Fractal and Fractional

16 pages, 5641 KiB

Open AccessArticle

Multifractal Structures and the Energy-Economic Efficiency of Chinese Cities: Using a Classification-Based Multifractal Method

by Jiaxin Wang, Bin Meng and Feng Lu

Fractal Fract. 2025, 9(2), 96; https://doi.org/10.3390/fractalfract9020096 (registering DOI) - 3 Feb 2025

Abstract

Improper urban spatial structure can lead to problems such as traffic congestion, long commuting times, and diseconomies of scale. Evaluating the efficiency of urban spatial structure is an important means to enhance the sustainable development of cities. The fractal method has been widely [...] Read more.

Improper urban spatial structure can lead to problems such as traffic congestion, long commuting times, and diseconomies of scale. Evaluating the efficiency of urban spatial structure is an important means to enhance the sustainable development of cities. The fractal method has been widely used in the identification and efficiency evaluation of urban spatial structure due to its sufficient characterization of urban complexity. However, the identification of urban fractal structures has expanded from monofractal structures to multifractal structures, while the efficiency evaluation of urban fractal structures remains limited to the single-dimensional efficiency evaluations of single fractals, seriously affecting the reliability of urban fractal structure evaluation. Therefore, this study identifies and evaluates urban spatial structure within the unified framework of multifractal analysis. Specifically, a classification-based multifractal method is introduced to identify the multifractal structure of 290 cities in China. An iterative application of the geographic detector method is used to evaluate the comprehensive energy-economic efficiency of urban multifractal structures. The results indicate that the 290 Chinese cities include 6 typical multifractal structures. The explanatory power of these six typical multifractal structures for urban energy-economic efficiency is 16.27%. The advantageous multifractal structures of cities that achieve higher energy-economic efficiency rates satisfy a cubic polynomial form. By comparing them with the advantageous multifractal structures, the main problems affecting the efficiency of urban multifractal structures in the other five types of cities are shown to include overly strong or weak concentration capacity of high-level centers, weak hierarchical structures among centers, and the spreading of low-level centers. Full article

(This article belongs to the Special Issue Fractional Processes and Systems in Computer Science and Engineering)

► Show Figures

Figure 1

25 pages, 11115 KiB

Open AccessArticle

Enhancing Banking Transaction Security with Fractal-Based Image Steganography Using Fibonacci Sequences and Discrete Wavelet Transform

by Alina Iuliana Tabirca, Catalin Dumitrescu and Valentin Radu

Fractal Fract. 2025, 9(2), 95; https://doi.org/10.3390/fractalfract9020095 (registering DOI) - 2 Feb 2025

Viewed by 379

Abstract

The growing reliance on digital banking and financial transactions has brought significant security challenges, including data breaches and unauthorized access. This paper proposes a robust method for enhancing the security of banking and financial transactions. In this context, steganography—hiding information within digital media—is [...] Read more.

The growing reliance on digital banking and financial transactions has brought significant security challenges, including data breaches and unauthorized access. This paper proposes a robust method for enhancing the security of banking and financial transactions. In this context, steganography—hiding information within digital media—is valuable for improving data protection. This approach combines biometric authentication, using face and voice recognition, with image steganography to secure communication channels. A novel application of Fibonacci sequences is introduced within a direct-sequence spread-spectrum (DSSS) system for encryption, along with a discrete wavelet transform (DWT) for embedding data. The secret message, encrypted through Fibonacci sequences, is concealed within an image and tested for effectiveness using the Mean Square Error (MSE) and Peak Signal-to-Noise Ratio (PSNR). The experimental results demonstrate that the proposed method achieves a high PSNR, particularly for grayscale images, enhancing the robustness of security measures in mobile and online banking environments. Full article

(This article belongs to the Special Issue Application of Fractal Processes and Fractional Derivatives in Finance, 2nd Edition)

► Show Figures

Figure 1

18 pages, 424 KiB

Open AccessArticle

Finite-Time Stability of Fractional-Order Switched Systems Based on Lyapunov Method

by Tian Feng, Lizhen Wang and Yangquan Chen

Fractal Fract. 2025, 9(2), 94; https://doi.org/10.3390/fractalfract9020094 (registering DOI) - 2 Feb 2025

Viewed by 165

Abstract

This paper investigates the finite-time stability of a class of fractional-order switched systems with order

0 < α < 1

, employing the fractional Lyapunov direct method. First, based on the Mittag-Leffler function and Gronwall inequality, two corresponding sufficient conditions are presented to [...] Read more.

This paper investigates the finite-time stability of a class of fractional-order switched systems with order

0 < α < 1

, employing the fractional Lyapunov direct method. First, based on the Mittag-Leffler function and Gronwall inequality, two corresponding sufficient conditions are presented to ensure the finite-time stability of the considered system. Second, in consideration of the effectiveness of dwell time technique in switched systems, a sufficient condition is derived under a minimum average dwell time constraint. Finally, numerical simulations are performed to validate the effectiveness of the theoretical formulation. Full article

(This article belongs to the Special Issue Advances in Fractional-Order Chaotic and Complex Systems)

19 pages, 1328 KiB

Open AccessArticle

Fractal Characterization and Pore Evolution in Coal Under Tri-Axial Cyclic Loading–Unloading: Insights from Low-Field NMR Imaging and Analysis

by Zelin Liu, Senlin Xie, Yajun Yin and Teng Su

Fractal Fract. 2025, 9(2), 93; https://doi.org/10.3390/fractalfract9020093 (registering DOI) - 1 Feb 2025

Viewed by 194

Abstract

Coal resource extraction and utilization are essential for sustainable development and economic growth. This study integrates a pseudo-triaxial mechanical loading system with low-field nuclear magnetic resonance (NMR) to enable the preliminary visualization of coal’s pore-fracture structure (PFS) under mechanical stress. Pseudo-triaxial and cyclic [...] Read more.

