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Fractal and Fractional, Volume 8, Issue 9

2024 September - 54 articles

Cover Story: Neural fractional differential equations (FDEs) are a neural network architecture designed to fit the solutions of fractional differential equations to data. This architecture combines an analytical component, the fractional derivative, with a neural network component, forming an initial value problem. We investigate the non-uniqueness of the optimal order of the derivative and its interaction with the neural network. We examine how different initialisations and values of the order of the derivative (in the optimisation process) impact its final optimal value. Consequently, neural FDEs do not require a unique α value; instead, they can use a wide range of α values to fit data. This flexibility is beneficial when fitting to given data is required and the underlying physics is not known. View this paper
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Articles (54)

  • Article
  • Open Access
12 Citations
2,724 Views
16 Pages

Ultrasound imaging is widely used in medical diagnostics due to its non-invasive and real-time capabilities. However, existing methods often overlook the benefits of fractional-order filters for denoising and dehazing. Thus, this work introduces an e...

(This article belongs to the Section Life Science, Biophysics)
  • Article
  • Open Access
66 Citations
2,477 Views
22 Pages

In this study, a model-free  PIφ-sliding mode control ( PIφ-SMC) methodology is proposed to synchronize a specific class of chaotic fractional-order memristive neural network systems (FOMNNSs) with delays and input saturation. The fracti...

(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization, 2nd Edition)
  • Article
  • Open Access
6 Citations
1,548 Views
16 Pages

This paper derives the sharp bounds for Hermite–Hadamard inequalities in the context of Riemann–Liouville fractional integrals. A generalization of Jensen’s inequality called the Jensen–Mercer inequality is used for general po...

(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications, 2nd Edition)
  • Article
  • Open Access
2 Citations
1,940 Views
15 Pages

The ultrasonic vibration rock-breaking method has been successfully applied to hard rock due to its high efficiency and controllable energy, providing a novel approach for the development of a more efficient, intelligent, safe, and environmentally fr...

(This article belongs to the Special Issue Applications of Fractal Analysis in Underground Engineering)
  • Article
  • Open Access
8 Citations
3,159 Views
22 Pages

Event-triggering mechanisms reported in the existing prescribed-time (PT) control do not adequately capture the dynamic nature of network environments, and are not applied to distributed consensus tracking problems with unknown time delays. Therefore...

(This article belongs to the Section Engineering)
  • Article
  • Open Access
2 Citations
2,196 Views
23 Pages

A Novel Fractal Model for Contact Resistance Based on Axisymmetric Sinusoidal Asperity

  • Yue Liu,
  • Shihao Yang,
  • Weikun Wang,
  • Shuai Wang,
  • Qi An,
  • Min Huang and
  • Shuangfu Suo

In this paper, a novel fractal model for the contact resistance based on axisymmetric sinusoidal asperity is proposed, which focuses on the resistance characteristics of the rough interface at a microscopic scale. By introducing the unique geometric...

(This article belongs to the Section Engineering)
  • Article
  • Open Access
7 Citations
1,417 Views
27 Pages

We explore the existence, uniqueness, and multiplicity of positive solutions to a system of Hadamard fractional differential equations that contain fractional integral terms. Defined on a finite interval, this system is subject to general coupled non...

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 3rd Edition)
  • Article
  • Open Access
4 Citations
1,267 Views
14 Pages

Modelling Yeast Prion Dynamics: A Fractional Order Approach with Predictor–Corrector Algorithm

  • Daasara Keshavamurthy Archana,
  • Doddabhadrappla Gowda Prakasha and
  • Nasser Bin Turki

This work aims to comprehend the dynamics of neurodegenerative disease using a mathematical model of fractional-order yeast prions. In the context of the Caputo fractional derivative, we here study and examine the solution of this model using the Pre...

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)
  • Article
  • Open Access
11 Citations
1,631 Views
24 Pages

In this article, we develop multiplicative fractional versions of Simpson’s and Newton’s formula-type inequalities for differentiable generalized convex functions with the help of established identities. The main motivation for using gene...

(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 2nd Edition)
  • Article
  • Open Access
4 Citations
2,502 Views
24 Pages

The study investigates the nonlinear contagion, tail dependence, and Granger causality relations with TAR-TR-GARCH–copula causality methods for daily Bitcoin, Fintech, energy consumption, and CO2 emissions in addition to examining these series...

