Fractal and Fractional

23 pages, 3312 KB

Open AccessEditor’s ChoiceArticle

Automatic Picking Method for the First Arrival Time of Microseismic Signals Based on Fractal Theory and Feature Fusion

by Huicong Xu, Kai Li, Pengfei Shan, Xuefei Wu, Shuai Zhang, Zeyang Wang, Chenguang Liu, Zhongming Yan, Liang Wu and Huachuan Wang

Fractal Fract. 2025, 9(11), 679; https://doi.org/10.3390/fractalfract9110679 - 23 Oct 2025

Cited by 38 | Viewed by 1790

Abstract

Microseismic signals induced by mining activities often have low signal-to-noise ratios, and traditional picking methods are easily affected by noise, making accurate identification of P-wave arrivals difficult. To address this problem, this study proposes an adaptive denoising algorithm based on wavelet-threshold-enhanced Complete Ensemble [...] Read more.

Microseismic signals induced by mining activities often have low signal-to-noise ratios, and traditional picking methods are easily affected by noise, making accurate identification of P-wave arrivals difficult. To address this problem, this study proposes an adaptive denoising algorithm based on wavelet-threshold-enhanced Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) and develops an automatic P-wave arrival picking method incorporating fractal box dimension features, along with a corresponding accuracy evaluation framework. The raw microseismic signals are decomposed using the improved CEEMDAN method, with high-frequency intrinsic mode functions (IMFs) processed by wavelet-threshold denoising and low- and mid-frequency IMFs retained for reconstruction, effectively suppressing background noise and enhancing signal clarity. Fractal box dimension is applied to characterize waveform complexity over short and long-time windows, and by introducing fractal derivatives and short-long window differences, abrupt changes in local-to-global complexity at P-wave arrivals are revealed. Energy mutation features are extracted using the short-term/long-term average (STA/LTA) energy ratio, and noise segments are standardized via Z-score processing. A multi-feature weighted fusion scoring function is constructed to achieve robust identification of P-wave arrivals. Evaluation metrics, including picking error, mean absolute error, and success rate, are used to comprehensively assess the method’s performance in terms of temporal deviation, statistical consistency, and robustness. Case studies using microseismic data from a mining site show that the proposed method can accurately identify P-wave arrivals under different signal-to-noise conditions, with automatic picking results highly consistent with manual labels, mean errors within the sampling interval (2–4 ms), and a picking success rate exceeding 95%. The method provides a reliable tool for seismic source localization and dynamic hazard prediction in mining microseismic monitoring. Full article

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

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25 pages, 4931 KB

Open AccessEditor’s ChoiceArticle

Optical Multi-Peakon Dynamics in the Fractional Cubic–Quintic Nonlinear Pulse Propagation Model Using a Novel Integral Approach

by Ejaz Hussain, Aljethi Reem Abdullah, Khizar Farooq and Usman Younas

Fractal Fract. 2025, 9(10), 631; https://doi.org/10.3390/fractalfract9100631 - 28 Sep 2025

Cited by 15 | Viewed by 1348

Abstract

This study examines the soliton dynamics in the time-fractional cubic–quintic nonlinear non-paraxial propagation model, applicable to optical signal processing, nonlinear optics, fiber-optic communication, and biomedical laser–tissue interactions. The fractional framework exhibits a wide range of nonlinear effects, such as self-phase modulation, wave mixing, [...] Read more.

This study examines the soliton dynamics in the time-fractional cubic–quintic nonlinear non-paraxial propagation model, applicable to optical signal processing, nonlinear optics, fiber-optic communication, and biomedical laser–tissue interactions. The fractional framework exhibits a wide range of nonlinear effects, such as self-phase modulation, wave mixing, and self-focusing, arising from the balance between cubic and quintic nonlinearities. By employing the Multivariate Generalized Exponential Rational Integral Function (MGERIF) method, we derive an extensive catalog of analytic solutions, multi-peakon structures, lump solitons, kinks, and bright and dark solitary waves, while periodic and singular solutions emerge as special cases. These outcomes are systematically constructed within a single framework and visualized through 2D, 3D, and contour plots under both anomalous and normal dispersion regimes. The analysis also addresses modulation instability (MI), interpreted as a sideband amplification of continuous-wave backgrounds that generates pulse trains and breather-type structures. Our results demonstrate that cubic–quintic contributions substantially affect MI gain spectrum, broadening instability bands and permitting MI beyond the anomalous-dispersion regime. These findings directly connect the obtained solution classes to experimentally observed routes for solitary wave shaping, pulse propagation, and instability and instability-driven waveform formation in optical communication devices, photonic platforms, and laser technologies. Full article

(This article belongs to the Special Issue Recent Computational Methods for Fractal and Fractional Nonlinear Partial Differential Equations)

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26 pages, 32601 KB

Open AccessEditor’s ChoiceArticle

Dynamic Analysis and FPGA Implementation of a Fractional-Order Memristive Hopfield Neural Network with Hidden Chaotic Dual-Wing Attractors

by Shaoqi He, Fei Yu, Rongyao Guo, Mingfang Zheng, Tinghui Tang, Jie Jin and Chunhua Wang

Fractal Fract. 2025, 9(9), 561; https://doi.org/10.3390/fractalfract9090561 - 26 Aug 2025

Cited by 46 | Viewed by 2178

Abstract

To model the response of neural networks to electromagnetic radiation in real-world environments, this study proposes a memristive dual-wing fractional-order Hopfield neural network (MDW-FOMHNN) model, utilizing a fractional-order memristor to simulate neuronal responses to electromagnetic radiation, thereby achieving complex chaotic dynamics. Analysis reveals [...] Read more.

To model the response of neural networks to electromagnetic radiation in real-world environments, this study proposes a memristive dual-wing fractional-order Hopfield neural network (MDW-FOMHNN) model, utilizing a fractional-order memristor to simulate neuronal responses to electromagnetic radiation, thereby achieving complex chaotic dynamics. Analysis reveals that within specific ranges of the coupling strength, the MDW-FOMHNN lacks equilibrium points and exhibits hidden chaotic attractors. Numerical solutions are obtained using the Adomian Decomposition Method (ADM), and the system’s chaotic behavior is confirmed through Lyapunov exponent spectra, bifurcation diagrams, phase portraits, and time series. The study further demonstrates that the coupling strength and fractional order significantly modulate attractor morphologies, revealing diverse attractor structures and their coexistence. The complexity of the MDW-FOMHNN output sequence is quantified using spectral entropy, highlighting the system’s potential for applications in cryptography and related fields. Based on the polynomial form derived from ADM, a field programmable gate array (FPGA) implementation scheme is developed, and the expected chaotic attractors are successfully generated on an oscilloscope, thereby validating the consistency between theoretical analysis and numerical simulations. Finally, to link theory with practice, a simple and efficient MDW-FOMHNN-based encryption/decryption scheme is presented. Full article

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

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19 pages, 2887 KB

Open AccessEditor’s ChoiceArticle

Multifractal Characterization of Heterogeneous Pore Water Redistribution and Its Influence on Permeability During Depletion: Insights from Centrifugal NMR Analysis

by Fangkai Quan, Wei Lu, Yu Song, Wenbo Sheng, Zhengyuan Qin and Huogen Luo

Fractal Fract. 2025, 9(8), 536; https://doi.org/10.3390/fractalfract9080536 - 15 Aug 2025

Cited by 21 | Viewed by 1358

Abstract

The dynamic process of water depletion plays a critical role in both surface coalbed methane (CBM) development and underground gas extraction, reshaping water–rock interactions and inducing complex permeability responses. Addressing the limited understanding of the coupling mechanism between heterogeneous pore water evolution and [...] Read more.

The dynamic process of water depletion plays a critical role in both surface coalbed methane (CBM) development and underground gas extraction, reshaping water–rock interactions and inducing complex permeability responses. Addressing the limited understanding of the coupling mechanism between heterogeneous pore water evolution and permeability during dynamic processes, this study simulates reservoir transitions across four zones (prospective planning, production preparation, active production, and mining-affected zones) via centrifugal experiments. The results reveal a pronounced scale dependence in pore water distribution. During low-pressure stages (0–0.54 MPa), rapid drainage from fractures and seepage pores leads to a ~12% reduction in total water content. In contrast, high-pressure stages (0.54–3.83 MPa) promote water retention in adsorption pores, with their relative contribution rising to 95.8%, forming a dual-structure of macropore drainage and micropore retention. Multifractal analysis indicates a dual-mode evolution of movable pore space. Under low centrifugal pressure, D₋₁₀ and Δα decrease by approximately 34% and 36%, respectively, reflecting improved connectivity within large-pore networks. At high centrifugal pressure, an ~8% increase in D₀−D₂ suggests that pore-scale heterogeneity in adsorption pores inhibits further seepage. A quantitative coupling model establishes a quadratic relationship between fractal parameters and permeability, illustrating that permeability enhancement results from the combined effects of pore volume expansion and structural homogenization. As water saturation decreases from 1.0 to 0.64, permeability increases by more than 3.5 times. These findings offer theoretical insights into optimizing seepage pathways and improving gas recovery efficiency in dynamically evolving reservoirs. Full article

(This article belongs to the Special Issue Multiscale Fractal Analysis in Unconventional Reservoirs)

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16 pages, 1993 KB

Open AccessEditor’s ChoiceArticle

A Fractional Derivative Insight into Full-Stage Creep Behavior in Deep Coal

by Shuai Yang, Hongchen Song, Hongwei Zhou, Senlin Xie, Lei Zhang and Wentao Zhou

Fractal Fract. 2025, 9(7), 473; https://doi.org/10.3390/fractalfract9070473 - 21 Jul 2025

Cited by 19 | Viewed by 1904

Abstract

The time-dependent creep behavior of coal is essential for assessing long-term structural stability and operational safety in deep coal mining. Therefore, this work develops a full-stage creep constitutive model. By integrating fractional calculus theory with statistical damage mechanics, a nonlinear fractional-order (FO) damage [...] Read more.

The time-dependent creep behavior of coal is essential for assessing long-term structural stability and operational safety in deep coal mining. Therefore, this work develops a full-stage creep constitutive model. By integrating fractional calculus theory with statistical damage mechanics, a nonlinear fractional-order (FO) damage creep model is constructed through serial connection of elastic, viscous, viscoelastic, and viscoelastic–plastic components. Based on this model, both one-dimensional and three-dimensional (3D) fractional creep damage constitutive equations are acquired. Model parameters are identified using experimental data from deep coal samples in the mining area. The result curves of the improved model coincide with experimental data points, accurately describing the deceleration creep stage (DCS), steady-state creep stage (SCS), and accelerated creep stage (ACS). Furthermore, a sensitivity analysis elucidates the impact of model parameters on coal creep behavior, thereby confirming the model’s robustness and applicability. Consequently, the proposed model offers a solid theoretical basis for evaluating the sustained stability of deep coal mining and has great application potential in deep underground engineering. Full article

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27 pages, 8698 KB

Open AccessEditor’s ChoiceArticle

A Multidisciplinary Approach to Triangular Shapes: Philosophy, Art, Mathematical Properties, and Application Purposes for High-Frequency Signal Processing Using Sierpiński Geometry

by Romolo Marcelli

Fractal Fract. 2025, 9(7), 444; https://doi.org/10.3390/fractalfract9070444 - 4 Jul 2025

Cited by 1 | Viewed by 5855

Abstract

Triangular shapes have been studied from different perspectives over a wide temporal frame since ancient times. Initially, fundamental theorems were formulated to demonstrate their geometrical properties. Philosophy and art leveraged the peculiar aspects of triangles as building blocks for more complex geometrical shapes. [...] Read more.