Coal resource extraction and utilization are essential for sustainable development and economic growth. This study integrates a pseudo-triaxial mechanical loading system with low-field nuclear magnetic resonance (NMR) to enable the preliminary visualization of coal’s pore-fracture structure (PFS) under mechanical stress. Pseudo-triaxial and cyclic loading–unloading tests were combined with real-time NMR monitoring to model porosity recovery, pore size evolution, and energy dissipation, while also calculating the fractal dimensions of pores in relation to stress. The results show that during the compaction phase, primary pores are compressed with limited recovery after unloading. In the elastic phase, both adsorption and seepage pores transform significantly, with most recovering post-unloading. After yield stress, new fractures and pores form, and unloading enhances fracture connectivity. Seepage pore porosity shows a negative exponential relationship with axial strain before yielding, and a logarithmic relationship afterward. The fractal dimension of adsorption pores decreases during compaction and increases afterward, while the fractal dimension of seepage pores decreases before yielding and increases post-yielding. These findings provide new insights into the flow patterns of methane in coal seams. Full article

(This article belongs to the Special Issue Fractal Dimensions with Applications in the Real World)

31 pages, 812 KiB

Open AccessArticle

Qualitative Analysis of Generalized Power Nonlocal Fractional System with p-Laplacian Operator, Including Symmetric Cases: Application to a Hepatitis B Virus Model

by Mohamed S. Algolam, Mohammed A. Almalahi, Muntasir Suhail, Blgys Muflh, Khaled Aldwoah, Mohammed Hassan and Saeed Islam

Fractal Fract. 2025, 9(2), 92; https://doi.org/10.3390/fractalfract9020092 (registering DOI) - 1 Feb 2025

Viewed by 181

Abstract

This paper introduces a novel framework for modeling nonlocal fractional system with a p-Laplacian operator under power nonlocal fractional derivatives (PFDs), a generalization encompassing established derivatives like Caputo–Fabrizio, Atangana–Baleanu, weighted Atangana–Baleanu, and weighted Hattaf. The core methodology involves employing a PFD with a [...] Read more.

This paper introduces a novel framework for modeling nonlocal fractional system with a p-Laplacian operator under power nonlocal fractional derivatives (PFDs), a generalization encompassing established derivatives like Caputo–Fabrizio, Atangana–Baleanu, weighted Atangana–Baleanu, and weighted Hattaf. The core methodology involves employing a PFD with a tunable power parameter within a non-singular kernel, enabling a nuanced representation of memory effects not achievable with traditional fixed-kernel derivatives. This flexible framework is analyzed using fixed-point theory, rigorously establishing the existence and uniqueness of solutions for four symmetric cases under specific conditions. Furthermore, we demonstrate the Hyers–Ulam stability, confirming the robustness of these solutions against small perturbations. The versatility and generalizability of this framework is underscored by its application to an epidemiological model of transmission of Hepatitis B Virus (HBV) and numerical simulations for all four symmetric cases. This study presents findings in both theoretical and applied aspects of fractional calculus, introducing an alternative framework for modeling complex systems with memory processes, offering opportunities for more sophisticated and accurate models and new avenues for research in fractional calculus and its applications. Full article

(This article belongs to the Special Issue Advanced Numerical Methods for Fractional Functional Models)

23 pages, 13764 KiB

Open AccessArticle

Physics–Informed Fractional–Order Recurrent Neural Network for Fast Battery Degradation with Vehicle Charging Snippets

by Yanan Wang, Min Wei, Feng Dai, Daijiang Zou, Chen Lu, Xuebing Han, Yangquan Chen and Changwei Ji

Fractal Fract. 2025, 9(2), 91; https://doi.org/10.3390/fractalfract9020091 (registering DOI) - 1 Feb 2025

Viewed by 209

Abstract

To handle and manage battery degradation in electric vehicles (EVs), various capacity estimation methods have been proposed and can mainly be divided into traditional modeling methods and data-driven methods. For realistic conditions, data-driven methods take the advantage of simple application. However, state-of-the-art machine [...] Read more.

To handle and manage battery degradation in electric vehicles (EVs), various capacity estimation methods have been proposed and can mainly be divided into traditional modeling methods and data-driven methods. For realistic conditions, data-driven methods take the advantage of simple application. However, state-of-the-art machine learning (ML) algorithms are still kinds of black-box models; thus, the algorithms do not have a strong ability to describe the inner reactions or degradation information of batteries. Due to a lack of interpretability, machine learning may not learn the degradation principle correctly and may need to depend on big data quality. In this paper, we propose a physics–informed recurrent neural network (PIRNN) with a fractional–order gradient for fast battery degradation estimation in running EVs to provide a physics–informed neural network that can make algorithms learn battery degradation mechanisms. Incremental capacity analysis (ICA) was conducted to extract aging characteristics, which could be selected as the inputs of the algorithm. The fractional–order gradient descent (FOGD) method was also applied to improve the training convergence and embedding of battery information during backpropagation; then, the recurrent neural network was selected as the main body of the algorithm. A battery dataset with fast degradation from ten EVs with a total of 5697 charging snippets were constructed to validate the performance of the proposed algorithm. Experimental results show that the proposed PIRNN with ICA and the FOGD method could control the relative error within

5 %

for most snippets of the ten EVs. The algorithm could even achieve a stable estimation accuracy (relative error < 3%) during three-quarters of a battery’s lifetime, while for a battery with dramatic degradation, it was difficult to maintain such high accuracy during the whole battery lifetime. Full article

(This article belongs to the Special Issue Recent Advances in Fractional-Order Equations and Their Applications in Modern Energy Systems)

30 pages, 1095 KiB

Open AccessArticle

Probing Malware Propagation Model with Variable Infection Rates Under Integer, Fractional, and Fractal–Fractional Orders

by Nausheen Razi, Ambreen Bano, Umar Ishtiaq, Tayyab Kamran, Mubariz Garayev and Ioan-Lucian Popa

Fractal Fract. 2025, 9(2), 90; https://doi.org/10.3390/fractalfract9020090 (registering DOI) - 1 Feb 2025

Viewed by 214

Abstract

Malware software has become a pervasive threat in computer and mobile technology attacks. Attackers use this software to obtain information about users of the digital world to obtain benefits by hijacking their data. Antivirus software has been developed to prevent the propagation of [...] Read more.