(This article belongs to the Special Issue Fractional-Order Dynamics and Control in Green Energy Systems)
  • Article
  • Open Access
1 Citations
1,989 Views
22 Pages

Fractional-Order Modeling of the Depth of Analgesia as Reference Model for Control Purposes

  • Cristina I. Muresan,
  • Erwin T. Hegedüs,
  • Marcian D. Mihai,
  • Ghada Ben Othman,
  • Isabela Birs,
  • Dana Copot,
  • Eva Henrietta Dulf,
  • Robin De Keyser,
  • Clara M. Ionescu and
  • Martine Neckebroek

Little research has been carried out in terms of modeling and control of analgesia. However, emerging new technology and recent prototypes paved the way for several ideas on pain modeling for control. Recently, such an idea has been proposed for meas...

(This article belongs to the Section Life Science, Biophysics)
  • Article
  • Open Access
4 Citations
2,129 Views
12 Pages

Non-Local Problems for the Fractional Order Diffusion Equation and the Degenerate Hyperbolic Equation

  • Menglibay Ruziev,
  • Roman Parovik,
  • Rakhimjon Zunnunov and
  • Nargiza Yuldasheva

This research explores nonlocal problems associated with fractional diffusion equations and degenerate hyperbolic equations featuring singular coefficients in their lower-order terms. The uniqueness of the solution is established using the energy int...

(This article belongs to the Section Mathematical Physics)
  • Article
  • Open Access
12 Citations
2,381 Views
21 Pages

Analysis of Infection and Diffusion Coefficient in an SIR Model by Using Generalized Fractional Derivative

  • Ibtehal Alazman,
  • Manvendra Narayan Mishra,
  • Badr Saad Alkahtani and
  • Ravi Shanker Dubey

In this article, a diffusion component in an SIR model is introduced, and its impact is analyzed using fractional calculus. We have included the diffusion component in the SIR model. in order to illustrate the variations. Here, we have applied the ge...

(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)
  • Article
  • Open Access
3 Citations
2,038 Views
18 Pages

This study aims to predict the carbon sequestration capacity of Chinese grasslands to address climate change and achieve carbon neutrality goals. Grassland carbon sequestration is a crucial part of the global carbon cycle. However, its capacity is si...

(This article belongs to the Special Issue Applications of Fractional-Order Grey Models)
  • Article
  • Open Access
5 Citations
1,948 Views
36 Pages

In this work, some properties of the general convolutional operators of general fractional calculus (GFC), which satisfy analogues of the fundamental theorems of calculus, are described. Two types of general fractional (GF) operators on a finite inte...

(This article belongs to the Section General Mathematics, Analysis)
  • Article
  • Open Access
5 Citations
1,468 Views
22 Pages

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

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

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...

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)
  • Article
  • Open Access
5 Citations
2,438 Views
22 Pages

In this study, the optimal q-Homotopy Analysis Method (optimal q-HAM) has been used to investigate fractional Abel differential equations. This article is designed as a case study, where several forms of Abel equations, containing Bernoulli and Ricca...

(This article belongs to the Special Issue Fractional Mathematical Modelling: Theory, Methods and Applications)
  • Article
  • Open Access
9 Citations
2,270 Views
24 Pages

Leveraging the Performance of Integrated Power Systems with Wind Uncertainty Using Fractional Computing-Based Hybrid Method

  • Hani Albalawi,
  • Yasir Muhammad,
  • Abdul Wadood,
  • Babar Sattar Khan,
  • Syeda Taleeha Zainab and
  • Aadel Mohammed Alatwi

Reactive power dispatch (RPD) in electric power systems, integrated with renewable energy sources, is gaining popularity among power engineers because of its vital importance in the planning, designing, and operation of advanced power systems. The go...

(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering, 2nd Edition)
  • Article
  • Open Access
15 Citations
1,480 Views
14 Pages

Practical Stability of Observer-Based Control for Nonlinear Caputo–Hadamard Fractional-Order Systems

  • Rihab Issaoui,
  • Omar Naifar,
  • Mehdi Tlija,
  • Lassaad Mchiri and
  • Abdellatif Ben Makhlouf

This paper investigates the problem of observer-based control for a class of nonlinear systems described by the Caputo–Hadamard fractional-order derivative. Given the growing interest in fractional-order systems for their ability to capture com...

