Fractal and Fractional

22 pages, 421 KiB

Open AccessArticle

Solution of Time-Fractional Partial Differential Equations via the Natural Generalized Laplace Transform Decomposition Method

by Hassan Eltayeb and Shayea Aldossari

Fractal Fract. 2025, 9(9), 554; https://doi.org/10.3390/fractalfract9090554 - 22 Aug 2025

Abstract

This work presents an efficient analytical technique for obtaining approximate solutions to time-fractional system partial differential equations. The proposed method combines the Natural generalized Laplace transform with the decomposition method to construct a systematic solution framework. A general formulation of the method is [...] Read more.

This work presents an efficient analytical technique for obtaining approximate solutions to time-fractional system partial differential equations. The proposed method combines the Natural generalized Laplace transform with the decomposition method to construct a systematic solution framework. A general formulation of the method is developed for a broad class of time-fractional system equations. In particular, we check the validity and effectiveness of the approach by providing three illustrative examples, confirming its accuracy and applicability in solving both linear and nonlinear fractional problems. Full article

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

15 pages, 1082 KiB

Open AccessArticle

Fractal Modeling of Nonlinear Flexural Wave Propagation in Functionally Graded Beams: Solitary Wave Solutions and Fractal Dimensional Modulation Effects

by Kai Fan, Zhongqing Ma, Cunlong Zhou, Jiankang Liu and Huaying Li

Fractal Fract. 2025, 9(9), 553; https://doi.org/10.3390/fractalfract9090553 - 22 Aug 2025

Abstract

In this study, a new nonlinear dynamic model was established for functionally graded material (FGM) beams with layered/porous fractal microstructures, aiming to reveal the cross-scale propagation mechanism of flexural waves under large deflection conditions. The characteristics of layered/porous microstructures were equivalently mapped to [...] Read more.

In this study, a new nonlinear dynamic model was established for functionally graded material (FGM) beams with layered/porous fractal microstructures, aiming to reveal the cross-scale propagation mechanism of flexural waves under large deflection conditions. The characteristics of layered/porous microstructures were equivalently mapped to the fractal dimension index. In the framework of the fractal derivative, a fractal nonlinear wave governing equation integrating geometric nonlinear effects and microstructure characteristics was derived, and the coupling effect of finite deformation and fractal characteristics was clarified. Four groups of deflection gradient traveling wave analytical solutions were obtained by solving the equation through the extended minimal (G′/G) expansion method. Compared with the traditional (G′/G) expansion method, the new method, which is concise and expands the solution space, generates additional csch² soliton solutions and csc² singular-wave solutions. Numerical simulations showed that the spatiotemporal fractal dimension can dynamically modulate the amplitude attenuation, waveform steepness, and phase rotation characteristics of kink solitary waves in beams. At the same time, it was found that the decrease in the spatial fractal dimension will make the deflection curve of the beam more gentle, revealing that the fractal characteristics of the microstructure have an active control effect on the geometric nonlinearity. This model provides theoretical support for the prediction and regulation of the wave behavior of fractal microstructure FGM components, and has application potential in acoustic metamaterial design and engineering vibration control. Full article

(This article belongs to the Special Issue Fractal Theory and Models in Nonlinear Dynamics and Their Applications, 2nd Edition)

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56 pages, 6217 KiB

Open AccessArticle

Rheologic Fractional Oscillators or Creepers

by Katica R. (Stevanović) Hedrih

Fractal Fract. 2025, 9(8), 552; https://doi.org/10.3390/fractalfract9080552 - 21 Aug 2025

Abstract

Using the newly introduced, by the author, basic complex and hybrid complex rheologic models of the fractional type, the dynamics of a series of mechanical rheologic discrete dynamical systems of the fractional type (RDDSFT) of rheologic oscillators (ROFTs) or creepers (RCFTs), with corresponding [...] Read more.