Malware software has become a pervasive threat in computer and mobile technology attacks. Attackers use this software to obtain information about users of the digital world to obtain benefits by hijacking their data. Antivirus software has been developed to prevent the propagation of malware, but this problem is not yet under control. To develop this software, we have to check the propagation of malware. In this paper, we explore an advanced malware propagation model with a time-delay factor and a variable infection rate. To better understand this model, we use fractal–fractional theory. We use an exponential decay kernel for this. For theoretical purposes (existence, uniqueness, and stability), we use the results from fixed-point theory, and, for numerical purposes, a Lagrange two-point interpolation polynomial is used to develop an algorithm. Matlab R2016a is used for simulation, and the physical significance is assessed. We examine the impact of different fractal and fractional orders for various parameters. Moreover, we compare four different mathematical models (classical, fractional, fractal, and fractal–fractional). Also, constant and variable fractional and fractal orders are compared using graphs. We investigate the idea that significant perturbation in infected nodes might be due to minor changes. This work may help with developing antivirus strategies in real life. Full article

(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications, 3rd Edition)

30 pages, 1985 KiB

Open AccessArticle

Representation of Special Functions by Multidimensional A- and J-Fractions with Independent Variables

by Roman Dmytryshyn and Serhii Sharyn

Fractal Fract. 2025, 9(2), 89; https://doi.org/10.3390/fractalfract9020089 - 28 Jan 2025

Viewed by 383

Abstract

The paper deals with the problem of representing special functions by branched continued fractions, particularly multidimensional A- and J-fractions with independent variables, which are generalizations of associated continued fractions and Jacobi continued fractions, respectively. A generalized Gragg’s algorithm is constructed that [...] Read more.

The paper deals with the problem of representing special functions by branched continued fractions, particularly multidimensional A- and J-fractions with independent variables, which are generalizations of associated continued fractions and Jacobi continued fractions, respectively. A generalized Gragg’s algorithm is constructed that enables us to compute, by the coefficients of the given formal multiple power series, the coefficients of the corresponding multidimensional A- and J-fractions with independent variables. Presented below are numerical experiments for approximating some special functions by these branched continued fractions, which are similar to fractals. Full article

19 pages, 892 KiB

Open AccessArticle

Fixed/Preassigned Time Synchronization of Impulsive Fractional-Order Reaction–Diffusion Bidirectional Associative Memory (BAM) Neural Networks

by Rouzimaimaiti Mahemuti, Abdujelil Abdurahman and Ahmadjan Muhammadhaji

Fractal Fract. 2025, 9(2), 88; https://doi.org/10.3390/fractalfract9020088 - 28 Jan 2025

Viewed by 340

Abstract

This study delves into the synchronization issues of the impulsive fractional-order, mainly the Caputo derivative of the order between 0 and 1, bidirectional associative memory (BAM) neural networks incorporating the diffusion term at a fixed time (FXT) and a predefined time (PDT). Initially, [...] Read more.

This study delves into the synchronization issues of the impulsive fractional-order, mainly the Caputo derivative of the order between 0 and 1, bidirectional associative memory (BAM) neural networks incorporating the diffusion term at a fixed time (FXT) and a predefined time (PDT). Initially, this study presents certain characteristics of fractional-order calculus and several lemmas pertaining to the stability of general impulsive nonlinear systems, specifically focusing on FXT and PDT stability. Subsequently, we utilize a novel controller and Lyapunov functions to establish new sufficient criteria for achieving FXT and PDT synchronizations. Finally, a numerical simulation is presented to ascertain the theoretical dependency. Full article

23 pages, 1656 KiB

Open AccessArticle

A Comparative Study of Fractal Models Applied to Artificial and Natural Data

by Gil Silva, Fernando Pellon de Miranda, Mateus Michelon, Ana Ovídio, Felipe Venturelli, João Parêdes, João Ferreira, Letícia Moraes, Flávio Barbosa and Alexandre Cury

Fractal Fract. 2025, 9(2), 87; https://doi.org/10.3390/fractalfract9020087 - 28 Jan 2025

Viewed by 547

Abstract

This paper presents an original and comprehensive comparative analysis of eight fractal analysis methods, including Box Counting, Compass, Detrended Fluctuation Analysis, Dynamical Fractal Approach, Hurst, Mass, Modified Mass, and Persistence. These methods are applied to artificially generated fractal data, such as Weierstrass–Mandelbrot functions [...] Read more.

This paper presents an original and comprehensive comparative analysis of eight fractal analysis methods, including Box Counting, Compass, Detrended Fluctuation Analysis, Dynamical Fractal Approach, Hurst, Mass, Modified Mass, and Persistence. These methods are applied to artificially generated fractal data, such as Weierstrass–Mandelbrot functions and fractal Brownian motion, as well as natural datasets related to environmental and geophysical domains. The objectives of this research are to evaluate the methods’ capabilities in capturing fractal properties, their computational efficiency, and their sensitivity to data fluctuations. Main findings indicate that the Dynamical Fractal Approach consistently demonstrated the highest accuracy across different datasets, particularly for artificial data. Conversely, methods like Mass and Modified Mass showed limitations in complex fractal structures. For natural datasets, including meteorological and geological data, the fractal dimensions varied significantly across methods, reflecting their differing sensitivities to structural complexities. Computational efficiency analysis revealed that methods with linear or logarithmic complexity, such as Persistence and Compass, are most suited for larger datasets, while methods like DFA and Dynamic Fractal Approaches required higher computational resources. This study provides an original comparative study for researchers to select appropriate fractal analysis techniques based on dataset characteristics and computational limitations. Full article

(This article belongs to the Section Engineering)

► Show Figures

Figure 1

16 pages, 1487 KiB

Open AccessArticle

Hybrid Dynamic Event-Triggered Interval Observer Design for Nonlinear Cyber–Physical Systems with Disturbance

by Hongrun Wu, Jun Huang, Yong Qin and Yuan Sun

Fractal Fract. 2025, 9(2), 86; https://doi.org/10.3390/fractalfract9020086 - 26 Jan 2025

Viewed by 301

Abstract

This paper investigates the state estimation problem for nonlinear cyber–physical systems (CPSs). To conserve system resources, we propose a novel hybrid dynamic event-triggered mechanism (ETM) that prevents the occurrence of Zeno behavior. This work is based on designing an interval observer under the [...] Read more.