Triangular shapes have been studied from different perspectives over a wide temporal frame since ancient times. Initially, fundamental theorems were formulated to demonstrate their geometrical properties. Philosophy and art leveraged the peculiar aspects of triangles as building blocks for more complex geometrical shapes. This paper will review triangles by adopting a multidisciplinary approach, recalling ancient science and Plato’s arguments in relation to their connection with philosophy. It will then consider the artistic utilization of triangles, particularly in compositions created during the medieval era, as exemplified by the Cosmati Italian family’s masterpieces. Various scientific environments have explored triangular 2D and 3D shapes for different purposes, which will be briefly reviewed here. After that, Sierpiński geometry and its properties will be introduced, focusing on the equilateral shape and its internal complexity generated by subdividing the entire triangle into smaller sub-triangles. Finally, examples of triangular planar shapes that fulfill the Sierpiński geometry will be presented as an application in signal processing for high-frequency signals in the microwave and millimeter-wave range. Full article

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13 pages, 2490 KB

Open AccessEditor’s ChoiceArticle

Soliton Dynamics of the Nonlinear Kodama Equation with M-Truncated Derivative via Two Innovative Schemes: The Generalized Arnous Method and the Kudryashov Method

by Khizar Farooq, Ali. H. Tedjani, Zhao Li and Ejaz Hussain

Fractal Fract. 2025, 9(7), 436; https://doi.org/10.3390/fractalfract9070436 - 2 Jul 2025

Cited by 16 | Viewed by 1432

Abstract

The primary aim of this research article is to investigate the soliton dynamics of the M-truncated derivative nonlinear Kodama equation, which is useful for optical solitons on nonlinear media, shallow water waves over complex media, nonlocal internal waves, and fractional viscoelastic wave propagation. [...] Read more.

The primary aim of this research article is to investigate the soliton dynamics of the M-truncated derivative nonlinear Kodama equation, which is useful for optical solitons on nonlinear media, shallow water waves over complex media, nonlocal internal waves, and fractional viscoelastic wave propagation. We utilized two recently developed analytical techniques, the generalized Arnous method and the generalized Kudryashov method. First, the nonlinear Kodama equation is transformed into a nonlinear ordinary differential equation using the homogeneous balance principle and a traveling wave transformation. Next, various types of soliton solutions are constructed through the application of these effective methods. Finally, to visualize the behavior of the obtained solutions, three-dimensional, two-dimensional, and contour plots are generated using Maple (2023) mathematical software. Full article

(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics, 2nd Edition)

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26 pages, 23383 KB

Open AccessEditor’s ChoiceArticle

Multi-Focus Image Fusion Based on Dual-Channel Rybak Neural Network and Consistency Verification in NSCT Domain

by Ming Lv, Sensen Song, Zhenhong Jia, Liangliang Li and Hongbing Ma

Fractal Fract. 2025, 9(7), 432; https://doi.org/10.3390/fractalfract9070432 - 30 Jun 2025

Cited by 15 | Viewed by 2228

Abstract

In multi-focus image fusion, accurately detecting and extracting focused regions remains a key challenge. Some existing methods suffer from misjudgment of focus areas, resulting in incorrect focus information or the unintended retention of blurred regions in the fused image. To address these issues, [...] Read more.

In multi-focus image fusion, accurately detecting and extracting focused regions remains a key challenge. Some existing methods suffer from misjudgment of focus areas, resulting in incorrect focus information or the unintended retention of blurred regions in the fused image. To address these issues, this paper proposes a novel multi-focus image fusion method that leverages a dual-channel Rybak neural network combined with consistency verification in the nonsubsampled contourlet transform (NSCT) domain. Specifically, the high-frequency sub-bands produced by NSCT decomposition are processed using the dual-channel Rybak neural network and a consistency verification strategy, allowing for more accurate extraction and integration of salient details. Meanwhile, the low-frequency sub-bands are fused using a simple averaging approach to preserve the overall structure and brightness information. The effectiveness of the proposed method has been thoroughly evaluated through comprehensive qualitative and quantitative experiments conducted on three widely used public datasets: Lytro, MFFW, and MFI-WHU. Experimental results show that our method consistently outperforms several state-of-the-art image fusion techniques, including both traditional algorithms and deep learning-based approaches, in terms of visual quality and objective performance metrics (

Q_{A B / F}

,

Q_{C B}

,

Q_{E}

,

Q_{F M I}

,

Q_{M I}

,

Q_{M S E}

,

Q_{N C I E}

,

Q_{N M I}

,

Q_{P}

, and

Q_{P S N R}

). These results clearly demonstrate the robustness and superiority of the proposed fusion framework in handling multi-focus image fusion tasks. Full article

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

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19 pages, 1286 KB

Open AccessEditor’s ChoiceArticle

Adsorption–Desorption at Anomalous Diffusion: Fractional Calculus Approach

by Ivan Bazhlekov and Emilia Bazhlekova

Fractal Fract. 2025, 9(7), 408; https://doi.org/10.3390/fractalfract9070408 - 24 Jun 2025

Cited by 7 | Viewed by 2071

Abstract

A mathematical model of the anomalous diffusion of surfactant and the process of adsorption–desorption on an interface is analyzed using a fractional calculus approach. The model is based on time-fractional partial differential equations in the bulk phases and the corresponding time-fractional description of [...] Read more.

A mathematical model of the anomalous diffusion of surfactant and the process of adsorption–desorption on an interface is analyzed using a fractional calculus approach. The model is based on time-fractional partial differential equations in the bulk phases and the corresponding time-fractional description of the flux bulk–interface. The general case, when the surfactant is soluble in both phases, is considered under the assumption that the adsorption–desorption process is diffusion-controlled. Some of the most popular kinetic models of Henry, Langmuir, and Volmer are considered. Applying the Laplace transform, the partial differential model is transformed into a single multi-term time-fractional nonlinear ordinary differential equation for the surfactant concentration on the interface. Based on existing analytical solutions of linear time-fractional differential equations, the exact solution in the case of the Henry model is derived in terms of multinomial Mittag–Leffler functions, and its asymptotic behavior is studied. Further, the fractional differential model in the general nonlinear case is rewritten as an integral equation, which is a generalization of the well-known Ward–Tordai equation. For computer simulations, based on the obtained integral equation, a predictor–corrector numerical technique is developed. Numerical results are presented and analyzed. Full article

(This article belongs to the Special Issue Analysis of Heat Conduction and Anomalous Diffusion in Fractional Calculus)

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13 pages, 2314 KB

Open AccessEditor’s ChoiceArticle

A Novel Approach to Solving Fractional Navier–Stokes Equations Using the Elzaki Transform

by Tarig M. Elzaki and Eltaib M. Abd Elmohmoud

Fractal Fract. 2025, 9(6), 396; https://doi.org/10.3390/fractalfract9060396 - 19 Jun 2025

Cited by 3 | Viewed by 2259

Abstract

A clear method is provided to explain a new approach for solving systems of fractional Navier–Stokes equations (SFNSEs) using initial conditions (ICs) that rely on the Elzaki transform (ET). A few steps show the technique’s validity and utility for handling SFNSE solutions. For [...] Read more.

A clear method is provided to explain a new approach for solving systems of fractional Navier–Stokes equations (SFNSEs) using initial conditions (ICs) that rely on the Elzaki transform (ET). A few steps show the technique’s validity and utility for handling SFNSE solutions. For fractional derivatives, the Caputo sense is used. This method does not need discretization or limiting assumptions and may be used to solve both linear and nonlinear SFNSEs. By eliminating round-off mistakes, the technique reduces the need for numerical calculations. Using examples, the new technique’s accuracy and efficacy are illustrated. Full article

(This article belongs to the Special Issue Recent Computational Methods for Fractal and Fractional Nonlinear Partial Differential Equations)

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31 pages, 6790 KB

Open AccessEditor’s ChoiceArticle

Proposal for the Application of Fractional Operators in Polynomial Regression Models to Enhance the Determination Coefficient R² on Unseen Data

by Anthony Torres-Hernandez, Rafael Ramirez-Melendez and Fernando Brambila-Paz

Fractal Fract. 2025, 9(6), 393; https://doi.org/10.3390/fractalfract9060393 - 19 Jun 2025

Cited by 3 | Viewed by 6288

Abstract

Since polynomial regression models are generally quite reliable for data that can be handled using a linear system, it is important to note that in some cases, they may suffer from overfitting during the training phase. This can lead to negative values of [...] Read more.

Since polynomial regression models are generally quite reliable for data that can be handled using a linear system, it is important to note that in some cases, they may suffer from overfitting during the training phase. This can lead to negative values of the coefficient of determination

R^{2}

when applied to unseen data. To address this issue, this work proposes the partial implementation of fractional operators in polynomial regression models to construct a fractional regression model. The aim of this approach is to mitigate overfitting, which could potentially improve the

R^{2}

value for unseen data compared to the conventional polynomial model, under the assumption that this could lead to predictive models with better performance. The methodology for constructing these fractional regression models is presented along with examples applicable to both Riemann–Liouville and Caputo fractional operators, where some results show that regions with initially negative or near-zero

R^{2}

values exhibit remarkable improvements after the application of the fractional operator, with absolute relative increases exceeding

800 %

on unseen data. Finally, the importance of employing sets in the construction of the fractional regression model within this methodological framework is emphasized, since from a theoretical standpoint, one could construct an uncountable family of fractional operators derived from the Riemann–Liouville and Caputo definitions that, although differing in their formulation, would yield the same regression results as those shown in the examples presented in this work. Full article

(This article belongs to the Special Issue Fractional Calculus Applications for Signal Processing and Data Science)

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11 pages, 2066 KB

Open AccessEditor’s ChoiceArticle

Numerical and Analytical Study of the Magnetic Field Distribution in a Three-Solenoid System

by Mostafa Behtouei, Alberto Bacci, Martina Carillo, Moreno Comelli, Luigi Faillace, Mauro Migliorati, Livio Verra and Bruno Spataro

Fractal Fract. 2025, 9(6), 383; https://doi.org/10.3390/fractalfract9060383 - 16 Jun 2025

Viewed by 2270

Abstract

This study investigates the magnetic fields produced by a three-solenoid system configuration using both traditional numerical solvers and fractional integral methods. We focus on the role of mesh resolution in influencing simulation accuracy, examining coils with dimensions 80 mm × 160 mm and [...] Read more.

This study investigates the magnetic fields produced by a three-solenoid system configuration using both traditional numerical solvers and fractional integral methods. We focus on the role of mesh resolution in influencing simulation accuracy, examining coils with dimensions 80 mm × 160 mm and a radius of 15.5 mm, each carrying a current of 200 A. Magnetic field behavior is analyzed along a line parallel to the central axis at a distance equal to half the solenoid’s radius. The fractional integral formulation employed provides a refined understanding of field variations, especially in off-axis regions. Comparisons with the Poisson solver highlight consistency across methods and suggest pathways for further optimization. The results support the potential of fractional approaches in advancing electromagnetic field modeling, particularly in accelerator and beamline applications. Full article

(This article belongs to the Special Issue Applications of Fractals and Fractional Calculus in Nuclear Reactors)

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35 pages, 24700 KB

Open AccessEditor’s ChoiceArticle

Optimizing Load Frequency Control of Multi-Area Power Renewable and Thermal Systems Using Advanced Proportional–Integral–Derivative Controllers and Catch Fish Algorithm

by Saleh A. Alnefaie, Abdulaziz Alkuhayli and Abdullah M. Al-Shaalan

Fractal Fract. 2025, 9(6), 355; https://doi.org/10.3390/fractalfract9060355 - 29 May 2025

Cited by 20 | Viewed by 3156

Abstract

Renewable energy sources (RESs) are increasingly combined into the power system due to market liberalization and environmental and economic benefits, but their weather-dependent variability causes power production and demand mismatches, leading to issues like frequency and regional power transmission fluctuations. To maintain synchronization [...] Read more.