(This article belongs to the Section Engineering)
  • Article
  • Open Access
17 Citations
2,292 Views
22 Pages

We utilize Lyapunov exponents to quantitatively assess the hyperchaos and categorize the limit sets of complex dynamical systems. While there are numerous methods for computing Lyapunov exponents in integer-order systems, these methods are not suitab...

  • Article
  • Open Access
6 Citations
3,424 Views
16 Pages

Neural Fractional Differential Equations (Neural FDEs) represent a neural network architecture specifically designed to fit the solution of a fractional differential equation to given data. This architecture combines an analytical component, represen...

(This article belongs to the Special Issue Recent Advances in the Spatial and Temporal Discretizations of Fractional PDEs)
  • Article
  • Open Access
1,607 Views
17 Pages

In this paper, we consider a logarithmic fractional Schrödinger-Poisson system where the potential is a sign-changing function. When the potential is coercive, we get the existence of infinitely many solutions for the system. When the potential...

(This article belongs to the Special Issue Variational Problems and Fractional Differential Equations)
  • Article
  • Open Access
1 Citations
1,912 Views
16 Pages

A Novel Approach for Testing Fractional Cointegration in Panel Data Models with Fixed Effects

  • Saidat Fehintola Olaniran,
  • Oyebayo Ridwan Olaniran,
  • Jeza Allohibi and
  • Abdulmajeed Atiah Alharbi

Fractional cointegration in time series data has been explored by several authors, but panel data applications have been largely neglected. A previous study of ours discovered that the Chen and Hurvich fractional cointegration test for time series wa...

  • Article
  • Open Access
2,039 Views
13 Pages

In this article, we focus on examining the existence, uniqueness, and continuous dependence of solutions on initial data for a specific initial boundary value problem which mainly arises from one-dimensional quasi-static contact problems in nonlinear...

(This article belongs to the Special Issue Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods and Application)
  • Article
  • Open Access
5 Citations
1,746 Views
15 Pages

This paper studies synchronization behaviors of two sorts of non-linear fractional-order complex spatio-temporal networks modeled by hyperbolic space-varying PDEs (FCSNHSPDEs), respectively, with time-invariant delays and time-varying delays, includi...

(This article belongs to the Section Complexity)
  • Article
  • Open Access
4 Citations
2,172 Views
19 Pages

This paper presents an application of a fractional-order system on modeling an industrial process system with large inertia and time delay. The traditional integer-order model of the process system is extended to a fractional-order one in this work....

(This article belongs to the Special Issue Fractional Order Complex Systems: Advanced Control, Intelligent Estimation and Reinforcement Learning Image Processing​ Algorithms)
  • Article
  • Open Access
45 Citations
2,312 Views
21 Pages

Consensus of T-S Fuzzy Fractional-Order, Singular Perturbation, Multi-Agent Systems

  • Xiyi Wang,
  • Xuefeng Zhang,
  • Witold Pedrycz,
  • Shuang-Hua Yang and
  • Driss Boutat

Due to system complexity, research on fuzzy fractional-order, singular perturbation, multi-agent systems (FOSPMASs) remains limited in control theory. This article focuses on the leader-following consensus of fuzzy FOSPMASs with orders in the range o...

(This article belongs to the Section Engineering)
  • Article
  • Open Access
10 Citations
2,825 Views
17 Pages

The occurrence of landslide hazards significantly induces changes in slope surface displacement. This study conducts an in-depth analysis of the multifractal characteristics and displacement prediction of highway slope surface displacement sequences....

(This article belongs to the Special Issue Fractal and Fractional in Geotechnical Engineering)
  • Article
  • Open Access
1,496 Views
23 Pages

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...

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)
  • Article
  • Open Access
7 Citations
2,590 Views
24 Pages

In the field of nonlinear mathematical physics, Ablowitz et al.’s algorithm has recently made significant progress in the inverse scattering transform (IST) of fractional-order nonlinear evolution equations (fNLEEs). However, the solved fNLEEs...

(This article belongs to the Special Issue Recent Developments and Applications of Fractional Differential Equations in Mathematical Physics)
  • Article
  • Open Access
1,185 Views
23 Pages

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

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

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. Appl...

