractal Fract. 2024, 8(10), 596; https://doi.org/10.3390/fractalfract8100596 - 11 Oct 2024

Cited by 3 | Viewed by 841

Abstract

Ebola virus disease (EVD) is a severe and often fatal illness posing significant public health challenges. This study investigates EVD transmission dynamics using a novel fractional mathematical model with five distinct compartments: individuals with low susceptibility (

S_{1}

), individuals with high [...] Read more.

S_{1}

), individuals with high susceptibility (

S_{2}

), infected individuals (

I

), exposed individuals (

E

), and recovered individuals (

R

). To capture the complex dynamics of EVD, we employ a

Φ

-piecewise hybrid fractional derivative approach. We investigate the crossover effect and its impact on disease dynamics by dividing the study interval into two subintervals and utilize the

Φ

-Caputo derivative in the first interval and the

Φ

-ABC derivative in the second interval. The study determines the basic reproduction number

R_{0}

, analyzes the stability of the disease-free equilibrium and investigates the sensitivity of the parameters to understand how variations affect the system’s behavior and outcomes. Numerical simulations support the model and demonstrate consistent results with the theoretical analysis, highlighting the importance of fractional calculus in modeling infectious diseases. This research provides valuable information for developing effective control strategies to combat EVD. Full article

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)

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17 pages, 3536 KiB

Open AccessArticle

Exploring Soliton Solutions and Chaotic Dynamics in the (3+1)-Dimensional Wazwaz–Benjamin–Bona–Mahony Equation: A Generalized Rational Exponential Function Approach

by Amjad E. Hamza, Muntasir Suhail, Amer Alsulami, Alaa Mustafa, Khaled Aldwoah and Hicham Saber

Fractal Fract. 2024, 8(10), 592; https://doi.org/10.3390/fractalfract8100592 - 9 Oct 2024

Cited by 2 | Viewed by 1121

Abstract

This paper investigates the explicit, accurate soliton and dynamic strategies in the resolution of the Wazwaz–Benjamin–Bona–Mahony (WBBM) equations. By exploiting the ensuing wave events, these equations find applications in fluid dynamics, ocean engineering, water wave mechanics, and scientific inquiry. The two main goals of the study are as follows: Firstly, using the dynamic perspective, examine the chaos, bifurcation, Lyapunov spectrum, Poincaré section, return map, power spectrum, sensitivity, fractal dimension, and other properties of the governing equation. Secondly, we use a generalized rational exponential function (GREF) technique to provide a large number of analytical solutions to nonlinear partial differential equations (NLPDEs) that have periodic, trigonometric, and hyperbolic properties. We examining the wave phenomena using 2D and 3D diagrams along with a projection of contour plots. Through the use of the computational program Mathematica, the research confirms the computed solutions to the WBBM equations. Full article

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)

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14 pages, 328 KiB

Open AccessArticle

Oscillatory and Asymptotic Criteria for a Fifth-Order Fractional Difference Equation

by Qinghua Feng

Fractal Fract. 2024, 8(10), 590; https://doi.org/10.3390/fractalfract8100590 - 7 Oct 2024

Viewed by 784

Abstract

In this paper, using the properties of the conformable fractional difference and fractional sum, we initially establish some oscillatory and asymptotic criteria for a fifth-order fractional difference equation. Several critical inequalities, the Riccati transformation technique, and the integral technique are used in the deduction process. We provide some example to test the results. The established criteria are new results in the study of oscillation, and can be extended to other types of high-order fractional difference equations as well as fractional differential equations with more complicated forms. Full article

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)

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18 pages, 2623 KiB

Open AccessArticle

Design and Applicability of Two-Step Fractional Newton–Raphson Method

by Naseem Zulfiqar Ali, Awais Gul Khan, Muhammad Uzair Awan, Loredana Ciurdariu and Kamel Brahim

Fractal Fract. 2024, 8(10), 582; https://doi.org/10.3390/fractalfract8100582 - 2 Oct 2024

Viewed by 1135

Abstract

Developing two-step fractional numerical methods for finding the solution of nonlinear equations is the main objective of this research article. In addition, we present a detailed study of convergence analysis for the methods that have been proposed. By comparing numerically, we can see that the proposed methods significantly improve convergence rate and accuracy. Additionally, we demonstrate how our main results can be applied to basins of attraction. Full article

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)

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22 pages, 343 KiB

Open AccessArticle

Novel Ostrowski–Type Inequalities for Generalized Fractional Integrals and Diverse Function Classes

by Areej A. Almoneef, Abd-Allah Hyder, Mohamed A. Barakat and Hüseyin Budak

Fractal Fract. 2024, 8(9), 534; https://doi.org/10.3390/fractalfract8090534 - 13 Sep 2024

Viewed by 647

Abstract

In this work, novel Ostrowski-type inequalities for dissimilar function classes and generalized fractional integrals (FITs) are presented. We provide a useful identity for differentiable functions under FITs, which results in special expressions for functions whose derivatives have convex absolute values. A new condition for bounded variation functions is examined, as well as expansions to bounded and Lipschitzian derivatives. Our comprehension is improved by comparison with current findings, and recommendations for future study areas are given. Full article

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)

23 pages, 2648 KiB

Open AccessArticle

On the Numerical Investigations of a Fractional-Order Mathematical Model for Middle East Respiratory Syndrome Outbreak

by Faisal E. Abd Alaal, Adel R. Hadhoud, Ayman A. Abdelaziz and Taha Radwan

Fractal Fract. 2024, 8(9), 521; https://doi.org/10.3390/fractalfract8090521 - 4 Sep 2024

