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Fractal Fract
Volume 9
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Fractal Fract., Volume 9, Issue 8 (August 2025) – 65 articles

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21 pages, 1191 KiB  
Article
ABAQUS Subroutine-Based Implementation of a Fractional Consolidation Model for Saturated Soft Soils
by Tao Zeng, Tao Feng and Yansong Wang
Fractal Fract. 2025, 9(8), 542; https://doi.org/10.3390/fractalfract9080542 (registering DOI) - 17 Aug 2025
Abstract
This paper presents a finite element implementation of a fractional rheological consolidation model in ABQUS, in which the fractional Merchant model governs the mechanical behavior of the soil skeleton, and the water flow is controlled by the fractional Darcy’s law. The implementation generally [...] Read more.
This paper presents a finite element implementation of a fractional rheological consolidation model in ABQUS, in which the fractional Merchant model governs the mechanical behavior of the soil skeleton, and the water flow is controlled by the fractional Darcy’s law. The implementation generally involves two main parts: subroutine-based fractional constitutive models’ development and their coupling. Considering the formal similarity between the energy equation and the mass equation, the fractional Darcy’s law was implemented using the UMATHT subroutine. The fractional Merchant model was then realized through the UMAT subroutine. Both subroutines were individually verified and then successfully coupled. The coupling was achieved by modifying the stress update scheme based on Biot’s poroelastic theory and the effective stress principle in UMAT, enabling a finite element analysis of the fractional consolidation model. Finally, the model was applied to simulate the consolidation behavior of a multi-layered foundation. The proposed approach may serve as a reference for the finite element implementation of consolidation models incorporating a fractional seepage model in ABAQUS. Full article
(This article belongs to the Special Issue Fractional Derivatives in Mathematical Modeling and Applications)
28 pages, 4456 KiB  
Article
Neural Networks-Based Analytical Solver for Exact Solutions of Fractional Partial Differential Equations
by Shanhao Yuan, Yanqin Liu, Limei Yan, Runfa Zhang and Shunjun Wu
Fractal Fract. 2025, 9(8), 541; https://doi.org/10.3390/fractalfract9080541 (registering DOI) - 16 Aug 2025
Abstract
This paper introduces an innovative artificial neural networks-based analytical solver for fractional partial differential equations (fPDEs), combining neural networks (NNs) with symbolic computation. Leveraging the powerful function approximation ability of NNs and the exactness of symbolic methods, our approach achieves notable improvements in [...] Read more.
This paper introduces an innovative artificial neural networks-based analytical solver for fractional partial differential equations (fPDEs), combining neural networks (NNs) with symbolic computation. Leveraging the powerful function approximation ability of NNs and the exactness of symbolic methods, our approach achieves notable improvements in both computational speed and solution precision. The efficacy of the proposed method is validated through four numerical examples, with results visualized using three-dimensional surface plots, contour mappings, and density distributions. Numerical experiments demonstrate that the proposed framework successfully derives exact solutions for fPDEs without relying on data samples. This research provides a novel methodological framework for solving fPDEs, with broad applicability across scientific and engineering fields. Full article
(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 3rd Edition)
30 pages, 3058 KiB  
Article
Existence, Uniqueness, and Stability of Weighted Fuzzy Fractional Volterra–Fredholm Integro-Differential Equation
by Sahar Abbas, Abdul Ahad Abro, Syed Muhammad Daniyal, Hanaa A. Abdallah, Sadique Ahmad, Abdelhamied Ashraf Ateya and Noman Bin Zahid
Fractal Fract. 2025, 9(8), 540; https://doi.org/10.3390/fractalfract9080540 (registering DOI) - 16 Aug 2025
Abstract
This paper investigates a novel class of weighted fuzzy fractional Volterra–Fredholm integro-differential equations (FWFVFIDEs) subject to integral boundary conditions. The analysis is conducted within the framework of Caputo-weighted fractional calculus. Employing Banach’s and Krasnoselskii’s fixed-point theorems, we establish the existence and uniqueness of [...] Read more.
This paper investigates a novel class of weighted fuzzy fractional Volterra–Fredholm integro-differential equations (FWFVFIDEs) subject to integral boundary conditions. The analysis is conducted within the framework of Caputo-weighted fractional calculus. Employing Banach’s and Krasnoselskii’s fixed-point theorems, we establish the existence and uniqueness of solutions. Stability is analyzed in the Ulam–Hyers (UHS), generalized Ulam–Hyers (GUHS), and Ulam–Hyers–Rassias (UHRS) senses. A modified Adomian decomposition method (MADM) is introduced to derive explicit solutions without linearization, preserving the problem’s original structure. The first numerical example validates the theoretical findings on existence, uniqueness, and stability, supplemented by graphical results obtained via the MADM. Further examples illustrate fuzzy solutions by varying the uncertainty level (r), the variable (x), and both parameters simultaneously. The numerical results align with the theoretical analysis, demonstrating the efficacy and applicability of the proposed method. Full article
(This article belongs to the Special Issue Recent Developments in Multidimensional Fractional Differential Equations and Fractional Difference Equations)
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29 pages, 5533 KiB  
Article
Automated First-Arrival Picking and Source Localization of Microseismic Events Using OVMD-WTD and Fractal Box Dimension Analysis
by Guanqun Zhou, Shiling Luo, Yafei Wang, Yongxin Gao, Xiaowei Hou, Weixin Zhang and Chuan Ren
Fractal Fract. 2025, 9(8), 539; https://doi.org/10.3390/fractalfract9080539 (registering DOI) - 16 Aug 2025
Abstract
Microseismic monitoring has become a critical technology for hydraulic fracturing in unconventional oil and gas reservoirs, owing to its high temporal and spatial resolution. It plays a pivotal role in tracking fracture propagation and evaluating stimulation effectiveness. However, the automatic picking of first-arrival [...] Read more.
