Fractal and Fractional

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

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)

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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

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

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

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

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

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

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

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

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)

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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

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)

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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

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

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)

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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

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

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

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

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

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)

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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

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)

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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

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)

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