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)
  • Article
  • Open Access
5 Citations
1,702 Views
24 Pages

This article investigates the dynamic behaviors of delayed fractional-order memristive fuzzy cellular neural networks via the Lyapunov method. To address the delay terms of fractional-order systems, a novel lemma is provided to make the solutions of...

(This article belongs to the Special Issue Modeling, Optimization, and Control of Fractional-Order Neural Networks and Nonlinear Systems)
  • Article
  • Open Access
11 Citations
2,222 Views
32 Pages

Function spaces play a crucial role in the study and application of mathematical inequalities. They provide a structured framework within which inequalities can be formulated, analyzed, and applied. They allow for the extension of inequalities from f...

(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications, 2nd Edition)
  • Article
  • Open Access
1 Citations
2,088 Views
12 Pages

Lp(Lq)-Maximal Regularity for Damped Equations in a Cylindrical Domain

  • Edgardo Alvarez,
  • Stiven Díaz and
  • Carlos Lizama

We show maximal regularity estimates for the damped hyperbolic and strongly damped wave equations with periodic initial conditions in a cylindrical domain. We prove that this property strongly depends on a critical combination on the parameters of th...

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)
  • Article
  • Open Access
8 Citations
1,896 Views
18 Pages

We looked at the (3+1)-dimensional fractional Kadomtsev–Petviashvili–Boussinesq (KP-B) equation, which comes up in fluid dynamics, plasma physics, physics, and superfluids, as well as when connecting the optical model and hydrodynamic dom...

(This article belongs to the Special Issue Recent Developments and Applications of Fractional Differential Equations in Mathematical Physics)
  • Article
  • Open Access
1 Citations
2,199 Views
16 Pages

Multifractal Analysis of Neuronal Morphology in the Human Dorsal Striatum: Age-Related Changes and Spatial Differences

  • Zorana Nedeljković,
  • Bojana Krstonošić,
  • Nebojša Milošević,
  • Olivera Stanojlović,
  • Dragan Hrnčić and
  • Nemanja Rajković

Multifractal analysis offers a sophisticated method to examine the complex morphology of neurons, which traditionally have been analyzed using monofractal techniques. This study investigates the multifractal properties of two-dimensional neuron proje...

(This article belongs to the Special Issue Advances in Pattern Recognition—Image and Time Series Analyses—through Fractal Geometry and Complexity Theory)
  • Brief Report
  • Open Access
1 Citations
1,561 Views
9 Pages

Our recent analysis of empirical limit order flow data in financial markets reveals a power-law distribution in limit order cancellation times. These times are modeled using a discrete probability mass function derived from the Tsallis q-exponential...

(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)
  • Article
  • Open Access
13 Citations
1,879 Views
15 Pages

In this manuscript, we survey a numerical algorithm based on the combination of the homotopy perturbation method and the Sadik transform for solving the time-fractional nonlinear modified shallow water waves (called Kawahara equation) within the fram...

  • Article
  • Open Access
12 Citations
3,725 Views
33 Pages

Enhancing Medical Image Quality Using Fractional Order Denoising Integrated with Transfer Learning

  • Abirami Annadurai,
  • Vidhushavarshini Sureshkumar,
  • Dhayanithi Jaganathan and
  • Seshathiri Dhanasekaran

In medical imaging, noise can significantly obscure critical details, complicating diagnosis and treatment. Traditional denoising techniques often struggle to maintain a balance between noise reduction and detail preservation. To address this challen...

(This article belongs to the Section Optimization, Big Data, and AI/ML)
  • Article
  • Open Access
1 Citations
1,869 Views
22 Pages

Study of a Coupled Ψ–Liouville–Riemann Fractional Differential System Characterized by Mixed Boundary Conditions

  • Brahim Tellab,
  • Abdelkader Amara,
  • Mohammed El-Hadi Mezabia,
  • Khaled Zennir and
  • Loay Alkhalifa

This research is concerned with the existence and uniqueness of solutions for a coupled system of Ψ–Riemann–Liouville fractional differential equations. To achieve this objective, we establish a set of necessary conditions by formulat...

(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)
  • Communication
  • Open Access
9 Citations
1,694 Views
9 Pages

In this paper, we defined a new family of meromorphic functions whose analytic characterization was motivated by the definition of the multiplicative derivative. Replacing the ordinary derivative with a multiplicative derivative in the subclass of st...