Microseismic monitoring has become a critical technology for hydraulic fracturing in unconventional oil and gas reservoirs, owing to its high temporal and spatial resolution. It plays a pivotal role in tracking fracture propagation and evaluating stimulation effectiveness. However, the automatic picking of first-arrival times and accurate source localization remain challenging under complex noise conditions, which constrain the reliability of fracture parameter inversion and reservoir assessment. To address these limitations, we propose a hybrid approach that combines optimized variational mode decomposition (OVMD), wavelet thresholding denoising (WTD), and an adaptive fractal box-counting dimension algorithm for enhanced first-arrival picking and source localization. Specifically, OVMD is first employed to adaptively decompose seismic signals and isolate noise-dominated components. Subsequently, WTD is applied in the multi-scale frequency domain to suppress residual noise. An adaptive fractal dimension strategy is then utilized to detect change points and accurately determine the first-arrival time. These results are used as inputs to a particle swarm optimization (PSO) algorithm for source localization. Both numerical simulations and laboratory experiments demonstrate that the proposed method exhibits high robustness and localization accuracy under severe noise conditions. It significantly outperforms conventional approaches such as short-time Fourier transform (STFT) and continuous wavelet transform (CWT). The proposed framework offers reliable technical support for dynamic fracture monitoring, detailed reservoir characterization, and risk mitigation in the development of unconventional reservoirs. Full article
(This article belongs to the Special Issue Multiscale Fractal Analysis in Unconventional Reservoirs)
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4 pages, 163 KiB  
Editorial
Fractal and Fractional Theories in Advancing Geotechnical Engineering Practices
by Shao-Heng He, Zhi Ding and Panpan Guo
Fractal Fract. 2025, 9(8), 537; https://doi.org/10.3390/fractalfract9080537 (registering DOI) - 16 Aug 2025
Abstract
Fractal and fractional theories have developed over several decades and have gradually grown in popularity, with significant applications in geotechnical engineering [...] Full article
(This article belongs to the Special Issue Fractal and Fractional in Geotechnical Engineering)
24 pages, 4026 KiB  
Article
Quantitative Analysis of Multi-Angle Correlation Between Fractal Dimension of Anthracite Surface and Its Coal Quality Indicators in Different Regions
by Shoule Zhao and Dun Wu
Fractal Fract. 2025, 9(8), 538; https://doi.org/10.3390/fractalfract9080538 - 15 Aug 2025
Abstract
The nanoporous structure of coal is crucial for the occurrence and development of coalbed methane (CBM). This study, leveraging the combined characterization of atomic force microscopy (AFM) and Gwyddion software, investigated six anthracite samples with varying degrees of metamorphism (Ro = 2.11–3.36%). [...] Read more.
The nanoporous structure of coal is crucial for the occurrence and development of coalbed methane (CBM). This study, leveraging the combined characterization of atomic force microscopy (AFM) and Gwyddion software, investigated six anthracite samples with varying degrees of metamorphism (Ro = 2.11–3.36%). It revealed the intrinsic relationships between their nanoporous structures, surface morphologies, fractal characteristics, and coalification processes. The research found that as Ro increases, the surface relief of coal decreases significantly, with pore structures evolving from being macropore-dominated to micropore-enriched, and the surface tending towards smoothness. Surface roughness parameters (Ra, Rq) exhibit a negative correlation with Ro. Quantitative data indicate that area porosity, pore count, and shape factor positively correlate with metamorphic grade, while mean pore diameter negatively correlates with it. The fractal dimensions calculated using the variance partition method, cube-counting method, triangular prism measurement method, and power spectrum method all show nonlinear correlations with Ro, moisture (Mad), ash content (Aad), and volatile matter (Vdaf). Among these, the fractal dimension obtained by the triangular prism measurement method has the highest correlation with Ro, Aad, and Vdaf, while the variance partition method shows the highest correlation with Mad. This study clarifies the regulatory mechanisms of coalification on the evolution of nanoporous structures and surface properties, providing a crucial theoretical foundation for the precise evaluation and efficient exploitation strategies of CBM reservoirs. Full article
(This article belongs to the Special Issue Applications of Fractal Dimensions in Rock Mechanics and Geomechanics)
19 pages, 2887 KiB  
Article
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
Viewed by 39
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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30 pages, 8331 KiB  
Article
Fracture Complexity and Mineral Damage in Shale Hydraulic Fracturing Based on Microscale Fractal Analysis
by Xin Liu, Jiaqi Zhang, Tianjiao Li, Zhengzhao Liang, Siwei Meng, Licai Zheng and Na Wu
Fractal Fract. 2025, 9(8), 535; https://doi.org/10.3390/fractalfract9080535 - 15 Aug 2025
Viewed by 33
Abstract
The geological structural complexity and microscale heterogeneity of shale reservoirs, characterized by the brittleness index and natural fracture density, exert a decisive effect on hydraulic fracturing’s effectiveness. However, the mechanisms underlying the true microscale heterogeneity of shale structures, which is neglected in conventional [...] Read more.
The geological structural complexity and microscale heterogeneity of shale reservoirs, characterized by the brittleness index and natural fracture density, exert a decisive effect on hydraulic fracturing’s effectiveness. However, the mechanisms underlying the true microscale heterogeneity of shale structures, which is neglected in conventional models and influences fracture evolution, remain unclear. Here, high-resolution scanning electron microscopy (SEM) was employed to obtain realistic distributions of mineral components and natural fractures, and hydraulic–mechanical coupled simulation models were developed within the Realistic Failure Process Analysis (RFPA) simulator using digital rock techniques. The analysis examined how the brittleness index and natural fracture density affect the fracture morphology’s complexity, mineral failure behavior, and flow conductivity. Numerical simulations show that the main fractures preferentially propagate toward areas with high local brittleness and dense natural fractures. Both the fracture’s fractal dimension and the stimulated reservoir volume increased with the brittleness index. A moderate natural fracture density promotes the fracture network’s complexity, whereas excessive densities may suppress the main fracture’s propagation. Microscopically, organic matter and silicate minerals are more prone to damage, predominantly tensile failures under external loading. These findings highlight the dominant role of microscale heterogeneity in shale fracturing and provide theoretical support for fracture control and stimulation optimization in complex reservoirs. Full article
(This article belongs to the Special Issue Multiscale Fractal Analysis in Unconventional Reservoirs)
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17 pages, 4064 KiB  
Article
Study on Multi-Scale Damage Evolution of Sandstone Under Freeze–Thaw Cycles: A Computational Perspective Based on Pore Structure and Fractal Dimension
by Jianhui Qiu, Keping Zhou, Guanglin Tian and Taoying Liu
Fractal Fract. 2025, 9(8), 534; https://doi.org/10.3390/fractalfract9080534 - 15 Aug 2025
Viewed by 63
Abstract
Understanding the intrinsic relationship between microscopic structures and macroscopic mechanical properties of rock under freeze–thaw (F-T) conditions is essential for ensuring the safety and stability of geotechnical engineering in cold regions. In this study, a series of F-T cycle tests, nuclear magnetic resonance [...] Read more.