Viewed by 771

Abstract

Middle East Respiratory Syndrome (MERS) is a human coronavirus subtype that poses a significant public health concern due to its ability to spread between individuals. This research aims to develop a fractional-order mathematical model to investigate the MERS pandemic and to subsequently develop two numerical methods to solve this model numerically to evaluate and comprehend the analysis results. The fixed-point theorem has been used to demonstrate the existence and uniqueness of the solution to the suggested model. We approximate the solutions of the proposed model using two numerical methods: the mean value theorem and the implicit trapezoidal method. The stability of these numerical methods is studied using various results and primary lemmas. Finally, we compare the results of our methods to demonstrate their efficiency and conduct a numerical simulation of the obtained results. A comparative study based on real data from Riyadh, Saudi Arabia is provided. The study’s conclusions demonstrate the computational efficiency of our approaches in studying nonlinear fractional differential equations that arise in daily life problems. Full article

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)

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23 pages, 6260 KiB

Open AccessArticle

Interaction Solutions for the Fractional KdVSKR Equations in (1+1)-Dimension and (2+1)-Dimension

by Lihua Zhang, Zitong Zheng, Bo Shen, Gangwei Wang and Zhenli Wang

Fractal Fract. 2024, 8(9), 517; https://doi.org/10.3390/fractalfract8090517 - 30 Aug 2024

Viewed by 568

Abstract

We extend two KdVSKR models to fractional KdVSKR models with the Caputo derivative. The KdVSKR equation in (2+1)-dimension, which is a recent extension of the KdVSKR equation in (1+1)-dimension, can model the soliton resonances in shallow water. Applying the Hirota bilinear method, finite symmetry group method, and consistent Riccati expansion method, many new interaction solutions have been derived. Soliton and elliptical function interplaying solution for the fractional KdVSKR model in (1+1)-dimension has been derived for the first time. For the fractional KdVSKR model in (2+1)-dimension, two-wave interaction solutions and three-wave interaction solutions, including dark-soliton-sine interaction solution, bright-soliton-elliptic interaction solution, and lump-hyperbolic-sine interaction solution, have been derived. The effect of the order

γ

on the dynamical behaviors of the solutions has been illustrated by figures. The three-wave interaction solution has not been studied in the current references. The novelty of this paper is that the finite symmetry group method is adopted to construct interaction solutions of fractional nonlinear systems. This research idea can be applied to other fractional differential equations. Full article

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)

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31 pages, 450 KiB

Open AccessArticle

Derivation of Closed-Form Expressions in Apéry-like Series Using Fractional Calculus and Applications

by Ampol Duangpan, Ratinan Boonklurb, Udomsak Rakwongwan and Phiraphat Sutthimat

Fractal Fract. 2024, 8(7), 406; https://doi.org/10.3390/fractalfract8070406 - 11 Jul 2024

Viewed by 865

Abstract

This paper explores the Apéry-like series and demonstrates the derivation of closed-form expressions using fractional calculus. We consider a variety of Apéry-like functions, which were categorized by their functional forms and coefficients by applying the Riemann–Liouville fractional integral and derivative to examine their properties across various domains. The study focuses on establishing rigorous mathematical frameworks that unveil new insights into the behaviors of these series, contributing to a deeper understanding of number theory and mathematical analysis. Key results include proofs of convergence and divergence within specified intervals and the derivation of closed-form solutions through fractional integration and differentiation. This paper also introduces a method aimed at conjecturing mathematical constants through continued fractions as an application of our results. Finally, we provide the proof of validation for three unproven conjectures of continued fractions obtained from the Ramanujan Machine. Full article

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)

18 pages, 11966 KiB

Open AccessArticle

Method for Measuring the Fractional Derivative of a Two-Dimensional Magnetic Field Based on Taylor–Riemann Series

by Ruijian Wang and Yangyi Sui

Fractal Fract. 2024, 8(7), 375; https://doi.org/10.3390/fractalfract8070375 - 26 Jun 2024

Cited by 1 | Viewed by 1173

Abstract

In magnetic data processing, a fractional derivative can enhance details without excessively amplifying high-frequency noise. To obtain a fractional derivative numerically, a large number of survey points are required. This article demonstrates how a few survey points can be used to obtain the fractional derivative of a two-dimensional magnetic field through the application of Taylor–Riemann series. First, we derive the measurement method for the fractional gradient. This method is achieved by measuring the magnetic field at several survey points on a circle, then constructing analytical functions and finally calculating the fractional derivative. Next, an experiment is designed and simulated to demonstrate the impact of the fractional derivative start point and the ability to suppress Gaussian noise. Finally, the experiment is performed, which verifies the feasibility of the proposed method in a two-dimensional magnetic field. Full article

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)

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14 pages, 3979 KiB

Open AccessArticle

Exploring the Exact Solution of the Space-Fractional Stochastic Regularized Long Wave Equation: A Bifurcation Approach

by Bashayr Almutairi, Muneerah Al Nuwairan and Anwar Aldhafeeri

Fractal Fract. 2024, 8(5), 298; https://doi.org/10.3390/fractalfract8050298 - 18 May 2024

Viewed by 1135

Abstract

This study explores the effects of using space-fractional derivatives and adding multiplicative noise, modeled by a Wiener process, on the solutions of the space-fractional stochastic regularized long wave equation. New fractional stochastic solutions are constructed, and the consistency of the obtained solutions is examined using the transition between phase plane orbits. Their bifurcation and dependence on initial conditions are investigated. Some of these solutions are shown graphically, illustrating both the individual and combined influences of fractional order and noise on selected solutions. These effects appear as alterations in the amplitude and width of the solutions, and as variations in their smoothness. Full article

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)

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