Renewable energy sources (RESs) are increasingly combined into the power system due to market liberalization and environmental and economic benefits, but their weather-dependent variability causes power production and demand mismatches, leading to issues like frequency and regional power transmission fluctuations. To maintain synchronization in power systems, frequency must remain constant; disruptions in the proper balance of production and load might produce frequency variations, risking serious issues. Therefore, a mechanism known as load frequency control (LFC) or automated generation control (AGC) is needed to keep the frequency and tie-line power within predefined stable limits. In this study, advanced proportional–integral–derivative PID controllers such as fractional-order PID (FOPID), cascaded PI(PDN), and PI(1+DD) for LFC in a two-area power system integrated with RES are optimized using the catch fish optimization algorithm (CFA). The controllers’ optimal gains are attained through using the integral absolute error (IAE) and ITAE objective functions. The performance of LFC with CFA-tuned PID, PI, cascaded PI(PDN), and FOPID, PI(1+DD) controllers is compared to other optimization techniques, including sine cosine algorithm (SCA), particle swarm optimization (PSO), brown bear algorithm (BBA), and grey wolf optimization (GWO), in a two-area power system combined with RESs under various conditions. Additionally, by contrasting the performance of the PID, PI, cascaded PI(PDN), and FOPID, PI(1+DD) controllers, the efficiency of the CFA is confirmed. Additionally, a sensitivity analysis that considers simultaneous modifications of the frequency bias coefficient (B) and speed regulation (R) within a range of ±25% validates the efficacy and dependability of the suggested CFA-tuned PI(1+DD). In the complex dynamics of a two-area interconnected power system, the results show how robust the suggested CFA-tuned PI(1+DD) control strategy is and how well it can stabilize variations in load frequency and tie-line power with a noticeably shorter settling time. Finally, the results of the simulation show that CFA performs better than the GWO, BBA, SCA, and PSO strategies. Full article

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15 pages, 3612 KB

Open AccessEditor’s ChoiceArticle

The Effect of pH Solution in the Sol–Gel Process on the Process of Formation of Fractal Structures in Thin SnO₂ Films

by Ekaterina Bondar, Igor Lebedev, Anastasia Fedosimova, Elena Dmitriyeva, Sayora Ibraimova, Anton Nikolaev, Aigul Shongalova, Ainagul Kemelbekova and Mikhail Begunov

Fractal Fract. 2025, 9(6), 353; https://doi.org/10.3390/fractalfract9060353 - 28 May 2025

Cited by 5 | Viewed by 2105

Abstract

In this paper, we investigated fractal cluster structures of colloidal particles in tin dioxide films obtained from lyophilic film-forming systems SnCl₄/EtOH/NH₄OH with different pH levels. It was revealed that at the ratio Sn > Cl₂ > O₂ [...] Read more.

In this paper, we investigated fractal cluster structures of colloidal particles in tin dioxide films obtained from lyophilic film-forming systems SnCl₄/EtOH/NH₄OH with different pH levels. It was revealed that at the ratio Sn > Cl₂ > O₂, N₂ = 0, and pH = 1.42, the growth of cross-shaped and flower-shaped structures of various sizes from several μm to tens of μm is observed. At the ratio Cl₂ > Sn > O₂ > N₂ and pH = 1.44, triangular and hexagonal structures are observed, the sizes of which are on the order of several tens of micrometers. The growth of hexagonal structures is probably affected by the presence of nitrogen in the film, according to the elemental analysis data. At the ratio Sn > Cl₂ > O₂ > N₂ and solution pH of 1.49, the growth of hexagonal and cross-shaped structures is observed, whereas flower-shaped structures are not observed. Hierarchical flower-like and cross-shaped structures are fractal. The shape of microstructures is directly related to the shape of the elementary cells of SnO₂ and NH₄Cl. A direct dependence of the formation of hierarchical structures on the volume of ammonium hydroxide additive was found. This allows for controlling the shape and size of the synthesized structures when changing the ratio of the initial precursors and influencing the final physicochemical characteristics of the obtained samples. Full article

(This article belongs to the Special Issue Advanced Research in Fractal Properties of Nanoparticle and Its Application)

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27 pages, 78121 KB

Open AccessEditor’s ChoiceArticle

Graph-Based Stock Volatility Forecasting with Effective Transfer Entropy and Hurst-Based Regime Adaptation

by Sangheon Lee and Poongjin Cho

Fractal Fract. 2025, 9(6), 339; https://doi.org/10.3390/fractalfract9060339 - 24 May 2025

Cited by 3 | Viewed by 5858

Abstract

This study proposes a novel hybrid model for stock volatility forecasting by integrating directional and temporal dependencies among financial time series and market regime changes into a unified modeling framework. Specifically, we design a novel Hurst Exponent Effective Transfer Entropy Graph Neural Network [...] Read more.

This study proposes a novel hybrid model for stock volatility forecasting by integrating directional and temporal dependencies among financial time series and market regime changes into a unified modeling framework. Specifically, we design a novel Hurst Exponent Effective Transfer Entropy Graph Neural Network (H-ETE-GNN) model that captures directional and asymmetric interactions based on Effective Transfer Entropy (ETE), and incorporates regime change detection using the Hurst exponent to reflect evolving global market conditions. To assess the effectiveness of the proposed approach, we compared the forecast performance of the hybrid GNN model with GNN models constructed using Transfer Entropy (TE), Granger causality, and Pearson correlation—each representing different measures of causality and correlation among time series. The empirical analysis was based on daily price data of 10 major country-level ETFs over a 19-year period (2006–2024), collected via Yahoo Finance. Additionally, we implemented recurrent neural network (RNN)-based models such as Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU) under the same experimental conditions to evaluate their performance relative to the GNN-based models. The effect of incorporating regime changes was further examined by comparing the model performance with and without Hurst-exponent-based detection. The experimental results demonstrated that the hybrid GNN-based approach effectively captured the structure of information flow between time series, leading to substantial improvements in the forecast performance for one-day-ahead realized volatility. Furthermore, incorporating regime change detection via the Hurst exponent enhanced the model’s adaptability to structural shifts in the market. This study highlights the potential of H-ETE-GNN in jointly modeling interactions between time series and market regimes, offering a promising direction for more accurate and robust volatility forecasting in complex financial environments. Full article

(This article belongs to the Special Issue Fractal and Multifractal Analyses in Financial Markets and Economics, 2nd Edition)

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19 pages, 2581 KB

Open AccessEditor’s ChoiceArticle

Analytical and Dynamical Study of Solitary Waves in a Fractional Magneto-Electro-Elastic System

by Sait San, Beenish and Fehaid Salem Alshammari

Fractal Fract. 2025, 9(5), 309; https://doi.org/10.3390/fractalfract9050309 - 10 May 2025

Cited by 15 | Viewed by 1217

Abstract

Magneto-electro-elastic materials, a novel class of smart materials, exhibit remarkable energy conversion properties, making them highly suitable for applications in nanotechnology. This study focuses on various aspects of the fractional nonlinear longitudinal wave equation (FNLWE) that models wave propagation in a magneto-electro-elastic circular [...] Read more.

Magneto-electro-elastic materials, a novel class of smart materials, exhibit remarkable energy conversion properties, making them highly suitable for applications in nanotechnology. This study focuses on various aspects of the fractional nonlinear longitudinal wave equation (FNLWE) that models wave propagation in a magneto-electro-elastic circular rod. Using the direct algebraic method, several new soliton solutions were derived under specific parameter constraints. In addition, Galilean transformation was employed to explore the system’s sensitivity and quasi-periodic dynamics. The study incorporates 2D, 3D, and time-series visualizations as effective tools for analyzing quasi-periodic behavior. The results contribute to a deeper understanding of the nonlinear dynamical features of such systems and demonstrate the robustness of the applied methodologies. This research not only extends existing knowledge of nonlinear wave equations but also introduces a substantial number of new solutions with broad applicability. Full article

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

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8 pages, 799 KB

Open AccessEditor’s ChoiceArticle

Optical Solutions of the Nonlinear Kodama Equation with the M-Truncated Derivative via the Extended (G^′/G)-Expansion Method

by Zhao Li

Fractal Fract. 2025, 9(5), 300; https://doi.org/10.3390/fractalfract9050300 - 5 May 2025

Cited by 26 | Viewed by 1841

Abstract

The main purpose of this article is to study the optical solutions of the nonlinear Kodama equation with the M-truncated derivative by using the extended

(G^{'} / G)

-expansion method. Firstly, the nonlinear Kodama equation with the M-truncated derivative is [...] Read more.

The main purpose of this article is to study the optical solutions of the nonlinear Kodama equation with the M-truncated derivative by using the extended

(G^{'} / G)

-expansion method. Firstly, the nonlinear Kodama equation with the M-truncated derivative is transformed into a nonlinear ordinary differential equation based on the principle of homogeneous equilibrium and the traveling wave transformation. Secondly, the optical solutions of the nonlinear Kodama equation with the M-truncated derivative are constructed by using the extended

(G^{'} / G)

-expansion method. Finally, three-dimensional, two-dimensional, and contour maps of partial solutions are obtained by using Matlab R2023b mathematical software. Full article

(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics, 2nd Edition)

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17 pages, 22156 KB

Open AccessEditor’s ChoiceArticle

Variable-Fractional-Order Nosé–Hoover System: Chaotic Dynamics and Numerical Simulations

by D. K. Almutairi, Dalal M. AlMutairi, Nidal E. Taha, Mohammed E. Dafaalla and Mohamed A. Abdoon

Fractal Fract. 2025, 9(5), 277; https://doi.org/10.3390/fractalfract9050277 - 25 Apr 2025

Cited by 9 | Viewed by 1960

Abstract

This study explores the variable-order fractional Nosé–Hoover system, investigating the evolution of its chaotic and stable states under variable-order derivatives. Variable-order derivatives introduce greater complexity and adaptability into a system’s dynamics. The main objective is to examine these effects through numerical simulations, showcasing [...] Read more.

This study explores the variable-order fractional Nosé–Hoover system, investigating the evolution of its chaotic and stable states under variable-order derivatives. Variable-order derivatives introduce greater complexity and adaptability into a system’s dynamics. The main objective is to examine these effects through numerical simulations, showcasing how changes in the order function influence a system’s behavior. The variable-order behavior is shown by phase space orbits and time series for various variable orders

α

. We look at how the system acts by using numerical solutions and numerical simulations. The phase space orbits and time series for different

α

show variable-order effects. The findings emphasize the role of variable-order derivatives in enhancing chaotic behavior, offering novel insights into their impact on dynamical systems. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 3rd Edition)

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22 pages, 11068 KB

Open AccessEditor’s ChoiceArticle

Relationship Between Pore-Size Distribution and 1D Compressibility of Different Reconstituted Clays Based on Fractal Theory

by Yanhao Zheng, Tanfang Zhu, Junqi Chen, Kun Shan and Junru Li

Fractal Fract. 2025, 9(4), 235; https://doi.org/10.3390/fractalfract9040235 - 9 Apr 2025

Cited by 29 | Viewed by 2520

Abstract

This paper first examines the evolution of pore-size distribution (PSD) in four types of reconstituted clays during one-dimensional (1D) compression, utilising mercury intrusion porosimetry. Central to this work, fractal theory is then applied to quantify the complexity of the pore structure through fractal [...] Read more.