Understanding the intrinsic relationship between microscopic structures and macroscopic mechanical properties of rock under freeze–thaw (F-T) conditions is essential for ensuring the safety and stability of geotechnical engineering in cold regions. In this study, a series of F-T cycle tests, nuclear magnetic resonance (NMR) measurements, and uniaxial compression tests were conducted on sandstone samples. The mechanisms by which F-T cycles influence pore structure and mechanical behavior were analyzed, revealing their internal correlation. A degradation model for peak strength was developed using mesopore porosity as the key influencing parameter. The results showed that with increasing F-T cycles, the total porosity and mesopore and macropore porosities all exhibited increasing trends, whereas the micropore and different fractal dimensions decreased. The compaction stage in the stress–strain curves became increasingly prominent with more F-T cycles. Meanwhile, the peak strength and secant modulus decreased, while the peak strain increased. When the frost heave pressure induced by water–ice phase transitions exceeded the ultimate bearing capacity of pore walls, smaller pores progressively evolved into larger ones, leading to an increase in the mesopores and macropores. Notably, mesopores and macropores demonstrated significant fractal characteristics. The transformation in pore size disrupted the power-law distribution of pore radii and reduced fractal dimensions. A strong correlation was observed between peak strength and both the mesopore and mesopore fractal dimensions. The increase in mesopores and macropores enhanced the compaction stage of the stress–strain curve. Moreover, the expansion and interconnection of mesopores under loading conditions degraded the deformation resistance and load-bearing capacity, thereby reducing both the secant modulus and peak strength. The degradation model for peak strength, developed based on changes in mesopore ratio, proved effective for evaluating the mechanical strength when subjected to different numbers of F-T cycles. Full article
(This article belongs to the Special Issue Applications of Fractal Dimensions in Rock Mechanics and Geomechanics)
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22 pages, 4009 KiB  
Article
A Multi-Dimensional Feature Extraction Model Fusing Fractional-Order Fourier Transform and Convolutional Information
by Haijing Sun, Wen Zhou, Jiapeng Yang, Yichuan Shao, Le Zhang and Zhiqiang Mao
Fractal Fract. 2025, 9(8), 533; https://doi.org/10.3390/fractalfract9080533 - 14 Aug 2025
Viewed by 197
Abstract
In the field of deep learning, the traditional Vision Transformer (ViT) model has some limitations when dealing with local details and long-range dependencies; especially in the absence of sufficient training data, it is prone to overfitting. Structures such as retinal blood vessels and [...] Read more.
In the field of deep learning, the traditional Vision Transformer (ViT) model has some limitations when dealing with local details and long-range dependencies; especially in the absence of sufficient training data, it is prone to overfitting. Structures such as retinal blood vessels and lesion boundaries have distinct fractal properties in medical images. The Fractional Convolution Vision Transformer (FCViT) model is proposed in this paper, which effectively compensates for the deficiency of ViT in local feature capture by fusing convolutional information. The ability to classify medical images is enhanced by analyzing frequency domain features using fractional-order Fourier transform and capturing global information through a self-attention mechanism. The three-branch architecture enables the model to fully understand the data from multiple perspectives, capturing both local details and global context, which in turn improves classification performance and generalization. The experimental results showed that the FCViT model achieved 93.52% accuracy, 93.32% precision, 92.79% recall, and a 93.04% F1-score on the standardized fundus glaucoma dataset. The accuracy on the Harvard Dataverse-V1 dataset reached 94.21%, with a precision of 93.73%, recall of 93.67%, and F1-score of 93.68%. The FCViT model achieves significant performance gains on a variety of neural network architectures and tasks with different source datasets, demonstrating its effectiveness and utility in the field of deep learning. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Fourier Transforms and Applications, 2nd Edition)
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18 pages, 6449 KiB  
Article
Analysis of the Microscopic Pore Structure Characteristics of Sandstone Based on Nuclear Magnetic Resonance Experiments and Nuclear Magnetic Resonance Logging Technology
by Shiqin Li, Chuanqi Tao, Haiyang Fu, Huan Miao and Jiutong Qiu
Fractal Fract. 2025, 9(8), 532; https://doi.org/10.3390/fractalfract9080532 - 14 Aug 2025
Viewed by 119
Abstract
This study focuses on the complex pore structure and pronounced heterogeneity of tight sandstone reservoirs in the Linxing area of the Ordos Basin and develops a multi-scale quantitative characterization approach to investigate the coupling mechanism between pore structure and reservoir properties. Six core [...] Read more.
This study focuses on the complex pore structure and pronounced heterogeneity of tight sandstone reservoirs in the Linxing area of the Ordos Basin and develops a multi-scale quantitative characterization approach to investigate the coupling mechanism between pore structure and reservoir properties. Six core samples were selected from the Shiqianfeng Formation (depth interval: 1326–1421 m) for detailed analysis. Cast thin sections and scanning electron microscopy (SEM) experiments were employed to characterize pore types and structural features. Nuclear magnetic resonance (NMR) experiments were conducted to obtain T2 spectra, which were used to classify bound and movable pores, and their corresponding fractal dimensions were calculated separately. In addition, NMR logging data from the corresponding well intervals were integrated to assess the applicability and consistency of the fractal characteristics at the logging scale. The results reveal that the fractal dimension of bound pores shows a positive correlation with porosity, whereas that of movable pores is negatively correlated with permeability, indicating that different scales of pore structural complexity exert distinct influences on reservoir performance. Mineral composition affects the evolution of pore structures through mechanisms such as framework support, dissolution, and pore-filling, thereby further enhancing reservoir heterogeneity. The consistency between logging responses and experimental observations verifies the regional applicability of fractal analysis. Bound pores dominate within the studied interval, and the vertical variation of the PMF/BVI ratio aligns closely with both the NMR T2 spectra and fractal results. This study demonstrates that fractal dimension is an effective descriptor of structural characteristics across different pore types and provides a theoretical foundation and methodological support for the evaluation of pore complexity and heterogeneity in tight sandstone reservoirs. Full article
(This article belongs to the Special Issue Multiscale Fractal Analysis in Unconventional Reservoirs)
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20 pages, 6947 KiB  
Article
Fractal Evolution Characteristics of Weakly Cemented Overlying Rock Fractures in Extra-Thick Coal Seams Mining in Western Mining Areas
by Cun Zhang, Zhaopeng Ren, Jun He and Xiangyu Zhao
Fractal Fract. 2025, 9(8), 531; https://doi.org/10.3390/fractalfract9080531 - 14 Aug 2025
Viewed by 178
Abstract
Coal mining disturbance induces progressive damage and fracturing in overlying rock (OLR), forming a complex fracture network. This process triggers groundwater depletion, ecological degradation, and severely compromises mine safety. Based on field drilling sampling and mechanical experiments, this paper reveals the occurrence properties [...] Read more.