This paper investigates the state estimation problem for nonlinear cyber–physical systems (CPSs). To conserve system resources, we propose a novel hybrid dynamic event-triggered mechanism (ETM) that prevents the occurrence of Zeno behavior. This work is based on designing an interval observer under the hybrid dynamic ETM to solve the state reconstruction problem of Lipschitz nonlinear CPSs subject to disturbances. That is, the designed triggering mechanism is integrated into the design of the Interval Observer (IO), resulting in a hybrid dynamic event-triggered interval observer (HDETIO), and the system stability and robustness are proved using a Lyapunov function, demonstrating that the observer can effectively provide interval estimation for CPSs with nonlinearity and disturbances. Compared to existing work, the primary contribution of this work is its ability to pre-specify the minimum inter-event time (MIET) and apply it to interval state estimation, enhancing its practicality for real-world physical systems. Finally, the correctness and effectiveness of the designed hybrid dynamic ETM and IO framework are validated with an example. Full article

(This article belongs to the Special Issue Fractional- and Integer-Order System: Control Theory and Applications, 2nd Edition)

► Show Figures

Figure 1

15 pages, 303 KiB

Open AccessArticle

Asymptotic Periodicity of Bounded Mild Solutions for Evolution Equations with Non-Densely Defined and Fractional Derivative

by Jiabin Zuo, Abdellah Taqbibt, Mohamed Chaib and M’hamed Elomari

Fractal Fract. 2025, 9(2), 85; https://doi.org/10.3390/fractalfract9020085 - 26 Jan 2025

Viewed by 261

Abstract

In the present article, we establish conditions for the asymptotic periodicity of bounded mild solutions in two distinct cases of evolution equations. The first class involves non-densely defined operators, while the second class incorporates densely defined operators with fractional derivatives that generate a [...] Read more.

In the present article, we establish conditions for the asymptotic periodicity of bounded mild solutions in two distinct cases of evolution equations. The first class involves non-densely defined operators, while the second class incorporates densely defined operators with fractional derivatives that generate a semigroup of contractions. Our method integrates the theory of spectral properties of uniformly bounded continuous functions defined on the positive real semi-axis. Additionally, we apply extrapolation theory to evolution equations with non-densely defined operators. To illustrate our main results, we provide a concrete example. Full article

(This article belongs to the Special Issue Recent Advances in Nonlocal Problems Involving the Fractional Laplacian Operators)

12 pages, 3493 KiB

Open AccessArticle

On a Preloaded Compliance System of Fractional Order: Numerical Integration

by Marius-F. Danca

Fractal Fract. 2025, 9(2), 84; https://doi.org/10.3390/fractalfract9020084 - 26 Jan 2025

Viewed by 306

Abstract

In this paper, the use of a class of fractional-order dynamical systems with discontinuous right-hand side defined with Caputo’s derivative is considered. The existence of the solutions is analyzed. For this purpose, differential inclusions theory is used to transform, via the Filippov regularization, [...] Read more.

In this paper, the use of a class of fractional-order dynamical systems with discontinuous right-hand side defined with Caputo’s derivative is considered. The existence of the solutions is analyzed. For this purpose, differential inclusions theory is used to transform, via the Filippov regularization, the discontinuous right-hand side into a set-valued function. Next, via Cellina’s Theorem, the obtained set-valued differential inclusion of fractional order can be restarted as a single-valued continuous differential equation of fractional order, to which the existing numerical schemes for fractional differential equations can be applied. In this way, the delicate problem of integrating discontinuous problems of fractional order, as well as integer order, is solved by transforming the discontinuous problem into a continuous one. Also, it is noted that even the numerical methods for fractional-order differential equations can be applied abruptly to the discontinuous problem, without considering the underlying discontinuity, so the results could be incorrect. The technical example of a single-degree-of-freedom preloaded compliance system of fractional order is presented. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 2nd Edition)

► Show Figures

Figure 1

19 pages, 2472 KiB

Open AccessArticle

Mechanical Response of Mudstone Based on Acoustic Emission Fractal Features

by Xianyin Chang, Yunpei Liang and Qican Ran

Fractal Fract. 2025, 9(2), 83; https://doi.org/10.3390/fractalfract9020083 - 25 Jan 2025

Viewed by 272

Abstract

In this study, the effect of the stress amplitude on the mechanical behavior of mudstone was systematically investigated by cyclic loading and unloading experiments and acoustic emission (AE) monitoring. The results show that at low-stress amplitudes, mudstone specimens show better elastic recovery ability, [...] Read more.

In this study, the effect of the stress amplitude on the mechanical behavior of mudstone was systematically investigated by cyclic loading and unloading experiments and acoustic emission (AE) monitoring. The results show that at low-stress amplitudes, mudstone specimens show better elastic recovery ability, lower damage accumulation and higher structural stability. At high-stress amplitudes, the irreversible damage of the mudstone increases significantly, the internal fractures gradually expand and penetrate through, and the risk of instability increases significantly. This is manifested by the gradual increase in cumulative irreversible strain of mudstone at different stress amplitudes, up to 0.144%. In addition, different stress amplitudes have significant effects on energy evolution characteristics, with low-stress amplitudes mainly showing elastic deformation and a high percentage of recoverable energy, while high-stress amplitudes show a high percentage of dissipated energy. Under the condition of high-stress amplitude, such as the mudstone specimen #4, the percentage of tensile failure is 81.15%. Tensile failure dominates at all stress amplitudes, where the failure mechanism within mudstone is mainly characterized by the extension of tensile-type fractures. Through the multifractal analysis of AE signals, this study reveals the effect of the stress amplitude on the fracture extension mode and failure mechanism of mudstone. As the stress amplitude increases, Δα and Δf show an increasing trend. This indicates that the fracture extension process transforms from a relatively homogeneous and simple mode to a more inhomogeneous and complex mode. This transformation reflects the nonlinear and multiscale fracture characteristics of mudstone under high-stress conditions. The results of this study help to understand the mechanical behavior of mudstone under cyclic loading during coal mining and provide theoretical support for safe coal production. Full article

(This article belongs to the Special Issue Applications of Fractal Analysis in Underground Engineering)

17 pages, 340 KiB

Open AccessArticle

Some Results of R-Matrix Functions and Their Fractional Calculus

by Mohra Zayed and Ahmed Bakhet

Fractal Fract. 2025, 9(2), 82; https://doi.org/10.3390/fractalfract9020082 - 25 Jan 2025

Viewed by 284

Abstract

In this study, we explore various fractional integral properties of R-matrix functions using the Hilfer fractional derivative operator within the framework of fractional calculus. We introduce the

θ

integral operator and extend its definition to include the R matrix functions. The composition [...] Read more.