This paper first examines the evolution of pore-size distribution (PSD) in four types of reconstituted clays during one-dimensional (1D) compression, utilising mercury intrusion porosimetry. Central to this work, fractal theory is then applied to quantify the complexity of the pore structure through fractal dimensions, followed by being correlated with 1D compressibility. The key findings are as follows: (1) The 1D compressibility of the four clays exhibits significant variability, with montmorillonite demonstrating the highest compressibility and Shenzhen clay, dominated by chlorite, the lowest. This is associated with distinct pore size evolution patterns under 1D loading while also emphasising the crucial role of mesopores in macroscopic clay deformation. (2) Fractal dimensions increase with loading, reflecting the progressive refinement of the pore structure, with natural Shenzhen clay demonstrating the most pronounced change. (3) A mathematical relationship between fractal dimension and 1D compressibility is established for each clay type, providing a quantitative tool for predicting 1D compressibility based on fractal dimensions. (4) The testing procedures and methods to ensure the representativeness of pore structure analysis are elaborated, ensuring reliable PSD data. By employing fractal theory, this study provides new insights into the correlation between pore structure complexity and compressibility in reconstituted clays. Full article

(This article belongs to the Special Issue Fractal and Fractional Models in Soil Mechanics)

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17 pages, 2057 KB

Open AccessEditor’s ChoiceArticle

A Fractional Time–Space Stochastic Advection–Diffusion Equation for Modeling Atmospheric Moisture Transport at Ocean–Atmosphere Interfaces

by Behrouz Parsa Moghaddam, Mahmoud A. Zaky, António Mendes Lopes and Alexandra Galhano

Fractal Fract. 2025, 9(4), 211; https://doi.org/10.3390/fractalfract9040211 - 28 Mar 2025

Cited by 24 | Viewed by 3294

Abstract

This study introduces a novel one-dimensional Fractional Time–Space Stochastic Advection–Diffusion Equation that revolutionizes the modeling of moisture transport within atmospheric boundary layers adjacent to oceanic surfaces. By synthesizing fractional calculus, advective transport mechanisms, and pink noise stochasticity, the proposed model captures the intricate [...] Read more.

This study introduces a novel one-dimensional Fractional Time–Space Stochastic Advection–Diffusion Equation that revolutionizes the modeling of moisture transport within atmospheric boundary layers adjacent to oceanic surfaces. By synthesizing fractional calculus, advective transport mechanisms, and pink noise stochasticity, the proposed model captures the intricate interplay between temporal memory effects, non-local turbulent diffusion, and the correlated-fluctuations characteristic of complex ocean–atmosphere interactions. The framework employs the Caputo fractional derivative to represent temporal persistence and the fractional Laplacian to model non-local turbulent diffusion, and incorporates a stochastic term with a

1 / f

power spectral density to simulate environmental variability. An efficient numerical solution methodology is derived utilizing complementary Fourier and Laplace transforms, which elegantly converts spatial fractional operators into algebraic expressions and yields closed-form solutions via Mittag–Leffler functions. This method’s application to a benchmark coastal domain demonstrates that stronger advection significantly increases the spatial extent of conditions exceeding fog formation thresholds, revealing advection’s critical role in moisture transport dynamics. Numerical simulations demonstrate the model’s capacity to reproduce both anomalous diffusion phenomena and realistic stochastic variability, while convergence analysis confirms the numerical scheme’s robustness against varying noise intensities. This integrated fractional stochastic framework substantially advances atmospheric moisture modeling capabilities, with direct applications to meteorological forecasting, coastal climate assessment, and environmental engineering. Full article

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

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20 pages, 7177 KB

Open AccessEditor’s ChoiceArticle

Slope Deformation Prediction Combining Particle Swarm Optimization-Based Fractional-Order Grey Model and K-Means Clustering

by Zhenzhu Meng, Yating Hu, Shunqiang Jiang, Sen Zheng, Jinxin Zhang, Zhenxia Yuan and Shaofeng Yao

Fractal Fract. 2025, 9(4), 210; https://doi.org/10.3390/fractalfract9040210 - 28 Mar 2025

Cited by 19 | Viewed by 1581

Abstract

Slope deformation poses significant risks to infrastructure, ecosystems, and human safety, making early and accurate predictions essential for mitigating slope failures and landslides. In this study, we propose a novel approach that integrates a fractional-order grey model (FOGM) with particle swarm optimization (PSO) [...] Read more.

Slope deformation poses significant risks to infrastructure, ecosystems, and human safety, making early and accurate predictions essential for mitigating slope failures and landslides. In this study, we propose a novel approach that integrates a fractional-order grey model (FOGM) with particle swarm optimization (PSO) to determine the optimal fractional order, thereby enhancing the model’s accuracy, even with limited and fluctuating data. Additionally, we employ a k-means clustering technique to account for both temporal and spatial variations in multi-point monitoring data, which improves the model’s ability to capture the relationships between monitoring points and increases prediction relevance. The model was validated using displacement data collected from 12 monitoring points on a slope located in Qinghai Province near the Yellow River, China. The results demonstrate that the proposed model outperforms the traditional statistical model and artificial neural networks, achieving a significantly higher coefficient of determination

R^{2}

up to 0.9998 for some monitoring points. Our findings highlight that the model maintains robust performance even when confronted with data of varying quality—a notable advantage over conventional approaches that typically struggle under such conditions. Overall, the proposed model offers a robust and data-efficient solution for slope deformation prediction, providing substantial potential for early warning systems and risk management. Full article

(This article belongs to the Special Issue Applications of Fractional-Order Grey Models)

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24 pages, 28364 KB

Open AccessEditor’s ChoiceArticle

Uncertainty-Aware Self-Attention Model for Time Series Prediction with Missing Values

by Jiabao Li, Chengjun Wang, Wenhang Su, Dongdong Ye and Ziyang Wang

Fractal Fract. 2025, 9(3), 181; https://doi.org/10.3390/fractalfract9030181 - 16 Mar 2025

Cited by 8 | Viewed by 6482

Abstract

Missing values in time series data present a significant challenge, often degrading the performance of downstream tasks such as classification and forecasting. Traditional approaches address this issue by first imputing the missing values and then independently solving the predictive tasks. Recent methods have [...] Read more.

Missing values in time series data present a significant challenge, often degrading the performance of downstream tasks such as classification and forecasting. Traditional approaches address this issue by first imputing the missing values and then independently solving the predictive tasks. Recent methods have leveraged self-attention models to enhance imputation quality and accelerate inference. These models, however, predict values based on all input observations—including the missing values—thereby potentially compromising the fidelity of the imputed data. In this paper, we propose the Uncertainty-Aware Self-Attention (UASA) model to overcome these limitations. Our approach introduces two novel techniques: (i) A self-attention mechanism with a partially observed diagonal that effectively captures complex non-local dependencies in time series data—a characteristic also observed in fractional-order systems. This approach draws inspiration from fractional calculus, where non-integer-order derivatives better characterize complex dynamical systems with long-memory effects, providing a more comprehensive mathematical framework for handling temporal data. And (ii) uncertainty quantification in data imputation to better inform downstream tasks. The UASA model comprises an upstream component for data imputation and a downstream component for time series prediction, trained jointly in an end-to-end fashion to optimize both imputation accuracy and task-specific objectives simultaneously. For classification tasks, the UASA model demonstrates remarkable performance even under high missing data rates, achieving a ROC-AUC of

99.5 %

, a PR-AUC of

58.5 %

, and an F1-SCORE of

49.3 %

. For forecasting tasks on the AUST-Gait dataset, the UASA model achieves a Mean Squared Error (MSE) of 0.72 under

0 %

missing data conditions (i.e., complete data input). Under the end-to-end training strategy evaluated across all missing data rates, the model achieves an average MSE of 0.74, showcasing its adaptability and robustness across diverse missing data scenarios. Full article

(This article belongs to the Special Issue Applied Fractional Calculus in Machine Learning and Biomedical Engineering)

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23 pages, 19331 KB

Open AccessEditor’s ChoiceArticle

Multi-Focus Image Fusion Based on Fractal Dimension and Parameter Adaptive Unit-Linking Dual-Channel PCNN in Curvelet Transform Domain

by Liangliang Li, Sensen Song, Ming Lv, Zhenhong Jia and Hongbing Ma

Fractal Fract. 2025, 9(3), 157; https://doi.org/10.3390/fractalfract9030157 - 3 Mar 2025

Cited by 24 | Viewed by 2887

Abstract

Multi-focus image fusion is an important method for obtaining fully focused information. In this paper, a novel multi-focus image fusion method based on fractal dimension (FD) and parameter adaptive unit-linking dual-channel pulse-coupled neural network (PAUDPCNN) in the curvelet transform (CVT) domain is proposed. [...] Read more.

Multi-focus image fusion is an important method for obtaining fully focused information. In this paper, a novel multi-focus image fusion method based on fractal dimension (FD) and parameter adaptive unit-linking dual-channel pulse-coupled neural network (PAUDPCNN) in the curvelet transform (CVT) domain is proposed. The source images are decomposed into low-frequency and high-frequency sub-bands by CVT, respectively. The FD and PAUDPCNN models, along with consistency verification, are employed to fuse the high-frequency sub-bands, the average method is used to fuse the low-frequency sub-band, and the final fused image is generated by inverse CVT. The experimental results demonstrate that the proposed method shows superior performance in multi-focus image fusion on Lytro, MFFW, and MFI-WHU datasets. Full article

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

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19 pages, 1606 KB

Open AccessEditor’s ChoiceArticle

Chaos in Fractional-Order Glucose–Insulin Models with Variable Derivatives: Insights from the Laplace–Adomian Decomposition Method and Generalized Euler Techniques

by Sayed Saber, Emad Solouma, Rasmiyah A. Alharb and Ahmad Alalyani

Fractal Fract. 2025, 9(3), 149; https://doi.org/10.3390/fractalfract9030149 - 27 Feb 2025

Cited by 27 | Viewed by 2508

Abstract

This study investigates the complex dynamics and control mechanisms of fractional-order glucose–insulin regulatory systems, incorporating memory-dependent properties through fractional derivatives. Employing the Laplace–Adomian Decomposition Method (LADM) and the Generalized Euler Method (GEM), the research models glucose–insulin interactions with time-varying fractional orders to simulate [...] Read more.