Coal mining disturbance induces progressive damage and fracturing in overlying rock (OLR), forming a complex fracture network. This process triggers groundwater depletion, ecological degradation, and severely compromises mine safety. Based on field drilling sampling and mechanical experiments, this paper reveals the occurrence properties and characteristics of weakly cemented overlying rock (WCOLR). At the same time, similar simulation experiments, DIC speckle analysis system, and fractal theory are used to explain the development and evolution mechanism of mining-induced fractures under this special geological condition. The OLR fracture is determined based on the grid fractal dimension (D) distribution. A stress arch-bed separation (BS) co-evolution model is established based on dynamic cyclic BS development and stress arch characteristics, enabling identification of BS horizons. The results show that the overlying weak and extremely weak rock accounts for more than 90%. During the process of longwall face (LF) advancing, the D undergoes oscillatory evolution through five distinct stages: rapid initial growth, constrained slow growth under thick, soft strata (TSS), dimension reduction induced by fracturing and compaction of TSS, secondary growth from newly generated fractures, and stabilization upon reaching full extraction. Grid-based D analysis further categorizes fracture zones, indicating a water conducting fracture zone (WCFZ) height of 160~180 m. Mining-induced fractures predominantly concentrate at dip angles of 0–10°, 40–50°, and 170–180°. Horizontally BS fractures account for 70.2% of the total fracture population, vertically penetrating fractures constitute 13.1% and transitional fractures make up the remaining 16.7%. The stress arch height is 314.4 m, and the stable BS horizon is 260 m away from the coal seam. Finally, an elastic foundation theory-based model was used to predict BS development under top-coal caving operations. This research provides scientific foundations for damage-reduced mining in ecologically vulnerable Western China coalfields. Full article
(This article belongs to the Special Issue Applications of Fractal Analysis in Underground Engineering, Second Edition)
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26 pages, 3065 KiB  
Article
A Kangaroo Escape Optimizer-Enabled Fractional-Order PID Controller for Enhancing Dynamic Stability in Multi-Area Power Systems
by Sulaiman Z. Almutairi and Abdullah M. Shaheen
Fractal Fract. 2025, 9(8), 530; https://doi.org/10.3390/fractalfract9080530 - 14 Aug 2025
Viewed by 194
Abstract
In this study, we propose a novel metaheuristic algorithm named Kangaroo Escape optimization Technique (KET), inspired by the survival-driven escape strategies of kangaroos in unpredictable environments. The algorithm integrates a chaotic logistic energy adaptation strategy to balance a two-phase exploration process—zigzag motion and [...] Read more.
In this study, we propose a novel metaheuristic algorithm named Kangaroo Escape optimization Technique (KET), inspired by the survival-driven escape strategies of kangaroos in unpredictable environments. The algorithm integrates a chaotic logistic energy adaptation strategy to balance a two-phase exploration process—zigzag motion and long-jump escape—and an adaptive exploitation phase with local search guided by either nearby elite solutions or random peers. A unique decoy drop mechanism is introduced to prevent premature convergence and ensure dynamic diversity. KET is applied to optimize the parameters of a fractional-order Proportional Integral Derivative (PID) controller for Load Frequency Control (LFC) in interconnected power systems. The designed fractional-order PID controller-based KET optimization extends the conventional PID by introducing fractional calculus into the integral and derivative terms, allowing for more flexible and precise control dynamics. This added flexibility enables enhanced robustness and tuning capability, particularly useful in complex and uncertain systems such as modern power systems. Comparative results with existing state-of-the-art algorithms demonstrate the superior robustness, convergence speed, and control accuracy of the proposed approach under dynamic scenarios. The proposed KET-fractional order PID controller offers 29.6% greater robustness under worst-case conditions and 36% higher consistency across multiple runs compared to existing techniques. It achieves optimal performance faster than the Neural Network Algorithm (NNA), achieving its best Integral of Time Absolute Error (ITAE) value within the first 20 iterations, demonstrating its superior learning rate and early-stage search efficiency. In addition to LFC, the robustness and generality of the proposed KET were validated on a standard speed reducer design problem, demonstrating superior optimization performance and consistent convergence when compared to several recent metaheuristics. Full article
(This article belongs to the Special Issue Advances in Fractional-Order Control with Special Focus to Power Systems)
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14 pages, 356 KiB  
Article
Pointwise Error Analysis of the Corrected L1 Scheme for the Multi-Term Subdiffusion Equation
by Qingzhao Li and Chaobao Huang
Fractal Fract. 2025, 9(8), 529; https://doi.org/10.3390/fractalfract9080529 - 14 Aug 2025
Viewed by 182
Abstract
This paper considers the multi-term subdiffusion equation with weakly singular solutions. In order to use sparser meshes near the initial time, a novel scheme (which we call the corrected L1 scheme) on graded meshes is constructed to estimate the multi-term Caputo fractional derivative [...] Read more.
This paper considers the multi-term subdiffusion equation with weakly singular solutions. In order to use sparser meshes near the initial time, a novel scheme (which we call the corrected L1 scheme) on graded meshes is constructed to estimate the multi-term Caputo fractional derivative by investigating a corrected term for the nonuniform L1 scheme. Combining this nonuniform corrected L1 scheme in the temporal direction and the finite element method (FEM) in the spatial direction, a fully discrete scheme for solving the multi-term subdiffusion equation is developed. The stability result of the developed scheme is given. Furthermore, the optimal pointwise-in-time error estimate of the developed scheme is derived. Finally, several numerical experiments are conducted to verify our theoretical findings. Full article
(This article belongs to the Special Issue Exploration and Analysis of Higher-Order Numerical Methods for Fractional Differential Equations)
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22 pages, 1566 KiB  
Article
Design and Control of Fractional-Order Systems Based on Fractal Operators
by Zhimo Jian, Chaoqian Luo and Yajun Yin
Fractal Fract. 2025, 9(8), 528; https://doi.org/10.3390/fractalfract9080528 - 13 Aug 2025
Viewed by 103
Abstract
In recent years, we have abstracted physical fractal space from biological structures and movements within living organisms, revealing the profound intrinsic connections between fractional order time and fractional-dimensional space, and providing partial explanations for the sources and orders of fractional order. We have [...] Read more.