In this study, we explore various fractional integral properties of R-matrix functions using the Hilfer fractional derivative operator within the framework of fractional calculus. We introduce the

θ

integral operator and extend its definition to include the R matrix functions. The composition of Riemann–Liouville fractional integral and differential operators is determined using the

θ

-integral operator. Additionally, we investigate the compositional properties of

θ

-integral operators, and we establish their inversion, offering new insights into their structural and functional characteristics. Full article

(This article belongs to the Special Issue Fractional Differential Operators with Classical and New Memory Kernels)

19 pages, 334 KiB

Open AccessArticle

Criterion of the Existence of a Strongly Continuous Resolving Family for a Fractional Differential Equation with the Hilfer Derivative

by Vladimir E. Fedorov, Wei-Shih Du, Marko Kostić, Marina V. Plekhanova and Anton S. Skorynin

Fractal Fract. 2025, 9(2), 81; https://doi.org/10.3390/fractalfract9020081 - 25 Jan 2025

Viewed by 281

Abstract

In the qualitative theory of differential equations in Banach spaces, the resolving families of operators of such equations play an important role. We obtained necessary and sufficient conditions for the existence of strongly continuous resolving families of operators for a linear homogeneous equation [...] Read more.

In the qualitative theory of differential equations in Banach spaces, the resolving families of operators of such equations play an important role. We obtained necessary and sufficient conditions for the existence of strongly continuous resolving families of operators for a linear homogeneous equation resolved with respect to the Hilfer derivative. These conditions have the form of estimates on derivatives of the resolvent of a linear closed operator from the equation and generalize the Hille–Yosida conditions for infinitesimal generators of

C_{0}

-semigroups of operators. Unique solvability theorems are proved for the corresponding inhomogeneous equations. Illustrative examples of the operators from the considered classes are constructed. Full article

21 pages, 626 KiB

Open AccessArticle

Fractional Mathieu Equation with Two Fractional Derivatives and Some Applications

by Ahmed Salem, Hunida Malaikah and Naif Alsobhi

Fractal Fract. 2025, 9(2), 80; https://doi.org/10.3390/fractalfract9020080 - 24 Jan 2025

Viewed by 433

Abstract

The importance of this research comes from the several applications of the Mathieu equation and its generalizations in many scientific fields. Two models of fractional Mathieu equations are provided using Katugampola fractional derivatives in the sense of Riemann-Liouville and Caputo. Each model contains [...] Read more.

The importance of this research comes from the several applications of the Mathieu equation and its generalizations in many scientific fields. Two models of fractional Mathieu equations are provided using Katugampola fractional derivatives in the sense of Riemann-Liouville and Caputo. Each model contains two fractional derivatives with unique fractional orders, periodic forcing of the cosine stiffness coefficient, and many extensions and generalizations. The Banach contraction principle is used to prove that each model under consideration has a unique solution. Our results are applied to four real-life problems: the nonlinear Mathieu equation for parametric damping and the Duffing oscillator, the quadratically damped Mathieu equation, the fractional Mathieu equation’s transition curves, and the tempered fractional model of the linearly damped ion motion with an octopole. Full article

(This article belongs to the Section General Mathematics, Analysis)

15 pages, 3457 KiB

Open AccessArticle

Fractional Dynamical Behaviour Modelling Using Convolution Models with Non-Singular Rational Kernels: Some Extensions in the Complex Domain

by Jocelyn Sabatier

Fractal Fract. 2025, 9(2), 79; https://doi.org/10.3390/fractalfract9020079 - 24 Jan 2025

Viewed by 433

Abstract

This paper introduces a convolution model with non-singular rational kernels in which coefficients are considered complex. An interlacing property of the poles and zeros in these rational kernels permits the accurate approximation of the power law function

t^{- ν}

in a predefined [...] Read more.

This paper introduces a convolution model with non-singular rational kernels in which coefficients are considered complex. An interlacing property of the poles and zeros in these rational kernels permits the accurate approximation of the power law function

t^{- ν}

in a predefined time range, where

ν

can be complex or real. This class of model can be used to model fractional (dynamical) behaviours in order to avoid fractional calculus-based models which are now associated with several limitations. This is an extension of a previous study by the author. In the real case, this allows a better approximation, close to the limits of the approximation interval, compared to the author’s previous work. In the complex case, this extends the scope of application of the convolution models proposed by the author. Full article

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

► Show Figures

Figure 1

20 pages, 1657 KiB

Open AccessArticle

An Efficient Petrov–Galerkin Scheme for the Euler–Bernoulli Beam Equation via Second-Kind Chebyshev Polynomials

by Youssri Hassan Youssri, Waleed Mohamed Abd-Elhameed, Amr Ahmed Elmasry and Ahmed Gamal Atta

Fractal Fract. 2025, 9(2), 78; https://doi.org/10.3390/fractalfract9020078 - 24 Jan 2025

Viewed by 501

Abstract

The current article introduces a Petrov–Galerkin method (PGM) to address the fourth-order uniform Euler–Bernoulli pinned–pinned beam equation. Utilizing a suitable combination of second-kind Chebyshev polynomials as a basis in spatial variables, the proposed method elegantly and simultaneously satisfies pinned–pinned and clamped–clamped boundary conditions. [...] Read more.