This study investigates the complex dynamics and control mechanisms of fractional-order glucose–insulin regulatory systems, incorporating memory-dependent properties through fractional derivatives. Employing the Laplace–Adomian Decomposition Method (LADM) and the Generalized Euler Method (GEM), the research models glucose–insulin interactions with time-varying fractional orders to simulate long-term physiological processes. Key aspects include the derivation of Lyapunov exponents, bifurcation diagrams, and phase diagrams to explore system stability and chaotic behavior. A novel control strategy using simple linear controllers is introduced to stabilize chaotic oscillations. The effectiveness of this approach is validated through numerical simulations, where Lyapunov exponents are reduced from positive values (

λ_{1} = 0.123

) in the uncontrolled system to negative values (

λ_{1} = - 0.045

) post-control application, indicating successful stabilization. Additionally, bifurcation analysis demonstrates a shift from chaotic to periodic behavior when control is applied, and time-series plots confirm a significant reduction in glucose–insulin fluctuations. These findings underscore the importance of fractional calculus in accurately modeling nonlinear and memory-dependent glucose–insulin dynamics, paving the way for improved predictive models and therapeutic strategies. The proposed framework provides a foundation for personalized diabetes management, real-time glucose monitoring, and intelligent insulin delivery systems. Full article

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

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28 pages, 4379 KB

Open AccessEditor’s ChoiceArticle

A New Approach Based on Metaheuristic Optimization Using Chaotic Functional Connectivity Matrices and Fractal Dimension Analysis for AI-Driven Detection of Orthodontic Growth and Development Stage

by Orhan Cicek, Yusuf Bahri Özçelik and Aytaç Altan

Fractal Fract. 2025, 9(3), 148; https://doi.org/10.3390/fractalfract9030148 - 26 Feb 2025

Cited by 56 | Viewed by 3157

Abstract

Accurate identification of growth and development stages is critical for orthodontic diagnosis, treatment planning, and post-treatment retention. While hand–wrist radiographs are the traditional gold standard, the associated radiation exposure necessitates alternative imaging methods. Lateral cephalometric radiographs, particularly the maturation stages of the second, [...] Read more.

Accurate identification of growth and development stages is critical for orthodontic diagnosis, treatment planning, and post-treatment retention. While hand–wrist radiographs are the traditional gold standard, the associated radiation exposure necessitates alternative imaging methods. Lateral cephalometric radiographs, particularly the maturation stages of the second, third, and fourth cervical vertebrae (C2, C3, and C4), have emerged as a promising alternative. However, the nonlinear dynamics of these images pose significant challenges for reliable detection. This study presents a novel approach that integrates chaotic functional connectivity (FC) matrices and fractal dimension analysis to address these challenges. The fractal dimensions of C2, C3, and C4 vertebrae were calculated from 945 lateral cephalometric radiographs using three methods: fast Fourier transform (FFT), box counting, and a pre-processed FFT variant. These results were used to construct chaotic FC matrices based on correlations between the calculated fractal dimensions. To effectively model the nonlinear dynamics, chaotic maps were generated, representing a significant advance over traditional methods. Feature selection was performed using a wrapper-based approach combining k-nearest neighbors (kNN) and the Puma optimization algorithm, which efficiently handles the chaotic and computationally complex nature of cervical vertebrae images. This selection minimized the number of features while maintaining high classification performance. The resulting AI-driven model was validated with 10-fold cross-validation and demonstrated high accuracy in identifying growth stages. Our results highlight the effectiveness of integrating chaotic FC matrices and AI in orthodontic practice. The proposed model, with its low computational complexity, successfully handles the nonlinear dynamics in C2, C3, and C4 vertebral images, enabling accurate detection of growth and developmental stages. This work represents a significant step in the detection of growth and development stages and provides a practical and effective solution for future orthodontic diagnosis. Full article

(This article belongs to the Special Issue Fractional Calculus in Signal, Imaging Processing and Machine Learning)

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14 pages, 311 KB

Open AccessEditor’s ChoiceArticle

Existence of Positive Solutions for a Class of Nabla Fractional Boundary Value Problems

by Nikolay D. Dimitrov and Jagan Mohan Jonnalagadda

Fractal Fract. 2025, 9(2), 131; https://doi.org/10.3390/fractalfract9020131 - 19 Feb 2025

Cited by 18 | Viewed by 1249

Abstract

In this manuscript, we study a class of equations with two different Riemann–Liouville-type orders of nabla difference operators. We show some new and fundamental properties of the related Green’s function. Depending on the values of the orders of the operators, we split our [...] Read more.

In this manuscript, we study a class of equations with two different Riemann–Liouville-type orders of nabla difference operators. We show some new and fundamental properties of the related Green’s function. Depending on the values of the orders of the operators, we split our research into two main cases, and for each one of them, we obtain suitable conditions under which we prove that the considered problem possesses a positive solution. We consider the latter to be the main novelty in this work. Our main tool in both cases of our study is Guo–Krasnoselskii’s fixed point theorem. In the end, we give particular examples in order to offer a concrete demonstration of our new theoretical findings, as well as some possible future work in this direction. Full article

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

20 pages, 7717 KB

Open AccessEditor’s ChoiceArticle

Dynamic Analysis and Implementation of FPGA for a New 4D Fractional-Order Memristive Hopfield Neural Network

by Fei Yu, Shankou Zhang, Dan Su, Yiya Wu, Yumba Musoya Gracia and Huige Yin

Fractal Fract. 2025, 9(2), 115; https://doi.org/10.3390/fractalfract9020115 - 13 Feb 2025

Cited by 59 | Viewed by 3842

Abstract

Memristor-based fractional-order chaotic systems can record information from the past, present, and future, and describe the real world more accurately than integer-order systems. This paper proposes a novel memristor model and verifies its characteristics through the pinched loop (PHL) method. Subsequently, a new [...] Read more.

Memristor-based fractional-order chaotic systems can record information from the past, present, and future, and describe the real world more accurately than integer-order systems. This paper proposes a novel memristor model and verifies its characteristics through the pinched loop (PHL) method. Subsequently, a new fractional-order memristive Hopfield neural network (4D-FOMHNN) is introduced to simulate induced current, accompanied by Caputo’s definition of fractional order. An Adomian decomposition method (ADM) is employed for system solution. By varying the parameters and order of the 4D-FOMHNN, rich dynamic behaviors including transient chaos, chaos, and coexistence attractors are observed using methods such as bifurcation diagrams and Lyapunov exponent analysis. Finally, the proposed FOMHNN system is implemented on a field-programmable gate array (FPGA), and the oscilloscope observation results are consistent with the MATLAB numerical simulation results, which further validate the theoretical analysis of the FOMHNN system and provide a theoretical basis for its application in the field of encryption. Full article

(This article belongs to the Special Issue Analysis and Modeling of Fractional-Order Dynamical Networks)

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19 pages, 6533 KB

Open AccessEditor’s ChoiceArticle

Vibration Suppression of the Vehicle Mechatronic ISD Suspension Using the Fractional-Order Biquadratic Electrical Network

by Yujie Shen, Zhaowei Li, Xiang Tian, Kai Ji and Xiaofeng Yang

Fractal Fract. 2025, 9(2), 106; https://doi.org/10.3390/fractalfract9020106 - 10 Feb 2025

Cited by 35 | Viewed by 2208

Abstract

In order to break the bottleneck of the integer-order transfer function in vehicle ISD (inerter-spring-damper) suspension design, a positive real synthesis design method of vehicle mechatronic ISD suspension based on the fractional-order biquadratic transfer function is proposed. The emergence of the fractional-order components [...] Read more.

In order to break the bottleneck of the integer-order transfer function in vehicle ISD (inerter-spring-damper) suspension design, a positive real synthesis design method of vehicle mechatronic ISD suspension based on the fractional-order biquadratic transfer function is proposed. The emergence of the fractional-order components disrupts the equivalence relationship between the passivity of components and the positive realness of integer-order transfer functions in traditional networks. In this paper, the positive real condition of the fractional-order biquadratic transfer function is given. Then, a quarter dynamic model of the vehicle mechatronic ISD suspension is established, and the parameters of the fractional-order biquadratic transfer function and vehicle suspension are obtained by an NSGA-II multi-objective genetic algorithm. Moreover, the structure of the external circuit and the parameters of the electrical components are obtained by the fractional-order passive network synthesis theory. The simulation results show that under the condition of random road input and vehicle speed of 20 m/s, the root-mean-square (RMS) value of the vehicle body acceleration and the dynamic tire load of the fractional-order ISD suspension are reduced by 7.98% and 18.75% compared with the traditional passive suspension, while under the same condition, the integer-order ISD suspension can only reduce by 5.34% and 16.07%, respectively. The results show that employing a fractional-order biquadratic electrical network in the vehicle mechatronic ISD suspension enhances vibration isolation performance compared with the suspension using an integer-order biquadratic electrical network. Full article

(This article belongs to the Special Issue Applications of Fractional-Order Systems to Automatic Control)

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18 pages, 4844 KB

Open AccessEditor’s ChoiceArticle

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

Cited by 22 | Viewed by 1563

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)

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20 pages, 1657 KB

Open AccessEditor’s ChoiceArticle

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

Cited by 14 | Viewed by 2928

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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27 pages, 16020 KB

Open AccessEditor’s ChoiceArticle

Pore Structure and Its Fractal Dimension: A Case Study of the Marine Shales of the Niutitang Formation in Northwest Hunan, South China

by Wei Jiang, Yang Zhang, Tianran Ma, Song Chen, Yang Hu, Qiang Wei and Dingxiang Zhuang

Fractal Fract. 2025, 9(1), 49; https://doi.org/10.3390/fractalfract9010049 - 17 Jan 2025

Cited by 22 | Viewed by 2535

Abstract

To analyze the pore structure and fractal characteristics of marine shale in the lower Cambrian Niutitang Formation in northwestern Hunan Province, China, the pore characteristics of shale were characterized using total organic carbon (TOC) content, field emission scanning electron microscopy (FESEM), X-ray diffraction [...] Read more.

To analyze the pore structure and fractal characteristics of marine shale in the lower Cambrian Niutitang Formation in northwestern Hunan Province, China, the pore characteristics of shale were characterized using total organic carbon (TOC) content, field emission scanning electron microscopy (FESEM), X-ray diffraction (XRD), low temperature nitrogen adsorption (LT-N₂GA) and methane adsorption experiments. The pore surface and pore space fractal dimensions of samples were calculated, respectively. The influencing factors of fractal dimensions and their impact on the adsorption of shale reservoirs were discussed. The results indicate the Niutitang Formation shale mainly develops four types of pores: organic pores, intragranular pores, intergranular pores and microcracks. The pores have a large specific surface area (SSA), primarily consisting of mesopores. The fractal dimensions are calculated using the FHH model and the XS model. The fractal dimensions (D₂ and D_f) are greater than D₁, indicating that the pore surface with larger pore size is rougher, and the pore structure of shale is complex. The pore volume (PV), SSA, and TOC show positive correlations with the fractal dimensions but negative correlations with APS. There is no obvious correlation between fractal dimensions and quartz content, while clay minerals show a negative correlation with D₂ and D_f. This is mainly because clay mineral particles are small in size and have weak resistance to compaction. The pyrite content is positively correlated with the fractal dimensions because pyrite promotes the development of organic, intergranular, and mold pores. According to Pearson correlation analysis, the main influencing factors of the pore surface fractal dimension are PV, SSA, and APS. The main influencing factors of the pore space fractal dimension are APS and the content of clay minerals. Further analysis of the influence of the fractal dimension on the adsorption capacity of shale reveals that the fractal dimensions are positively correlated with Langmuir volume, indicating that fractal dimensions can be used as a quantitative target for evaluating shale gas reservoirs. Full article

(This article belongs to the Special Issue Applications of General Fractional Calculus Models: Insights into Viscoelasticity and Wave Propagation)

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23 pages, 444 KB

Open AccessEditor’s ChoiceArticle

A Study on the Existence, Uniqueness, and Stability of Fractional Neutral Volterra-Fredholm Integro-Differential Equations with State-Dependent Delay

by Prabakaran Raghavendran, Tharmalingam Gunasekar, Junaid Ahmad and Walid Emam

Fractal Fract. 2025, 9(1), 20; https://doi.org/10.3390/fractalfract9010020 - 31 Dec 2024

Cited by 17 | Viewed by 2892

Abstract

This paper presents an analysis of the existence, uniqueness, and stability of solutions to fractional neutral Volterra-Fredholm integro-differential equations, incorporating Caputo fractional derivatives and semigroup operators with state-dependent delays. By employing Krasnoselskii’s fixed point theorem, conditions under which solutions exist are established. To [...] Read more.