In recent years, we have abstracted physical fractal space from biological structures and movements within living organisms, revealing the profound intrinsic connections between fractional order time and fractional-dimensional space, and providing partial explanations for the sources and orders of fractional order. We have confirmed that the topological invariants of fractal cells, the order of physical components, and the mismatch of spatiotemporal order are important factors determining the fractional order of operators. This paper is a continuation of the previous work. Inspired by bone fractal operators, this article attempts to identify other factors that affect the order of operators. Specifically, the following contents are included: (1) originating from the bone fractal operators, we present the construction process of the “apparent half-order” system; (2) using the Schiessel–Blumen model as the comparative object, we analyze the origin and characteristics of the “γ-order” system; (3) using the continued fraction theory and operatorization thought as the link, we establish the design and control method for general fractional-order systems, and discuss the factors affecting the order of fractional-order operators. Full article
(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing, 2nd Edition)
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17 pages, 294 KiB  
Article
Novel Fixed Point Results in Rectangular Gb-Metric Spaces and Some Applications on Fractional Differential Equations
by Rende Ramadan, Ozgur Ege and Rajagopalan Ramaswamy
Fractal Fract. 2025, 9(8), 527; https://doi.org/10.3390/fractalfract9080527 - 13 Aug 2025
Viewed by 215
Abstract
In this work, we prove some fixed point theorems in rectangular Gb-metric space, which is the generalization of rectangular metric space and Gb-metric space. Moreover, we give some examples to support our theoretical findings. Finally, using our main results, [...] Read more.
In this work, we prove some fixed point theorems in rectangular Gb-metric space, which is the generalization of rectangular metric space and Gb-metric space. Moreover, we give some examples to support our theoretical findings. Finally, using our main results, we present some applications to obtain solutions of Riemann–Liouville and Atangana–Baleanu fractional integral equations. Full article
(This article belongs to the Special Issue Nonlinear Fractional Differential Equation and Fixed-Point Theory, 2nd Edition)
31 pages, 1578 KiB  
Article
The Impact of Elastoplastic Deformation Behavior on the Apparent Gas Permeability of Deep Fractal Shale Rocks
by Xu Zhou, Zhaoqin Huang, Aifen Li, Jun Yao and Xu Zhang
Fractal Fract. 2025, 9(8), 526; https://doi.org/10.3390/fractalfract9080526 - 13 Aug 2025
Viewed by 136
Abstract
Deep shale gas reservoirs are vital sources of unconventional natural gas and present unique challenges for exploration and development due to their multiscale flow characteristics and elastoplastic deformation behavior of reservoir rocks. Accurately predicting permeability in these reservoirs is crucial. This study introduces [...] Read more.
Deep shale gas reservoirs are vital sources of unconventional natural gas and present unique challenges for exploration and development due to their multiscale flow characteristics and elastoplastic deformation behavior of reservoir rocks. Accurately predicting permeability in these reservoirs is crucial. This study introduces a novel model utilizing fractal theory and a thick-walled cylinder model to characterize stress-dependent apparent gas permeability. The model incorporates various flow mechanisms, including viscous flow, transition flow, Knudsen diffusion, surface diffusion, real gas effects, and gas slip effects. It enables predictions of how permeability changes with elastoplastic behavior and affects the pore volume fractions of different flow mechanisms. Experimental validation during elastic and elastoplastic deformations confirms the model’s accuracy, with each parameter having clear physical significance. Key findings reveal that, at the same effective stress, apparent gas permeability increases with pore radius fractal dimension, temperature, and Young’s modulus, while decreasing with capillary tortuosity fractal dimension. Additionally, during plastic deformation, greater magnitudes of plastic strain lead to more pronounced changes in apparent gas permeability compared to elastic deformation. These insights emphasize the importance of incorporating elastoplastic behavior in studies of deep shale gas reservoirs. Full article
(This article belongs to the Special Issue Fractal Dimensions and Their Implications in Characterization of Porous Media)
26 pages, 4171 KiB  
Article
Arithmetic Harris Hawks-Based Effective Battery Charging from Variable Sources and Energy Recovery Through Regenerative Braking in Electric Vehicles, Implying Fractional Order PID Controller
by Dola Sinha, Saibal Majumder, Chandan Bandyopadhyay and Haresh Kumar Sharma
Fractal Fract. 2025, 9(8), 525; https://doi.org/10.3390/fractalfract9080525 - 13 Aug 2025
Viewed by 212
Abstract
A significant application of the proportional–integral (PI) controller in the automotive sector is in electric motors, particularly brushless direct current (BLDC) motors utilized in electric vehicles (EVs). This paper presents a high-performance boost converter regulated by a fractional-order proportional–integral (FoPI) controller to ensure [...] Read more.
A significant application of the proportional–integral (PI) controller in the automotive sector is in electric motors, particularly brushless direct current (BLDC) motors utilized in electric vehicles (EVs). This paper presents a high-performance boost converter regulated by a fractional-order proportional–integral (FoPI) controller to ensure stable output voltage and power delivery to effectively charge the battery under fluctuating input conditions. The FoPI controller parameters, including gains and fractional order, are optimized using an Arithmetic Harris Hawks Optimization (AHHO) algorithm with an integral absolute error (IAE) as the objective function. The primary objective is to enhance the system’s robustness against input voltage fluctuation while charging from renewable sources. Conversely, regenerative braking is crucial for energy recovery during vehicle operation. This study implements a fractional-order PI controller (FOPI) for the smooth and exact regulation of speed and energy recuperation during regenerative braking. The proposed scheme underwent extensive simulations in the Simulink environment using the FOMCON toolbox version 2023b. The results were validated with the traditional Ziegler–Nichols method. The simulation findings demonstrate smooth and precise speed control and effective energy recovery during regenerative braking and a constant voltage output of 375 V, with fewer ripples and rapid transient responses during charging of batteries from variable input supply. Full article
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22 pages, 4428 KiB  
Article
Pore Structure Characteristics and Controlling Factors of the Lower Cambrian Niutitang Formation Shale in Northern Guizhou: A Case Study of Well QX1
by Yuanyan Yin, Niuniu Zou, Daquan Zhang, Yi Chen, Zhilong Ye, Xia Feng and Wei Du
Fractal Fract. 2025, 9(8), 524; https://doi.org/10.3390/fractalfract9080524 - 13 Aug 2025
Viewed by 188
Abstract
Shale pore architecture governs gas storage capacity, permeability, and production potential in reservoirs. Therefore, this study systematically investigates the pore structure features and influencing factors of the Niutitang Formation shale from the QX1 well in northern Guizhou using field emission scanning electron microscopy [...] Read more.