The current article introduces a Petrov–Galerkin method (PGM) to address the fourth-order uniform Euler–Bernoulli pinned–pinned beam equation. Utilizing a suitable combination of second-kind Chebyshev polynomials as a basis in spatial variables, the proposed method elegantly and simultaneously satisfies pinned–pinned and clamped–clamped boundary conditions. To make PGM application easier, explicit formulas for the inner product between these basis functions and their derivatives with second-kind Chebyshev polynomials are derived. This leads to a simplified system of algebraic equations with a recognizable pattern that facilitates effective inversion to produce an approximate spectral solution. Presentations are made regarding the method’s convergence analysis and the computational cost of matrix inversion. The efficiency of the method described in precisely solving the Euler–Bernoulli beam equation under different scenarios has been validated by numerical testing. Additionally, the procedure proposed in this paper is more effective compared to other existing techniques. Full article

(This article belongs to the Special Issue Analysis and Numerical Computations of Nonlinear Fractional and Classical Differential Equations)

► Show Figures

Figure 1

25 pages, 437 KiB

Open AccessArticle

Hermite–Hadamard-Type Inequalities for Harmonically Convex Functions via Proportional Caputo-Hybrid Operators with Applications

by Saad Ihsan Butt, Muhammad Umar, Dawood Khan, Youngsoo Seol and Sanja Tipurić-Spužević

Fractal Fract. 2025, 9(2), 77; https://doi.org/10.3390/fractalfract9020077 - 24 Jan 2025

Viewed by 444

Abstract

In this paper, we aim to establish new inequalities of Hermite–Hadamard (H.H) type for harmonically convex functions using proportional Caputo-Hybrid (P.C.H) fractional operators. Parameterized by

α

, these operators offer a unique flexibility: setting

α = 1

recovers the classical inequalities for harmonically [...] Read more.

In this paper, we aim to establish new inequalities of Hermite–Hadamard (H.H) type for harmonically convex functions using proportional Caputo-Hybrid (P.C.H) fractional operators. Parameterized by

α

, these operators offer a unique flexibility: setting

α = 1

recovers the classical inequalities for harmonically convex functions, while setting

α = 0

yields inequalities for differentiable harmonically convex functions. This framework allows us to unify classical and fractional cases within a single operator. To validate the theoretical results, we provide several illustrative examples supported by graphical representations, marking the first use of such visualizations for inequalities derived via P.C.H operators. Additionally, we demonstrate practical applications of the results by deriving new fractional-order recurrence relations for the modified Bessel function of type-1, which are useful in mathematical modeling, engineering, and physics. The findings contribute to the growing body of research in fractional inequalities and harmonic convexity, paving the way for further exploration of generalized convexities and higher-order fractional operators. Full article

(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications, 2nd Edition)

► Show Figures

Figure 1

30 pages, 129873 KiB

Open AccessArticle

Dynamic Analysis of a 10-Dimensional Fractional-Order Hyperchaotic System Using Advanced Hyperchaotic Metrics

by Muhammad Sarfraz, Jiang Zhou, Mazhar Islam, Akhter Rasheed and Qi Liu

Fractal Fract. 2025, 9(2), 76; https://doi.org/10.3390/fractalfract9020076 - 24 Jan 2025

Viewed by 394

Abstract

In this paper, we propose an innovative approach to fractional-order dynamics by introducing a 10-dimensional (10D) chaotic system that leverages the intrinsic memory characteristic of the Grünwald–Letnikov (G-L) derivative. We utilize Lyapunov exponents as a quantitative measure to characterize hyperchaotic behavior, and classify [...] Read more.

In this paper, we propose an innovative approach to fractional-order dynamics by introducing a 10-dimensional (10D) chaotic system that leverages the intrinsic memory characteristic of the Grünwald–Letnikov (G-L) derivative. We utilize Lyapunov exponents as a quantitative measure to characterize hyperchaotic behavior, and classify the nature of the suggested 10D fractional-order system (FOS). While several methods exist for calculating Lyapunov exponents (LEs) through the utilization of integer-order systems, these approaches are not applicable for FOS due to its non-local nature. Initially, the system dynamics are thoroughly examined through Lyapunov exponents and bifurcation analysis, considering the influence of both state variables and fractional orders. To assess the hyperchaotic behavior of the proposed model, sensitivity analyses are conducted by exploring changes in state variables under two distinct initial conditions, along with time history simulations for various parameter settings. Furthermore, we examine the impact of different fractional-order sets on the system’s dynamics. A comprehensive performance comparison is conducted between the proposed 10-dimensional fractional-order hyperchaotic system and several existing hyperchaotic systems. This comparison utilizes advanced metrics, including the Kolmogorov–Sinai (KS) entropy, Kaplan–Yorke dimension, the Perron effect analysis, and the 0-1 test for chaos. Simulation outcomes reveal that the proposed system surpasses existing algorithms, delivering improved precision and accuracy. Full article

16 pages, 399 KiB

Open AccessArticle

Fractional Calculus of Piecewise Continuous Functions

by Manuel Duarte Ortigueira

Fractal Fract. 2025, 9(2), 75; https://doi.org/10.3390/fractalfract9020075 - 24 Jan 2025

Viewed by 556

Abstract

The fractional derivative computation of piecewise continuous functions is treated with generality. It is shown why some applications give incorrect results and why Caputo derivative give strange results. Some examples are described. Full article

(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems)

► Show Figures

Figure 1

21 pages, 2014 KiB

Open AccessArticle

A New Chaotic Weak Signal Detection Method Based on a Simplified Fractional-Order Genesio–Tesi Chaotic System

by Hongcun Mao, Yuling Feng, Xiaoqian Wang, Chao Gao, Changhao Lin and Zhihai Yao

Fractal Fract. 2025, 9(2), 74; https://doi.org/10.3390/fractalfract9020074 - 24 Jan 2025

Viewed by 368

Abstract

The detection of weak signals is a well-established application in chaos theory. This theory leverages the inherent robustness of chaotic systems, enabling them to resist noise and thus serve as effective tools for identifying weak signals. However, challenges remain in selecting appropriate chaotic [...] Read more.