This paper presents an analysis of the existence, uniqueness, and stability of solutions to fractional neutral Volterra-Fredholm integro-differential equations, incorporating Caputo fractional derivatives and semigroup operators with state-dependent delays. By employing Krasnoselskii’s fixed point theorem, conditions under which solutions exist are established. To ensure uniqueness, the Banach Contraction Principle is applied, and the contraction condition is verified. Stability is analyzed using Ulam’s stability concept, emphasizing the resilience of solutions to perturbations and providing insights into their long-term behavior. An example is included, accompanied by graphical analysis that visualizes the solutions and their dynamic properties. Full article

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

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12 pages, 2318 KB

Open AccessEditor’s ChoiceArticle

An Adsorption Model Considering Fictitious Stress

by Xiaohua Tan, Xinjian Ma, Xiaoping Li and Yilong Li

Fractal Fract. 2025, 9(1), 17; https://doi.org/10.3390/fractalfract9010017 - 30 Dec 2024

Cited by 26 | Viewed by 2050

Abstract

The adsorption of coalbed methane alters the pore structure of reservoirs, subsequently affecting the coal seam’s gas adsorption capacity. However, traditional gas adsorption models often neglect this crucial aspect. In this article, we introduce a fractal capillary bundle model that accounts for the [...] Read more.

The adsorption of coalbed methane alters the pore structure of reservoirs, subsequently affecting the coal seam’s gas adsorption capacity. However, traditional gas adsorption models often neglect this crucial aspect. In this article, we introduce a fractal capillary bundle model that accounts for the expansion of coal seam adsorption. We utilize curvature fractal dimension and capillary fractal dimension to characterize the complexity of the coal seam’s pore structure. By incorporating the concept of fictitious stress, we have described the relationship between gas adsorption, matrix porosity, and permeability changes. We have developed a model that describes the changes in matrix porosity and permeability during the gas adsorption process. After fitting this model to experimental data, it demonstrated high accuracy in predictions. Furthermore, our investigation into how factors such as curvature fractal dimension, capillary fractal dimension, and fictitious stress influence gas adsorption capacity reveals several key findings. Firstly, the specific surface area within the pore structure of coal seams is the primary factor controlling gas adsorption capacity. Secondly, the virtual stress generated during the gas adsorption process alters the coal seam’s maximum gas adsorption capacity, a factor that cannot be overlooked. Lastly, we found that gas adsorption primarily affects the gas migration process, while under high-pressure conditions, gas desorption does not cause significant changes in the matrix porosity and permeability. Full article

(This article belongs to the Section Engineering)

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25 pages, 7489 KB

Open AccessEditor’s ChoiceArticle

Pore Structure and Fractal Characteristics of Inter-Layer Sandstone in Marine–Continental Transitional Shale: A Case Study of the Upper Permian Longtan Formation in Southern Sichuan Basin, South China

by Jianguo Wang, Jizhen Zhang, Xiao Xiao, Ya’na Chen and Denglin Han

Fractal Fract. 2025, 9(1), 11; https://doi.org/10.3390/fractalfract9010011 - 29 Dec 2024

Cited by 10 | Viewed by 2229

Abstract

With the evolution of unconventional oil and gas exploration concepts from source rocks and reservoirs to carrier beds, the inter-layer sandstone carrier bed within marine–continental transitional shale strata has emerged as a significant target for oil and gas exploration. The inter-layer sandstone is [...] Read more.

With the evolution of unconventional oil and gas exploration concepts from source rocks and reservoirs to carrier beds, the inter-layer sandstone carrier bed within marine–continental transitional shale strata has emerged as a significant target for oil and gas exploration. The inter-layer sandstone is closely associated with the source rock and differs from conventional tight sandstone in terms of sedimentary environment, matrix composition, and the characteristics of reservoir microscopic pore development. Preliminary exploration achievements display that the inter-layer sandstone is plentiful in gas content and holds promising prospects for exploration and development. Consequently, it is essential to investigate the gas-rich accumulation theory specific to the inter-layer sandstone reservoir in transitional facies. Pore development characteristics and heterogeneity are crucial aspects of oil and gas accumulation research, as they influence reservoir seepage performance and capacity. This paper employs total organic carbon analysis, X-ray diffraction, rock thin section examination, field emission scanning electron microscopy, physical analysis, high-pressure mercury intrusion analysis, gas adsorption experiments, and fractal theory to explore the reservoir development characteristics of the sandstone samples from the Upper Permian marine–continental transitional facies Longtan Formation in the southern Sichuan Basin. It also attempts to combine high-pressure mercury intrusion analysis and gas adsorption experiments to describe the structural and fractal characteristics of pores at different scales in a segmented manner. The findings reveal that the sandstone type of the Longtan Formation is mainly lithic sandstone. The pore size distribution of the sandstone primarily falls below 30 nm and above 1000 nm, with the main pore types being inter-granular pores and micro-fractures in clay minerals. The pore volume and specific surface area are largely attributed to the micropores and mesopores of clay minerals. The pore morphology is complex, exhibiting strong heterogeneity, predominantly characterized by slit-like and ink bottle-like features. Notably, there are discernible differences in reservoir structural characteristics and homogeneity between muddy sandstone and non-muddy sandstone. The pore morphology is complex, exhibiting strong heterogeneity, predominantly characterized by slit-like and ink bottle-like features. Notably, there are discernible differences in reservoir structural characteristics and homogeneity between muddy sandstone and non-muddy sandstone. Full article

(This article belongs to the Special Issue Pore Structure and Fractal Characteristics in Unconventional Oil and Gas Reservoirs)

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18 pages, 4434 KB

Open AccessEditor’s ChoiceArticle

Fractal Characteristics of Pore Structure of Longmaxi Shales with Different Burial Depths in Southern Sichuan and Its Geological Significance

by Manping Yang, Yanyan Pan, Hongye Feng, Qiang Yan, Yanjun Lu, Wanxin Wang, Yu Qi and Hongjian Zhu

Fractal Fract. 2025, 9(1), 2; https://doi.org/10.3390/fractalfract9010002 - 24 Dec 2024

Cited by 30 | Viewed by 2346

Abstract

Burial depth can significantly impact the pore structure characteristics of shale. The Lower Silurian Longmaxi Shale in the Weiyuan block of the Sichuan Basin is a marine formation that we studied for deep shale gas exploration. We used two groups of Longmaxi samples, [...] Read more.

Burial depth can significantly impact the pore structure characteristics of shale. The Lower Silurian Longmaxi Shale in the Weiyuan block of the Sichuan Basin is a marine formation that we studied for deep shale gas exploration. We used two groups of Longmaxi samples, outcrop shale and middle-deep shale, to investigate the pore structure fractal features at varying burial depths using a combination of mineralogy, organic geochemistry, scanning electron microscopy (SEM), and low-temperature gas (CO₂, N₂) adsorption. The V-S fractal model was used to determine the fractal dimension (Dc) of micropores, and the FHH fractal model was used to determine the fractal dimension (D_N) of mesopores. The findings indicate that the pore morphology of organic matter becomes irregular and more broken as the burial depth increases, as does the content and maturity of organic matter. The pore size of organic matter gradually decreases, the SSA (BET, DR) and PV (BJH, DA) of shale pores increase, the pore structure becomes more complex, and the average shale pore size decreases. According to this study, the organic matter content and its maturity show an increasing trend as burial depth increases. Meanwhile, the organic matter’s pore morphology tends to be irregular, and fracture rates rise, which causes the organic matter’s pore size to gradually decrease. In addition, the SSA (comprising the values assessed by BET and DR techniques) and PV (evaluated by BJH and DA methods) of shale pores grew, suggesting that the pore structure became more complex. Correspondingly, the average pore size of the shale decreased. The fractal dimensions of the micropores (D_C), mesoporous surface (D_N1), and mesoporous structure (D_N2) of outcrop shale are 2.6728~2.7245, 2.5612~2.5838, and 2.7911~2.8042, respectively. The mean values are 2.6987, 2.5725, and 2.7977, respectively. The D_C, D_N1, and D_N2 of middle-deep shale are 2.6221~2.7510, 2.6085~2.6390, and 2.8140~2.8357, respectively, and the mean values are 2.7050, 2.6243, and 2.8277, respectively. As the fractal dimension grows, the shale’s pore structure becomes more intricate, and the heterogeneity increases as the buried depth increases. The fractal dimension has a positive association with the pore structure parameters (SSA, PV), TOC, and R_o and a negative association with the mineral component (quartz, feldspar, clay mineral) contents. Minerals like quartz, feldspar, and clay will slow down the expansion of pores, but when SSA and PV increase, the pore heterogeneity will be greater and the pore structure more complex. Full article

(This article belongs to the Special Issue Pore Structure and Fractal Characteristics in Unconventional Oil and Gas Reservoirs)

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14 pages, 482 KB

Open AccessEditor’s ChoiceArticle

Novel GPID: Grünwald–Letnikov Fractional PID for Enhanced Adaptive Cruise Control

by Diaa Eldin Elgezouli, Hassan Eltayeb and Mohamed A. Abdoon

Fractal Fract. 2024, 8(12), 751; https://doi.org/10.3390/fractalfract8120751 - 20 Dec 2024

Cited by 18 | Viewed by 2265

Abstract

This study demonstrates that the Grünwald–Letnikov fractional proportional–integral–derivative (GPID) controller outperforms traditional PID controllers in adaptive cruise control systems, while conventional PID controllers struggle with nonlinearities, dynamic uncertainties, and stability, the GPID enhances robustness and provides more precise control across various driving conditions. [...] Read more.

This study demonstrates that the Grünwald–Letnikov fractional proportional–integral–derivative (GPID) controller outperforms traditional PID controllers in adaptive cruise control systems, while conventional PID controllers struggle with nonlinearities, dynamic uncertainties, and stability, the GPID enhances robustness and provides more precise control across various driving conditions. Simulation results show that the GPID improves the accuracy, reducing errors better than the PID controller. Additionally, the GPID maintains a more consistent speed and reaches the target speed faster, demonstrating superior speed control. The GPID’s performance across different fractional orders highlights its adaptability to changing road conditions, which is crucial for ensuring safety and comfort. By leveraging fractional calculus, the GPID also improves acceleration and deceleration profiles. These findings emphasize the GPID’s potential to revolutionize adaptive cruise control, significantly enhancing driving performance and comfort. Numerical results obtained in

α = 0.99

from the GPID controller have shown better accuracy and speed consistency, adapting to road conditions for improved safety and comfort. The GPID also demonstrated faster stabilization of speed at 60 km/h with smaller errors and reduced the error to 0.59 km/h at 50 s compared to 0.78 km/h for the PID. Full article

(This article belongs to the Special Issue Applications of Fractional-Order Tools in Engineering Technology and Physical Processes)

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21 pages, 26235 KB

Open AccessEditor’s ChoiceArticle

Fractal Dimensional Analysis of Building Facades: The Case of Office Buildings in Erbil City

by Lana Abubakr Ali and Faris Ali Mustafa

Fractal Fract. 2024, 8(12), 746; https://doi.org/10.3390/fractalfract8120746 - 17 Dec 2024

Cited by 6 | Viewed by 6203

Abstract

Fractal dimension is a characteristic parameter used to measure the complexity and irregularity of geometric shapes and patterns. It is applied in architecture to explore complexity and irregularity and to assess the aesthetic preferences in architectural design. Office building facade design pattern, as [...] Read more.