Shale pore architecture governs gas storage capacity, permeability, and production potential in reservoirs. Therefore, this study systematically investigates the pore structure features and influencing factors of the Niutitang Formation shale from the QX1 well in northern Guizhou using field emission scanning electron microscopy (FE-SEM), high-pressure mercury intrusion (HPMI), low-temperature nitrogen adsorption (LTNA), and nuclear magnetic resonance (NMR) experiments. The results show that ① The pore size of the QX1 well’s Niutitang Formation shale is primarily in the nanometer range, with pore types including intragranular pores, intergranular pores, organic matter pores, and microfractures, with the former two types constituting the primary pore network. ② Pore shapes are plate-shaped intersecting conical microfractures or plate-shaped intersecting ink bottles, ellipsoidal, and beaded pores. ③ The pore size distribution showed a multi-peak distribution, predominantly mesopores, followed by micropores, with the fewest macropores. ④ The fractal dimension D1 > D2 indicates that the shale pore system is characterized by a rough surface and some connectivity of the pore network. ⑤ Carbonate mineral abundances are the main controlling factors affecting the pore structure of shales in the study area, and total organic carbon (TOC) content also has some influence, while clay mineral content shows negligible statistical correlation. Full article
(This article belongs to the Special Issue Multiscale Fractal Analysis in Unconventional Reservoirs)
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18 pages, 4917 KiB  
Article
Rapid Estimation of Soil Copper Content Using a Novel Fractional Derivative Three-Band Index and Spaceborne Hyperspectral Data
by Shichao Cui, Guo Jiang and Jiawei Lu
Fractal Fract. 2025, 9(8), 523; https://doi.org/10.3390/fractalfract9080523 - 12 Aug 2025
Viewed by 221
Abstract
Rapid and large-scale monitoring of soil copper levels enables the quick identification of areas where copper concentrations significantly exceed safe thresholds. It allows for selecting regions that require treatment and protection and is essential for safeguarding environmental and human health. Widely adopted monitoring [...] Read more.
Rapid and large-scale monitoring of soil copper levels enables the quick identification of areas where copper concentrations significantly exceed safe thresholds. It allows for selecting regions that require treatment and protection and is essential for safeguarding environmental and human health. Widely adopted monitoring models that utilize ground- and uncrewed-aerial-vehicle-based spectral data are superior to labor-intensive and time-consuming traditional methods that rely on point sampling, chemical analysis, and spatial interpolation. However, these methods are unsuitable for large-scale observations. Therefore, this study investigates the potential of utilizing spaceborne GF-5 hyperspectral data for monitoring soil copper content. Ninety-five soil samples were collected from the Kalatage mining area in Xinjiang, China. Three-band indices were constructed using fractional derivative spectra, and estimation models were developed using spectral indices highly correlated with the copper content. The results show that the proposed three-band spectral index accurately identifies subtle spectral characteristics associated with the copper content. Although the model is relatively simple, selecting the correct fractional order is critical in constructing spectral indices. The three-band spectral index based on fractional derivatives with orders of less than 0.6 provides higher accuracy than higher-order fractional derivatives. The index with spectral wavelengths of 426.796 nm, 512.275 nm, and 974.245 nm with 0.35-order derivatives exhibits the optimal performance (R2 = 0.51, RPD = 1.46). Additionally, we proposed a novel approach that identifies the three-band indices exhibiting a strong correlation with the copper content. Subsequently, the selected indices were used as independent variables to develop new spectral indices for model development. This approach provides higher performance than models that use spectral indices derived from individual band values. The model utilizing the proposed spectral index achieved the best performance (R2 = 0.56, RPD = 1.52). These results indicate that utilizing GF-5 hyperspectral data for large-scale monitoring of soil copper content is feasible and practical. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 3rd Edition)
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14 pages, 4996 KiB  
Article
Fractional Wave Structures in a Higher-Order Nonlinear Schrödinger Equation with Cubic–Quintic Nonlinearity and β-Fractional Dispersion
by Mahmoud Soliman, Hamdy M. Ahmed, Niveen M. Badra, Islam Samir, Taha Radwan and Karim K. Ahmed
Fractal Fract. 2025, 9(8), 522; https://doi.org/10.3390/fractalfract9080522 - 11 Aug 2025
Viewed by 235
Abstract
This study employs the improved modified extended tanh method (IMETM) to derive exact analytical solutions of a higher-order nonlinear Schrödinger (HNLS) model, incorporating β-fractional derivatives in both time and space. Unlike classical methods such as the inverse scattering transform or Hirota’s bilinear [...] Read more.
This study employs the improved modified extended tanh method (IMETM) to derive exact analytical solutions of a higher-order nonlinear Schrödinger (HNLS) model, incorporating β-fractional derivatives in both time and space. Unlike classical methods such as the inverse scattering transform or Hirota’s bilinear technique, which are typically limited to integrable systems and integer-order operators, the IMETM offers enhanced flexibility for handling fractional models and higher-order nonlinearities. It enables the systematic construction of diverse solution types—including Weierstrass elliptic, exponential, Jacobi elliptic, and bright solitons—within a unified algebraic framework. The inclusion of fractional derivatives introduces richer dynamical behavior, capturing nonlocal dispersion and temporal memory effects. Visual simulations illustrate how fractional parameters α (space) and β (time) affect wave structures, revealing their impact on solution shape and stability. The proposed framework provides new insights into fractional NLS dynamics with potential applications in optical fiber communications, nonlinear optics, and related physical systems. Full article
(This article belongs to the Section Mathematical Physics)
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26 pages, 1345 KiB  
Article
Fractional Chebyshev Transformation for Improved Binarization in the Energy Valley Optimizer for Feature Selection
by Islam S. Fathi, Ahmed R. El-Saeed, Gaber Hassan and Mohammed Aly
Fractal Fract. 2025, 9(8), 521; https://doi.org/10.3390/fractalfract9080521 - 11 Aug 2025
Viewed by 269
Abstract
The feature selection (FS) procedure is a critical preprocessing step in data mining and machine learning, aiming to enhance model performance by eliminating redundant features and reducing dimensionality. The Energy Valley Optimizer (EVO), inspired by particle physics concepts of stability and decay, offers [...] Read more.
The feature selection (FS) procedure is a critical preprocessing step in data mining and machine learning, aiming to enhance model performance by eliminating redundant features and reducing dimensionality. The Energy Valley Optimizer (EVO), inspired by particle physics concepts of stability and decay, offers a novel metaheuristic approach. This study introduces an enhanced binary version of EVO, termed Improved Binarization in the Energy Valley Optimizer with Fractional Chebyshev Transformation (IBEVO-FC), and specifically designed for feature selection challenges. IBEVO-FC incorporates several key advancements over the original EVO. Firstly, it employs a novel fractional Chebyshev transformation function to effectively map the continuous search space of EVO to the binary domain required for feature selection, leveraging the unique properties of fractional orthogonal polynomials for improved binarization. Secondly, the Laplace crossover method is integrated into the initialization phase to improve population diversity and local search capabilities. Thirdly, a random replacement strategy is applied to enhance exploitation and mitigate premature convergence. The efficacy of IBEVO-FC is rigorously evaluated on 26 benchmark datasets from the UCI Repository and compared against 7 contemporary wrapper-based feature selection algorithms. Statistical analysis confirms the competitive performance of the proposed IBEVO-FC method in terms of classification accuracy and feature subset size. Full article
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28 pages, 1721 KiB  
Article
Stability and Convergence Analysis of Compact Finite Difference Method for High-Dimensional Time-Fractional Diffusion Equations with High-Order Accuracy in Time
by Jun-Ying Cao, Jian-Qiang Fang, Zhong-Qing Wang and Zi-Qiang Wang
Fractal Fract. 2025, 9(8), 520; https://doi.org/10.3390/fractalfract9080520 - 8 Aug 2025
Viewed by 267
Abstract
Based on the spatial compact finite difference (SCFD) method, an improved high-order temporal accuracy scheme for high-dimensional time-fractional diffusion equations (TFDEs) is presented in this work. Combining the temporal piecewise quadratic interpolation and the high-dimensional SCFD method, the proposed numerical method is described. [...] Read more.