The detection of weak signals is a well-established application in chaos theory. This theory leverages the inherent robustness of chaotic systems, enabling them to resist noise and thus serve as effective tools for identifying weak signals. However, challenges remain in selecting appropriate chaotic systems and in their practical implementation—areas that are still under-explored. In this paper, we analyze a simplified fractional-order Genesio–Tesi chaotic system, which exhibits a unique chaos-divergence characteristic. Based on this characteristic, we propose a new detection method that uses the chaos-divergence state as a criterion for determining the presence or absence of a signal when detecting weak signal amplitudes. This approach makes the simplified fractional-order Genesio–Tesi chaotic system more suitable for chaotic weak signal detection. Notably, the significant variance observed in the divergent state’s independent variables emerges as a key feature, enhancing the system’s ability to detect the frequencies of weak signals. Our numerical simulations focus on detecting weak cosine signals masked by three different types of noise. The results demonstrate successful detection of a weak signal at a frequency of 100 rad/s under the specified conditions, with the lowest detectable signal-to-noise ratio of −40.83 dB. Overall, these results highlight the effectiveness and feasibility of our proposed method for weak signal detection. Full article

(This article belongs to the Special Issue Design, Optimization and Applications for Fractional Chaotic System)

► Show Figures

Figure 1

19 pages, 2883 KiB

Open AccessArticle

Nonlinear Analysis of the U.S. Stock Market: From the Perspective of Multifractal Properties and Cross-Correlations with Comparisons

by Chenyu Han and Yingying Xu

Fractal Fract. 2025, 9(2), 73; https://doi.org/10.3390/fractalfract9020073 - 24 Jan 2025

Viewed by 325

Abstract

This study investigates the multifractal properties of daily returns of the Standard and Poor’s 500 Index (SPX), the Dow Jones Industrial Average (DJI), and the Nasdaq Composite Index (IXIC), the three main indices representing the U.S. stock market, from 1 January 2005 to [...] Read more.

This study investigates the multifractal properties of daily returns of the Standard and Poor’s 500 Index (SPX), the Dow Jones Industrial Average (DJI), and the Nasdaq Composite Index (IXIC), the three main indices representing the U.S. stock market, from 1 January 2005 to 1 November 2024. The multifractal detrended fluctuation analysis (MF-DFA) method is applied in this study. The origins of the multifractal properties of these returns are both long-range correlation and fat-tail distribution properties. Our findings show that the SPX exhibits the highest multifractal degree, and the DJI exhibits the lowest for the whole sample. This study also examines the multifractal behaviors of cross-correlations among the three major indices through the multifractal detrended cross-correlation analysis (MF-DCCA) method. It is concluded that the indices are cross-correlated and the cross-correlations also exhibit multifractal properties. Meanwhile, these returns exhibit different multifractal properties in different stages of the market, which shows some asymmetrical dynamics of the multifractal properties. These empirical results may have some important managerial and academic implications for investors, policy makers, and other market participants. Full article

(This article belongs to the Special Issue Application of Fractal Processes and Fractional Derivatives in Finance, 2nd Edition)

► Show Figures

Figure 1

31 pages, 17297 KiB

Open AccessArticle

Construction of the Closed Form Wave Solutions for TFSMCH and (1 1) Dimensional TFDMBBM Equations via the EMSE Technique⁺

by Md. Asaduzzaman and Farhana Jesmin

Fractal Fract. 2025, 9(2), 72; https://doi.org/10.3390/fractalfract9020072 - 24 Jan 2025

Viewed by 728

Abstract

The purpose of this study is to investigate a series of novel exact closed form traveling wave solutions for the TFSMCH equation and (1 + 1) dimensional TFDMBBM equation using the EMSE technique. The considered FONLEEs are used to delineate the characteristic of [...] Read more.

The purpose of this study is to investigate a series of novel exact closed form traveling wave solutions for the TFSMCH equation and (1 + 1) dimensional TFDMBBM equation using the EMSE technique. The considered FONLEEs are used to delineate the characteristic of diffusion in the creation of shapes in liquid beads arising in plasma physics and fluid flow and to estimate the external long waves in nonlinear dispersive media. These equations are also used to characterize various types of waves, such as hydromagnetic waves, acoustic waves, and acoustic gravity waves. Here, we utilize the Caputo-type fractional order derivative to fractionalize the considered FONLEEs. Some trigonometric and hyperbolic trigonometric functions have been used to represent the obtained closed form traveling wave solutions. Furthermore, here, we reveal that the EMSE technique is a suitable, significant, and dominant mathematical tool for finding the exact traveling wave solutions for various FONLEEs in mathematical physics. We draw some 3D, 2D, and contour plots to describe the various wave behaviors and analyze the physical consequence of the attained solutions. Finally, we make a numerical comparison of our obtained solutions and other analogous solutions obtained using various techniques. Full article

(This article belongs to the Special Issue Recent Developments and Applications of Fractional Differential Equations in Mathematical Physics)

► Show Figures

Figure 1

31 pages, 20594 KiB

Open AccessArticle

Dynamic Failure Mechanism and Fractal Features of Fractured Rocks Under Quasi-Triaxial Static Pressures and Repeated Impact Loading

by Peng Li, Yan Liu, Jie Zhang, Zhihong Dong, Xinghui Wu, Shengjun Miao and Meifeng Cai

Fractal Fract. 2025, 9(2), 71; https://doi.org/10.3390/fractalfract9020071 - 23 Jan 2025

Viewed by 471

Abstract

Mastering the dynamic mechanical behaviors of pre-stressed fractured rocks under repeated impact loads is crucial for safety management in rock engineering. To achieve this, repeated impact loading experiments were performed on produced fractured samples exposed to varying pre-applied axial and confining pressures using [...] Read more.

Mastering the dynamic mechanical behaviors of pre-stressed fractured rocks under repeated impact loads is crucial for safety management in rock engineering. To achieve this, repeated impact loading experiments were performed on produced fractured samples exposed to varying pre-applied axial and confining pressures using a split Hopkinson pressure bar test system in combination with a nuclear magnetic resonance imaging system, and the dynamic failure mechanism and fractal features were investigated. The results indicate that the dynamic stress–strain curves exemplify typical class II curves, and the strain rebound progressively diminishes with growing impact times. The impact times, axial pressure, and confining pressure all significantly affect the dynamic peak strength, average dynamic strength, dynamic deformation modulus, average dynamic deformation modulus, maximum strain, and impact resistance performance. Moreover, under low confining pressures, numerous shear cracks and tensile cracks develop, which are interconnected and converge to form large-scale macroscopic fracture surfaces. In contrast, specimens under a high confining pressure primarily experience tensile failure, accompanied by localized small-scale shear failure. Under low axial pressure, some shear cracks and tensile cracks emerge, while at high axial pressure, anti-wing cracks and secondary coplanar cracks occur, characterized predominantly by shear failure. In addition, as the confining pressure grows from 8 to 20 MPa, the fractal dimensions are 2.44, 2.32, 2.23, and 2.12, respectively. When the axial pressures are 8, 14, and 20 MPa, the fractal dimensions are 2.44, 2.46, and 2.52, respectively. Overall, the degree of fragmentation of the sample decreases with growing confining pressure and grows with rising axial pressure. Full article