(This article belongs to the Special Issue Fractional Calculus, Quantum Calculus and Special Functions in Complex Analysis)
  • Article
  • Open Access
4 Citations
2,484 Views
21 Pages

Extensions of Bicomplex Hypergeometric Functions and Riemann–Liouville Fractional Calculus in Bicomplex Numbers

  • Ahmed Bakhet,
  • Mohamed Fathi,
  • Mohammed Zakarya,
  • Ghada AlNemer and
  • Mohammed A. Saleem

In this paper, we present novel advancements in the theory of bicomplex hypergeometric functions and their applications. We extend the hypergeometric function to bicomplex parameters, analyse its convergence region, and define its integral and deriva...

(This article belongs to the Special Issue Fractional Calculus, Quantum Calculus and Special Functions in Complex Analysis)
  • Article
  • Open Access
3 Citations
1,493 Views
19 Pages

In this study, we examine Timoshenko systems with boundary conditions featuring two types of fractional dissipations. By applying semigroup theory, we demonstrate the existence and uniqueness of solutions. Our analysis shows that while the system exh...

(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)
  • Article
  • Open Access
15 Citations
2,327 Views
20 Pages

Fuzzy Adaptive Approaches for Robust Containment Control in Nonlinear Multi-Agent Systems under False Data Injection Attacks

  • Ammar Alsinai,
  • Mohammed M. Ali Al-Shamiri,
  • Waqar Ul Hassan,
  • Saadia Rehman and
  • Azmat Ullah Khan Niazi

This study addresses the problem of fractional-order nonlinear containment control of heterogeneous multi-agent systems within a leader–follower framework, focusing on the impact of False Data Injection (FDI) attacks. By employing adaptive mech...

(This article belongs to the Special Issue Advances in Fractional Order Systems and Robust Control, 2nd Edition)
  • Article
  • Open Access
23 Citations
1,903 Views
17 Pages

The fractal theory can effectively describe the complexity and multi-scale of concrete under impact load and provide a scientific basis for evaluating concrete’s impact resistance. Therefore, based on the fractal theory, this study carried out...

  • Article
  • Open Access
3 Citations
1,195 Views
10 Pages

Multifractal Characteristics of Gain Structures: A Universal Law of Polycrystalline Strain-Hardening Behaviors

  • Maoqing Fu,
  • Jiapeng Chen,
  • Zhaowen Huang,
  • Bin Chen,
  • Yangfan Hu and
  • Biao Wang

The quantitative relationship between material microstructures, such as grain distributions, and the nonlinear strain-hardening behaviors of polycrystalline metals has not yet been completely understood. This study finds that the grain correlation di...

  • Article
  • Open Access
6 Citations
2,183 Views
21 Pages

In order to obtain the contact resistance of relay contacts more accurately, a novel contact resistance model for the spherical–planar joint interface is constructed based on the three-dimensional fractal theory. In this model, three-dimensiona...

  • Article
  • Open Access
2 Citations
1,593 Views
12 Pages

This study investigates the initial value problem of high-order variable-order φ-Hilfer fractional implicit integro-differential equations. Due to the lack of the semigroup property in variable-order fractional integrals, solving these equations...

(This article belongs to the Section General Mathematics, Analysis)
  • Article
  • Open Access
6 Citations
1,535 Views
13 Pages

Applications of Caputo-Type Fractional Derivatives for Subclasses of Bi-Univalent Functions with Bounded Boundary Rotation

  • Kholood M. Alsager,
  • Gangadharan Murugusundaramoorthy,
  • Adriana Catas and
  • Sheza M. El-Deeb

In this article, for the first time by using Caputo-type fractional derivatives, we introduce three new subclasses of bi-univalent functions associated with bounded boundary rotation in an open unit disk to obtain non-sharp estimates of the first two...

(This article belongs to the Special Issue Fractional Calculus, Quantum Calculus and Special Functions in Complex Analysis)
  • Brief Report
  • Open Access
10 Citations
3,017 Views
13 Pages

Fractional Differential Equations with Impulsive Effects

  • Michal Fečkan,
  • Marius-F. Danca and
  • Guanrong Chen

This paper discusses impulsive effects on fractional differential equations. Two approaches are taken to obtain our results: either with fixed or changing lower limits in Caputo fractional derivatives. First, we derive an existence result for periodi...

(This article belongs to the Section Numerical and Computational Methods)

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Fractal Fract. - ISSN 2504-3110