Based on the spatial compact finite difference (SCFD) method, an improved high-order temporal accuracy scheme for high-dimensional time-fractional diffusion equations (TFDEs) is presented in this work. Combining the temporal piecewise quadratic interpolation and the high-dimensional SCFD method, the proposed numerical method is described. In order to establish the stability and convergence analysis, we introduce a norm ||·||H˜1, which is rigorously proved equivalent to the standard H1-norm. Considering that the coefficients of high-order numerical schemes are not entirely positive, we introduce an appropriate parameter to transform the numerical scheme into an equivalent form with positive coefficients. Based on the equivalent form, we prove that the temporal and spatial convergence orders are (3−γ) and 4 by applying the convergence of geometric progression. The proposed scheme ensures that the theoretical convergence accuracy at each time step is of order (3−γ) without requiring any additional processing techniques. Ultimately, the convergence of the proposed high-order accurate scheme is verified through numerical experiments involving (non-)linear high-dimensional TFDEs. Full article
(This article belongs to the Special Issue Recent Advances in the Spatial and Temporal Discretizations of Fractional PDEs, Second Edition)
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21 pages, 404 KiB  
Article
Wirtinger-Type Inequalities Involving Tempered Ψ-Fractional Derivatives with Applications
by Qingzhe Wu, Muming Zhang, Jing Shao, Muhammad Samraiz, Humaira Javaid and Saima Naheed
Fractal Fract. 2025, 9(8), 519; https://doi.org/10.3390/fractalfract9080519 - 8 Aug 2025
Viewed by 198
Abstract
In this work, we explore Wirtinger-type inequalities involving tempered Ψ-Caputo fractional derivatives by utilizing Taylor’s formula. We establish more general inequalities for the same operator in Lp norms for p>1 by using Hölder’s inequality. Special cases are discussed in the [...] Read more.
In this work, we explore Wirtinger-type inequalities involving tempered Ψ-Caputo fractional derivatives by utilizing Taylor’s formula. We establish more general inequalities for the same operator in Lp norms for p>1 by using Hölder’s inequality. Special cases are discussed in the form of remarks by highlighting their relationships with the existing literature. The derived results are also verified through illustrative examples, including tables and graphs. Moreover, applications of the obtained inequalities are discussed in the context of arithmetic and geometric means. Full article
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22 pages, 4572 KiB  
Article
Effects of Organic Matter Volume Fraction and Fractal Dimension on Tensile Crack Evolution in Shale Using Digital Core Numerical Models
by Xin Liu, Yuepeng Wang, Tianjiao Li, Zhengzhao Liang, Siwei Meng and Licai Zheng
Fractal Fract. 2025, 9(8), 518; https://doi.org/10.3390/fractalfract9080518 - 8 Aug 2025
Viewed by 260
Abstract
Organic matter plays a vital role in shale reservoirs as both a hydrocarbon storage medium and migration pathway. However, the quantitative relationship between the microstructural features of organic matter and the macroscopic mechanical and failure behaviors of shale remains unclear due to rock [...] Read more.
Organic matter plays a vital role in shale reservoirs as both a hydrocarbon storage medium and migration pathway. However, the quantitative relationship between the microstructural features of organic matter and the macroscopic mechanical and failure behaviors of shale remains unclear due to rock heterogeneity and opacity. In this study, high-resolution three-dimensional digital core models of shale were reconstructed using Focused Ion Beam Scanning Electron Microscopy (FIB-SEM) imaging. The digital models captured the spatial distribution of silicate minerals, clay minerals, and organic matter. Numerical simulations of uniaxial tensile failure were performed on these models, considering variations in the organic matter volume fraction and fractal dimension. The results indicate that an increased organic matter volume fraction and fractal dimension are associated with lower tensile strength, simpler fracture geometry, and reduced acoustic emission activity. Tensile cracks preferentially initiate at interfaces between minerals with contrasting elastic moduli, especially between organic matter and clay, and then propagate and coalesce under loading. These findings reveal that both the volume fraction and fractal structure of organic matter are reliable predictors of tensile strength and damage evolution in shale. This study provides new microscale insights into shale failure mechanisms and offers guidance for optimizing hydraulic fracturing in organic-rich formations. Full article
(This article belongs to the Special Issue Applications of Fractal Dimensions in Rock Mechanics and Geomechanics)
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13 pages, 1582 KiB  
Article
Assessment of Mandibular Trabecular Bone Structure in Hypodivergent Growth Patterns Using Fractal Analysis
by Gizem Boztaş Demir, Rüveyda Doğrugören, Kübra Gülnur Topsakal, Gökhan Serhat Duran and Serkan Görgülü
Fractal Fract. 2025, 9(8), 517; https://doi.org/10.3390/fractalfract9080517 - 8 Aug 2025
Viewed by 320
Abstract
The objective of this study is to evaluate the trabecular structure in hypodivergent individuals using fractal analysis, with a particular focus on specific mandibular regions. This study aims to assess the impact of hypodivergent growth patterns on bone microarchitecture. This research involved a [...] Read more.