Fractal dimension is a characteristic parameter used to measure the complexity and irregularity of geometric shapes and patterns. It is applied in architecture to explore complexity and irregularity and to assess the aesthetic preferences in architectural design. Office building facade design pattern, as an observation unit, has a positive connection with the aesthetic value. This study aims to evaluate facade design styles in terms of two aesthetic qualities, visual complexity and visual diversity, via applying fractal dimension to three design styles of office building facades in Erbil City. The study uses a combination of qualitative and quantitative evaluations to achieve this goal. It employs box-counting analysis through the ImageJ plugin to FracLac and the mathematical perplexity equation to evaluate visual complexity and diversity. The results indicate that the neo-classical office facade style, with a visual complexity value of 1.7008 and visual diversity of 21.27, presents an elevated level of aesthetics similar to the saccadic pattern facade. This study concluded that a neo-classical architectural style for office building facades is the most aesthetically preferable. Modern facade design is considered a secondary architectural style aimed at achieving aesthetic value. Ultimately, the high-tech style is the least attractive facade style. This study contributes to avoiding designs of unattractive office building facades due to a lack of architectural design vocabulary while avoiding overly complex designs that prove visually upsetting for viewers. Full article

(This article belongs to the Section Engineering)

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21 pages, 506 KB

Open AccessEditor’s ChoiceArticle

Study on Controllability for Ψ-Hilfer Fractional Stochastic Differential Equations

by Abdur Raheem, Fahad M. Alamrani, Javed Akhtar, Adel Alatawi, Esmail Alshaban, Areefa Khatoon and Faizan Ahmad Khan

Fractal Fract. 2024, 8(12), 727; https://doi.org/10.3390/fractalfract8120727 - 11 Dec 2024

Cited by 12 | Viewed by 2040

Abstract

The goal of this paper is to study the existence of a mild solution and controllability for a class of neutral stochastic differential equations (SDEs) involving the

Ψ

-Hilfer fractional derivatives, a generalization of the well-known Riemann–Liouville fractional derivative using almost sectorial operators. [...] Read more.

The goal of this paper is to study the existence of a mild solution and controllability for a class of neutral stochastic differential equations (SDEs) involving the

Ψ

-Hilfer fractional derivatives, a generalization of the well-known Riemann–Liouville fractional derivative using almost sectorial operators. Sufficient conditions for controllability are established using the notion of measure of noncompactness (MNC) and the Mönch fixed-point theorem. An example is given to illustrate the abstract findings. Full article

(This article belongs to the Special Issue Advances in Fractional Dynamical and Control Problems with Nonlocal Operators)

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28 pages, 12115 KB

Open AccessEditor’s ChoiceArticle

Energy Evolution and Fractal Characteristics of Sandstones Under True Triaxial Cyclic Loading and Unloading

by Qihang Zhang, Xiangrui Meng and Guangming Zhao

Fractal Fract. 2024, 8(12), 714; https://doi.org/10.3390/fractalfract8120714 - 30 Nov 2024

Cited by 16 | Viewed by 2188

Abstract

To study the fractal characteristics and energy evolution of sandstones under true three-dimensional stress states, a true triaxial compression test and a cyclic loading and unloading test of sandstone specimens under different loads were carried out using a self-developed true triaxial disturbance testing [...] Read more.

To study the fractal characteristics and energy evolution of sandstones under true three-dimensional stress states, a true triaxial compression test and a cyclic loading and unloading test of sandstone specimens under different loads were carried out using a self-developed true triaxial disturbance testing system. Based on the evolution law of true triaxial cyclic loading and unloading stress–strain, the types of loading and unloading in the cyclic loading and unloading test were delineated, and the reasons for the change in peak maximum principal stress intensity under different paths were analyzed. By analyzing the crushing characteristics of rock samples under different paths, it was found that the staged cyclic loading and unloading caused the greatest damage to the rock mass, while the equal-amplitude and unequal-lower-limit staged loading and unloading caused the least damage to the rock mass. Based on fractal theory, it was found that the rock samples under path V had the highest fractal dimension D. The elastic energy density, dissipated energy density, and input energy density of true triaxial cyclic loading and unloading under different paths were calculated by graphical area integration and superposition methods, respectively, to analyze the evolution of the three with the increase in the loading and unloading cycles and the energy distribution during the loading and unloading process. True triaxial cyclic loading and unloading tests revealed a linear relationship between the elastic energy density and total input energy density of the rock mass, and the energy storage coefficient exceeded 0.5, regardless of the loading path. Full article

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

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19 pages, 375 KB

Open AccessEditor’s ChoiceArticle

ADI Compact Difference Scheme for the Two-Dimensional Integro-Differential Equation with Two Fractional Riemann–Liouville Integral Kernels

by Ziyi Chen, Haixiang Zhang and Hu Chen

Fractal Fract. 2024, 8(12), 707; https://doi.org/10.3390/fractalfract8120707 - 29 Nov 2024

Cited by 21 | Viewed by 1847

Abstract

In this paper, a numerical method of a two-dimensional (2D) integro-differential equation with two fractional Riemann–Liouville (R-L) integral kernels is investigated. The compact difference method is employed in the spatial direction. The integral terms are approximated by a second-order convolution quadrature formula. The [...] Read more.

In this paper, a numerical method of a two-dimensional (2D) integro-differential equation with two fractional Riemann–Liouville (R-L) integral kernels is investigated. The compact difference method is employed in the spatial direction. The integral terms are approximated by a second-order convolution quadrature formula. The alternating direction implicit (ADI) compact difference scheme reduces the CPU time for two-dimensional problems. Simultaneously, the stability and convergence of the proposed ADI compact difference scheme are demonstrated. Finally, two numerical examples are provided to verify the established ADI compact difference scheme. Full article

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10 pages, 269 KB

Open AccessEditor’s ChoiceArticle

On the Fixed Point Theorem for Large Contraction Mappings with Applications to Delay Fractional Differential Equations

by Mouataz Billah Mesmouli, Elvan Akın, Loredana Florentina Iambor, Osman Tunç and Taher S. Hassan

Fractal Fract. 2024, 8(12), 703; https://doi.org/10.3390/fractalfract8120703 - 28 Nov 2024

Cited by 12 | Viewed by 2867

Abstract

This paper explores a new class of mappings and presents several fixed-point results for these mappings. We define these mappings by combining well-known mappings in the literature, specifically the large contraction mapping and Chatterjea’s mapping. This combination allows us to achieve significant fixed-point [...] Read more.

This paper explores a new class of mappings and presents several fixed-point results for these mappings. We define these mappings by combining well-known mappings in the literature, specifically the large contraction mapping and Chatterjea’s mapping. This combination allows us to achieve significant fixed-point results in complete metric spaces, both in a continuous and a non-continuous sense. Additionally, we provide an explicit example to validate our findings. Furthermore, we discuss a general model for fractional differential equations using the Caputo derivative. Finally, we outline the benefits of our study and suggest potential areas for future research. Full article

20 pages, 341 KB

Open AccessEditor’s ChoiceArticle

Fixed Point and Stability Analysis of a Tripled System of Nonlinear Fractional Differential Equations with n-Nonlinear Terms

by Mohamed S. Algolam, Osman Osman, Arshad Ali, Alaa Mustafa, Khaled Aldwoah and Amer Alsulami

Fractal Fract. 2024, 8(12), 697; https://doi.org/10.3390/fractalfract8120697 - 26 Nov 2024

Cited by 13 | Viewed by 1942

Abstract

This research article investigates a tripled system of nonlinear fractional differential equations with n terms. The study explores this novel class of differential equations to establish existence and stability results. Utilizing Schaefer’s and Banach’s fixed point theorems, we derive sufficient conditions for the [...] Read more.

This research article investigates a tripled system of nonlinear fractional differential equations with n terms. The study explores this novel class of differential equations to establish existence and stability results. Utilizing Schaefer’s and Banach’s fixed point theorems, we derive sufficient conditions for the existence of at least one solution, as well as a unique solution. Furthermore, we apply Hyers–Ulam stability analysis to establish criteria for the stability of the system. To demonstrate the applicability of the main results, a detailed example is provided. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Advanced Results for Cases with Singularities)

17 pages, 14485 KB

Open AccessEditor’s ChoiceArticle

Quantitative Analysis of Crack Propagation Behavior in Recycled Concrete Subjected to Axial Compression Using Digital Image Correlation (DIC) Technology and Fractal Theory

by Cheng-Gong Lu, Xiu-Cheng Zhang, Wei-Zhi Chen and Xue-Fei Chen

Fractal Fract. 2024, 8(12), 686; https://doi.org/10.3390/fractalfract8120686 - 23 Nov 2024

Cited by 17 | Viewed by 3587

Abstract

The current research endeavors to explore the mechanical properties of recycled concrete cubic specimens, predominantly concentrating on macroscopic attributes such as compressive strength and splitting tensile strength. However, at the mesoscopic scale, the internal structure of recycled concrete becomes increasingly intricate due to [...] Read more.

The current research endeavors to explore the mechanical properties of recycled concrete cubic specimens, predominantly concentrating on macroscopic attributes such as compressive strength and splitting tensile strength. However, at the mesoscopic scale, the internal structure of recycled concrete becomes increasingly intricate due to the adherence of substantial mortar on the surface of recycled coarse aggregates, ultimately influencing its macroscopic mechanical behavior and crack propagation trajectories. To address this complexity, Digital Image Correlation (DIC) technology is harnessed to quantitatively analyze crack tip displacement fields and crack propagation behavior in recycled concrete, considering variations in aggregate replacement ratios and particle sizes. This analysis encompasses quantitative assessments of pre-cracking damage, post-cracking crack morphology characteristics, and size. Utilizing the advanced 3D-DIC algorithmic technique, this study delves into the evolution of crack propagation patterns in recycled concrete during the loading process, evaluating the impact of both aggregate replacement ratios and particle sizes on crack propagation behavior. Ultimately, fractal theory is employed to provide a quantitative analysis of the crack morphology on the surface of recycled concrete. Full article

(This article belongs to the Section Engineering)

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24 pages, 400 KB

Open AccessEditor’s ChoiceArticle

Theory on New Fractional Operators Using Normalization and Probability Tools

by Marc Jornet

Fractal Fract. 2024, 8(11), 665; https://doi.org/10.3390/fractalfract8110665 - 15 Nov 2024

Cited by 19 | Viewed by 2354

Abstract

We show how a rescaling of fractional operators with bounded kernels may help circumvent their documented deficiencies, for example, the inconsistency at zero or the lack of inverse integral operator. On the other hand, we build a novel class of linear operators with [...] Read more.