(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering)

► Show Figures

Figure 1

13 pages, 282 KiB

Open AccessArticle

New Results on a Nonlocal Sturm–Liouville Eigenvalue Problem with Fractional Integrals and Fractional Derivatives

by Yunyang Zhang, Shaojie Chen and Jing Li

Fractal Fract. 2025, 9(2), 70; https://doi.org/10.3390/fractalfract9020070 - 23 Jan 2025

Viewed by 476

Abstract

In this paper, we investigate the eigenvalue properties of a nonlocal Sturm–Liouville equation involving fractional integrals and fractional derivatives under different boundary conditions. Based on these properties, we obtained the geometric multiplicity of eigenvalues for the nonlocal Sturm–Liouville problem with a non-Dirichlet boundary [...] Read more.

In this paper, we investigate the eigenvalue properties of a nonlocal Sturm–Liouville equation involving fractional integrals and fractional derivatives under different boundary conditions. Based on these properties, we obtained the geometric multiplicity of eigenvalues for the nonlocal Sturm–Liouville problem with a non-Dirichlet boundary condition. Furthermore, we discussed the continuous dependence of the eigenvalues on the potential function for a nonlocal Sturm–Liouville equation under a Dirichlet boundary condition. Full article

(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications, 2nd Edition)

35 pages, 2120 KiB

Open AccessArticle

Fractional Transfer Entropy Networks: Short- and Long-Memory Perspectives on Global Stock Market Interactions

by Ömer Akgüller, Mehmet Ali Balcı, Larissa Margareta Batrancea and Lucian Gaban

Fractal Fract. 2025, 9(2), 69; https://doi.org/10.3390/fractalfract9020069 - 23 Jan 2025

Viewed by 500

Abstract

This study addresses the challenge of capturing both short-run volatility and long-run dependencies in global stock markets by introducing fractional transfer entropy (FTE), a new framework that embeds fractional calculus into transfer entropy. FTE allows analysts to tune memory parameters and thus observe [...] Read more.

This study addresses the challenge of capturing both short-run volatility and long-run dependencies in global stock markets by introducing fractional transfer entropy (FTE), a new framework that embeds fractional calculus into transfer entropy. FTE allows analysts to tune memory parameters and thus observe how different temporal emphases reshape the network of directional information flows among major financial indices. Empirical evidence reveals that when short-memory effects dominate, markets swiftly incorporate recent news, creating networks that adapt quickly but remain vulnerable to transient shocks. In contrast, balanced memory parameters yield a more stable equilibrium, blending immediate reactions with persistent structural ties. Under long-memory configurations, historically entrenched relationships prevail, enabling established market leaders to remain central despite ongoing fluctuations. These findings demonstrate that FTE uncovers nuanced dynamics overlooked by methods focusing solely on either current events or deep-rooted patterns. Although the method relies on price returns and does not differentiate specific shock types, it offers a versatile tool for investors, policymakers, and researchers to gauge financial stability, evaluate contagion risk, and better understand how ephemeral signals and historical legacies jointly govern global market connectivity. Full article

(This article belongs to the Special Issue Complexity, Fractality and Fractional Dynamics Applied to Science and Engineering)

► Show Figures

Figure 1

25 pages, 1605 KiB

Open AccessArticle

Analysis of an Acute Diarrhea Piecewise Modified ABC Fractional Model: Optimal Control, Stability and Simulation

by Yasir A. Madani, Mohammed A. Almalahi, Osman Osman, Blgys Muflh, Khaled Aldwoah, Khidir Shaib Mohamed and Nidal Eljaneid

Fractal Fract. 2025, 9(2), 68; https://doi.org/10.3390/fractalfract9020068 - 23 Jan 2025

Viewed by 379

Abstract

Acute diarrhea poses a significant global health challenge, especially in settings with poor sanitation. This study develops a mathematical model of diarrhea, employing a piecewise modified ABC (pmABC) fractional derivative to capture the disease’s transmission dynamics, including crossover effects between classical and fractional [...] Read more.

Acute diarrhea poses a significant global health challenge, especially in settings with poor sanitation. This study develops a mathematical model of diarrhea, employing a piecewise modified ABC (pmABC) fractional derivative to capture the disease’s transmission dynamics, including crossover effects between classical and fractional behaviors. We analyze the local and global stability of the disease-free equilibrium and calculate the basic reproduction number

R_{0}

using the next-generation matrix method. Furthermore, we formulate an optimal control model that incorporates both strategies to reduce contact between susceptible and infected individuals, and to treat infected patients. Numerical simulations demonstrate the model’s behavior, illustrating that enhanced hygiene compliance reduces

R_{0}

by decreasing contact rates, while increased effective contact rates elevate

R_{0}

. Additionally, the simulations reveal a positive correlation between higher concentrations of acute diarrhea bacteria and increased rates of subsequent infections. Full article

(This article belongs to the Special Issue Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods and Application)

► Show Figures

Figure 1

5 pages, 163 KiB

Open AccessEditorial

Fractional-Order Complex Systems: Advanced Control, Intelligent Estimation and Reinforcement Learning Image-Processing Algorithms

by Jin-Xi Zhang, Xuefeng Zhang, Driss Boutat and Da-Yan Liu

Fractal Fract. 2025, 9(2), 67; https://doi.org/10.3390/fractalfract9020067 - 23 Jan 2025

Viewed by 424

Abstract

In this Special Issue on “Applications of Fractional Operators in Image Processing and Stability of Control Systems”, more than 20 high-quality papers have been published [...] Full article

(This article belongs to the Special Issue Fractional Order Complex Systems: Advanced Control, Intelligent Estimation and Reinforcement Learning Image Processing Algorithms)

Journal Menu

Journal Browser

Fractal Fract., Volume 9, Issue 2 (February 2025) – 41 articles

Further Information

Guidelines

MDPI Initiatives

Follow MDPI