The objective of this study is to evaluate the trabecular structure in hypodivergent individuals using fractal analysis, with a particular focus on specific mandibular regions. This study aims to assess the impact of hypodivergent growth patterns on bone microarchitecture. This research involved a methodological approach using panoramic radiographs to assess trabecular structure at specific regions of the mandible using fractal analyses. The dimensions of the fractals were calculated with the use of the box-counting technique by the software Image J (v1.53c; Bethesda, MD, USA, National Institutes of Health), while the statistical evaluations were carried out with the Jamovi Software (The Jamovi Project, version 2.3.21.0). The study found significant differences in fractal dimension values between hypodivergent individuals and the control group, particularly in the condyle and angulus regions, indicating a less complex trabecular structure in hypodivergent individuals. This study concludes that individuals with a hypodivergent growth pattern exhibit alterations in trabecular bone structure within the mandibular condyle and angulus regions, characterized by reduced complexity. These findings suggest that increased occlusal forces and mechanical stress associated with this growth pattern may contribute to changes in trabecular architecture. Understanding these variations is essential for orthodontic and maxillofacial diagnosis, treatment planning, and biomechanical considerations, particularly in cases requiring vertical dimension management or anchorage control. Full article
(This article belongs to the Special Issue Fractal Analysis in Biology and Medicine)
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16 pages, 319 KiB  
Article
Exploring Advanced Weighted Integral Inequalities via Extended Fractional Calculus Approaches
by Areej A. Almoneef, Abd-Allah Hyder, Hüseyin Budak and Mohamed A. Barakat
Fractal Fract. 2025, 9(8), 516; https://doi.org/10.3390/fractalfract9080516 - 7 Aug 2025
Viewed by 197
Abstract
This paper investigates weighted Milne-type (M−t) inequalities within the context of Riemann–Liouville (R−L) fractional integrals. We establish multiple versions of these inequalities, applicable to different function categories, such as convex functions with differentiability properties, bounded [...] Read more.
This paper investigates weighted Milne-type (M−t) inequalities within the context of Riemann–Liouville (R−L) fractional integrals. We establish multiple versions of these inequalities, applicable to different function categories, such as convex functions with differentiability properties, bounded functions, functions satisfying Lipschitz conditions, and those exhibiting bounded variation behavior. In particular, we present integral equalities that are essential to establish the main results, using non-negative weighted functions. The findings contribute to the extension of existing inequalities in the literature and provide a deeper understanding of their applications in fractional calculus. This work highlights the advantage of the established inequalities in extending classical results by accommodating a broader class of functions and yielding sharper bounds. It also explores potential directions for future research inspired by these findings. Full article
(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications, 2nd Edition)
22 pages, 8053 KiB  
Article
Rolling Bearing Fault Diagnosis Based on Fractional Constant Q Non-Stationary Gabor Transform and VMamba-Conv
by Fengyun Xie, Chengjie Song, Yang Wang, Minghua Song, Shengtong Zhou and Yuanwei Xie
Fractal Fract. 2025, 9(8), 515; https://doi.org/10.3390/fractalfract9080515 - 6 Aug 2025
Viewed by 248
Abstract
Rolling bearings are prone to failure, meaning that research on intelligent fault diagnosis is crucial in relation to this key transmission component in rotating machinery. The application of deep learning (DL) has significantly advanced the development of intelligent fault diagnosis. This paper proposes [...] Read more.
Rolling bearings are prone to failure, meaning that research on intelligent fault diagnosis is crucial in relation to this key transmission component in rotating machinery. The application of deep learning (DL) has significantly advanced the development of intelligent fault diagnosis. This paper proposes a novel method for rolling bearing fault diagnosis based on the fractional constant Q non-stationary Gabor transform (FCO-NSGT) and VMamba-Conv. Firstly, a rolling bearing fault experimental platform is established and the vibration signals of rolling bearings under various working conditions are collected using an acceleration sensor. Secondly, a kurtosis-to-entropy ratio (KER) method and the rotational kernel function of the fractional Fourier transform (FRFT) are proposed and applied to the original CO-NSGT to overcome the limitations of the original CO-NSGT, such as the unsatisfactory time–frequency representation due to manual parameter setting and the energy dispersion problem of frequency-modulated signals that vary with time. A lightweight fault diagnosis model, VMamba-Conv, is proposed, which is a restructured version of VMamba. It integrates an efficient selective scanning mechanism, a state space model, and a convolutional network based on SimAX into a dual-branch architecture and uses inverted residual blocks to achieve a lightweight design while maintaining strong feature extraction capabilities. Finally, the time–frequency graph is inputted into VMamba-Conv to diagnose rolling bearing faults. This approach reduces the number of parameters, as well as the computational complexity, while ensuring high accuracy and excellent noise resistance. The results show that the proposed method has excellent fault diagnosis capabilities, with an average accuracy of 99.81%. By comparing the Adjusted Rand Index, Normalized Mutual Information, F1 Score, and accuracy, it is concluded that the proposed method outperforms other comparison methods, demonstrating its effectiveness and superiority. Full article
(This article belongs to the Special Issue Implementations and Applications of Algorithms Based on Fractional Calculus to Engineering Problems)
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11 pages, 270 KiB  
Article
Boundary Feedback Control of a Cascade System of Fractional PDE-ODEs
by Dong-Xing Fu, Yi-Ying Feng, Zhi-Wen Zhu, Mi-Mi Hou and Jia Wang
Fractal Fract. 2025, 9(8), 514; https://doi.org/10.3390/fractalfract9080514 - 6 Aug 2025
Viewed by 218
Abstract
The fractional PDE-ODE coupled system with Neumann boundary condition is considered in this paper. We design a boundary state feedback controller using the Backstepping method to stabilize the considered system. According to operator semigroup theory, we obtain a unique solution of the investigated [...] Read more.
The fractional PDE-ODE coupled system with Neumann boundary condition is considered in this paper. We design a boundary state feedback controller using the Backstepping method to stabilize the considered system. According to operator semigroup theory, we obtain a unique solution of the investigated system. Based on this, by using the fractional Lyapunov scheme, we prove that the system is Mittag-Leffler stable. Full article
(This article belongs to the Section Engineering)
18 pages, 347 KiB  
Article
Existence of Positive Solutions for a Class of Nabla Fractional Difference Equations with Parameter-Dependent Summation Boundary Conditions
by Nikolay D. Dimitrov and Jagan Mohan Jonnalagadda
Fractal Fract. 2025, 9(8), 513; https://doi.org/10.3390/fractalfract9080513 - 5 Aug 2025
Viewed by 269
Abstract
In this manuscript, we study a class of nabla fractional difference equations with summation boundary conditions that depend on a parameter. We construct the Green’s function related to the linear problem and we deduce some of its properties. First, we obtain an upper [...] Read more.
In this manuscript, we study a class of nabla fractional difference equations with summation boundary conditions that depend on a parameter. We construct the Green’s function related to the linear problem and we deduce some of its properties. First, we obtain an upper bound of the sum of it, and use this property to give an existence result for the considered problem based on the Leray–Shauder nonlinear alternative. Then, we establish some bounds on the parameter in which the Green’s function is positive, and by using Krasnoselski–Zabreiko fixed-point theorem, we deduce another existence result. Finally, we give some particular examples in order to demonstrate our primary findings. Full article
(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 3rd Edition)
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