We show how a rescaling of fractional operators with bounded kernels may help circumvent their documented deficiencies, for example, the inconsistency at zero or the lack of inverse integral operator. On the other hand, we build a novel class of linear operators with memory effects to extend the L-fractional and the ordinary derivatives, using probability tools. A Mittag–Leffler-type function is introduced to solve linear problems, and nonlinear equations are addressed with power series, illustrating the methods for the SIR epidemic model. The inverse operator is constructed, and a fundamental theorem of calculus and an existence-and-uniqueness result for differintegral equations are proven. A conjecture on deconvolution is raised, which would permit completing the proposed theory. Full article

(This article belongs to the Special Issue Mittag-Leffler Function: Generalizations and Applications)

10 pages, 1157 KB

Open AccessEditor’s ChoiceArticle

Some Insights into the Sierpiński Triangle Paradox

by Miguel-Ángel Martínez-Cruz, Julián Patiño-Ortiz, Miguel Patiño-Ortiz and Alexander S. Balankin

Fractal Fract. 2024, 8(11), 655; https://doi.org/10.3390/fractalfract8110655 - 11 Nov 2024

Cited by 6 | Viewed by 11051

Abstract

We realize that a Sierpiński arrowhead curve (SAC) fills a Sierpiński gasket (SG) in the same manner as a Peano curve fills a square. Namely, in the limit of an infinite number of iterations, the fractal SAC remains self-avoiding, such that [...] Read more.

We realize that a Sierpiński arrowhead curve (SAC) fills a Sierpiński gasket (SG) in the same manner as a Peano curve fills a square. Namely, in the limit of an infinite number of iterations, the fractal SAC remains self-avoiding, such that

S A C \subset S G

. Therefore, SAC differs from SG in the same sense as the self-avoiding Peano curve

P C \subset {[0,1]}^{2}

differs from the square. In particular, the SG has three-line segments constituting a regular triangle as its border, whereas the border of SAC has the structure of a totally disconnected fat Cantor set. Thus, in contrast to the SG, which has loops at all scales, the SAC is loopless. Consequently, although both patterns have the same similarity dimension

D = \ln 3 / \ln 2

, their connectivity dimensions are different. Specifically, the connectivity dimension of the self-avoiding SAC is equal to its topological dimension

d_{l} (S A C) = d = 1

, whereas the connectivity dimension of the SG is equal to its similarity dimension, that is,

d_{l} (S G) = D

. Therefore, the dynamic properties of SG and SAC are also different. Some other noteworthy features of the Sierpiński triangle are also highlighted. Full article

(This article belongs to the Special Issue Fractal Geometry and Dynamical Systems)

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22 pages, 4049 KB

Open AccessEditor’s ChoiceArticle

Fractal-Fractional-Order Modeling of Liver Fibrosis Disease and Its Mathematical Results with Subinterval Transitions

by Amjad E. Hamza, Osman Osman, Arshad Ali, Amer Alsulami, Khaled Aldwoah, Alaa Mustafa and Hicham Saber

Fractal Fract. 2024, 8(11), 638; https://doi.org/10.3390/fractalfract8110638 - 29 Oct 2024

Cited by 34 | Viewed by 2515

Abstract

In this paper, we study human liver disease with a different approach of interval-based investigation by introducing subintervals. This investigation may be referred to as a short memory investigation. Such concepts are useful in problems where a transition is observed when transitioning from [...] Read more.

In this paper, we study human liver disease with a different approach of interval-based investigation by introducing subintervals. This investigation may be referred to as a short memory investigation. Such concepts are useful in problems where a transition is observed when transitioning from one subinterval to the other one. We use the classical and fractal-fractional-order derivative in each subinterval. We study the existence of solutions by using Banach’s and Krasnoselskii’s fixed-point theorems. Their stability is analyzed by adopting the Hyers–Ulam (H-U) stability approach. Also, using the extended Adams–Bashforth–Moulton (ABM) method, we simulate the results that visually present the numerical solutions for different fractal-fractional-order values. Full article

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

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30 pages, 4505 KB

Open AccessEditor’s ChoiceArticle

Hover Flight Improvement of a Quadrotor Unmanned Aerial Vehicle Using PID Controllers with an Integral Effect Based on the Riemann–Liouville Fractional-Order Operator: A Deterministic Approach

by Gustavo Delgado-Reyes, Jorge Salvador Valdez-Martínez, Pedro Guevara-López and Miguel Angel Hernández-Pérez

Fractal Fract. 2024, 8(11), 634; https://doi.org/10.3390/fractalfract8110634 - 28 Oct 2024

Cited by 12 | Viewed by 5020

Abstract

The hovering flight of a quadrotor Unmanned Aerial Vehicle (UAV) refers to maintaining the aircraft in a fixed position in the air, without lateral, vertical, or rotational movements, using only the vehicle’s control systems to maintain proper balance in all spatial dimensions. Algorithms [...] Read more.

The hovering flight of a quadrotor Unmanned Aerial Vehicle (UAV) refers to maintaining the aircraft in a fixed position in the air, without lateral, vertical, or rotational movements, using only the vehicle’s control systems to maintain proper balance in all spatial dimensions. Algorithms and control systems have been developed to continuously adjust motor speeds to counteract deviations from the desired position and achieve effective hovering flight. This paper proposes a set of PID controllers with an integral effect based on the Riemann–Liouville fractional-order approach to improve the hovering flight of a quadrotor UAV. This research innovates by introducing a set of fractional-order PID controllers for UAV hover stability, which offer better adaptability to non-linear dynamics and robustness than traditional PID controllers. Also presented is the development of new performance metrics (MSE, BQC-LR), which allow for more comprehensive control system evaluations. A thorough comparative analysis with conventional control methods demonstrates the superior performance of fractional-order control in real-world simulations. The numerical simulation results show the effectiveness of the proposed Fractional Integral Action PID Controller in the control of UAV hovering flight, while comparative analyses against a classical controller emphasize the benefits of the fractional-order approach in terms of control accuracy. Full article

(This article belongs to the Special Issue Applications of Fractional-Order Systems to Automatic Control)

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18 pages, 5666 KB

Open AccessEditor’s ChoiceArticle

The Investigation of Nonlinear Time-Fractional Models in Optical Fibers and the Impact Analysis of Fractional-Order Derivatives on Solitary Waves

by Muhammad Idrees Afridi, Tamanna Islam, Md Ali Akbar and Mohamed S. Osman

Fractal Fract. 2024, 8(11), 627; https://doi.org/10.3390/fractalfract8110627 - 24 Oct 2024

Cited by 18 | Viewed by 2403

Abstract

In this article, we investigate a couple of nonlinear time-fractional evolution equations, namely the cubic-quintic-septic-nonic equation and the Davey–Stewartson (DS) equation, both of which have significant applications in complex physical phenomena such as fiber optical communication, optical signal processing, and nonlinear optics. Using [...] Read more.

In this article, we investigate a couple of nonlinear time-fractional evolution equations, namely the cubic-quintic-septic-nonic equation and the Davey–Stewartson (DS) equation, both of which have significant applications in complex physical phenomena such as fiber optical communication, optical signal processing, and nonlinear optics. Using a powerful technique named the extended generalized Kudryashov approach, we extract different rich structured soliton solutions to these models, including bell-shaped, cuspon, parabolic soliton, singular soliton, and squeezed bell-shaped soliton. We also study the impact of fractional-order derivatives on these solutions, providing new insights into the dynamics of nonlinear models. The results are compared with the existing literature, revealing novel and distinct solutions that offer a deeper understanding of these fractional models. The results show that the implemented approach is useful, reliable, and compatible for examining fractional nonlinear evolution equations in applied science and engineering. Full article

(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application, 2nd Edition)

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19 pages, 6656 KB

Open AccessEditor’s ChoiceArticle

Dynamic Analysis and FPGA Implementation of Fractional-Order Hopfield Networks with Memristive Synapse

by Andrés Anzo-Hernández, Ernesto Zambrano-Serrano, Miguel Angel Platas-Garza and Christos Volos

Fractal Fract. 2024, 8(11), 628; https://doi.org/10.3390/fractalfract8110628 - 24 Oct 2024

Cited by 14 | Viewed by 2400

Abstract

Memristors have become important components in artificial synapses due to their ability to emulate the information transmission and memory functions of biological synapses. Unlike their biological counterparts, which adjust synaptic weights, memristor-based artificial synapses operate by altering conductance or resistance, making them useful [...] Read more.

Memristors have become important components in artificial synapses due to their ability to emulate the information transmission and memory functions of biological synapses. Unlike their biological counterparts, which adjust synaptic weights, memristor-based artificial synapses operate by altering conductance or resistance, making them useful for enhancing the processing capacity and storage capabilities of neural networks. When integrated into systems like Hopfield neural networks, memristors enable the study of complex dynamic behaviors, such as chaos and multistability. Moreover, fractional calculus is significant for their ability to model memory effects, enabling more accurate simulations of complex systems. Fractional-order Hopfield networks, in particular, exhibit chaotic and multistable behaviors not found in integer-order models. By combining memristors with fractional-order Hopfield neural networks, these systems offer the possibility of investigating different dynamic phenomena in artificial neural networks. This study investigates the dynamical behavior of a fractional-order Hopfield neural network (HNN) incorporating a memristor with a piecewise segment function in one of its synapses, highlighting the impact of fractional-order derivatives and memristive synapses on the stability, robustness, and dynamic complexity of the system. Using a network of four neurons as a case study, it is demonstrated that the memristive fractional-order HNN exhibits multistability, coexisting chaotic attractors, and coexisting limit cycles. Through spectral entropy analysis, the regions in the initial condition space that display varying degrees of complexity are mapped, highlighting those areas where the chaotic series approach a pseudo-random sequence of numbers. Finally, the proposed fractional-order memristive HNN is implemented on a Field-Programmable Gate Array (FPGA), demonstrating the feasibility of real-time hardware realization. Full article

(This article belongs to the Special Issue Fractional Calculus in the Design, Control and Implementation of Complex Systems)

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12 pages, 2154 KB

Open AccessEditor’s ChoiceArticle

Fractal Strategy for Improving Characterization of N₂ Adsorption–Desorption in Mesopores

by Kunpeng Feng, Gaofeng Liu, Zhen Zhang, Huan Liu, Runsheng Lv, Xiaoming Wang, Ping Chang, Jia Lin and George Barakos

Fractal Fract. 2024, 8(11), 617; https://doi.org/10.3390/fractalfract8110617 - 23 Oct 2024

Cited by 22 | Viewed by 4033

Abstract

The current studies primarily analyze the heterogeneity and complexity of mesopore structures based on low-temperature nitrogen (N₂) adsorption curves and the Frenkel–Halsey–Hill (FHH) fractal model. However, these studies ignore the fact that the low-temperature N₂ desorption curve can also reflect [...] Read more.

The current studies primarily analyze the heterogeneity and complexity of mesopore structures based on low-temperature nitrogen (N₂) adsorption curves and the Frenkel–Halsey–Hill (FHH) fractal model. However, these studies ignore the fact that the low-temperature N₂ desorption curve can also reflect the desorption performance of the mesopore structure. In this research, novel fractal indicators for characterizing the adsorption–desorption performance of mesopores based on the fractal dimension from the N₂ adsorption curves and N₂ desorption curves are proposed. The novel fractal indicators I₁ and I₂ are applied to evaluate the adsorption–desorption performance of mesopores with pore size 2–5 nm and pore size 5–50 nm, respectively. The fractal indicator I₁ shows an increasing trend with coalification, reflecting that the gas adsorption performance of 2–5 nm mesopores is enhanced with coalification. The fractal indicator I₂ exhibits a trend of first increasing and then decreasing with coalification, indicating the gas desorption performance of mesopores with pore size 5–50 nm decreases first and then increases. The proposed indicators provide novel analytical parameters for further understanding the gas adsorption–desorption mechanism of porous coal-based or carbon-based materials. Full article

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

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