Using the newly introduced, by the author, basic complex and hybrid complex rheologic models of the fractional type, the dynamics of a series of mechanical rheologic discrete dynamical systems of the fractional type (RDDSFT) of rheologic oscillators (ROFTs) or creepers (RCFTs), with corresponding independent generalized coordinates (IGCs) and external (IGCEDF) and internal (IGCIGF) degrees of freedom of movement, were studied. Laplace transformations of solutions for independent generalized coordinates (IGCs), as well as external (IGCEDFs) and internal (IGCIDF) degrees of freedom of system dynamics, were determined. On the studied specimens, it was shown that rheologic complex models of the fractional type introduce internal degrees of freedom into the dynamics of rheologic discrete dynamical systems. New challenges appear for mathematicians, such as translating the Laplace transformations of solutions for independent generalized coordinates (LTIGCs) into the time domain. A number of translations of Laplace transformations of solutions into the time domain were performed by the author of this paper. A series of characteristic surfaces of elongations of Laplace transformations of independent generalized coordinates (IGCs) of the dynamics of rheologic discrete dynamic systems of the rheologic oscillator type, i.e., the rheologic creeper type, is shown as a function of fractional order differentiation exponent and Laplace transformation parameter. This manuscript presents the scientific results of theoretical research on the dynamics of rheologic discrete dynamic systems of the fractional type that was conducted using new models and a rigorous mathematical analytical analysis with fractional-order differential equations (DEFOs) and Laplace transformations (LTs). These results can contribute to new experimental research and materials technologies. A separate section presents the theoretical foundations of the methods and methodologies used in this research, without the details that can be found in the author’s previously published works. Full article

(This article belongs to the Special Issue From Fractals to Fractional Calculus: Nonlinear Dynamics and Hybrid Complex Systems)

18 pages, 4907 KiB

Open AccessArticle

The Development of a Mesh-Free Technique for the Fractional Model of the Inverse Problem of the Rayleigh–Stokes Equation with Additive Noise

by Farzaneh Safari and Xingya Feng

Fractal Fract. 2025, 9(8), 551; https://doi.org/10.3390/fractalfract9080551 - 21 Aug 2025

Abstract

We are especially interested in the general framework and ability of a semi-analytic method (SAM) to use the trigonometric basis function (TBF) in different domains. Moreover, the stabilizing effect of increasing boundary nodes on the convergence of the method when a level of [...] Read more.

We are especially interested in the general framework and ability of a semi-analytic method (SAM) to use the trigonometric basis function (TBF) in different domains. Moreover, the stabilizing effect of increasing boundary nodes on the convergence of the method when a level of noise is added to the boundary data of the inverse boundary value problem for the nonlinear Rayleigh–Stokes (R-S) equation is investigated. The solution of the ill-conditioned Rayleigh–Stokes equation which the equation is reduced to the linear system

ℑ [C] = ♭

with corrupted boundary data by quasilinearization technical on nonlinear source terms relies on TBFs and radial basis functions (RBFs). Finally, the implementation of the scheme is supported by the numerical experiments. Full article

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43 pages, 5190 KiB

Open AccessArticle

Noise-Induced Transitions in Nonlinear Oscillators: From Quasi-Periodic Stability to Stochastic Chaos

by Adil Jhangeer and Atef Abdelkader

Fractal Fract. 2025, 9(8), 550; https://doi.org/10.3390/fractalfract9080550 - 21 Aug 2025

Abstract

This paper presents a comprehensive dynamical analysis of a nonlinear oscillator subjected to both deterministic and stochastic excitations. Utilizing a diverse suite of analytical tools—including phase portraits, Poincaré sections, Lyapunov exponents, recurrence plots, Fokker–Planck equations, and sensitivity diagnostics—we investigate the transitions between quasi-periodicity, [...] Read more.

This paper presents a comprehensive dynamical analysis of a nonlinear oscillator subjected to both deterministic and stochastic excitations. Utilizing a diverse suite of analytical tools—including phase portraits, Poincaré sections, Lyapunov exponents, recurrence plots, Fokker–Planck equations, and sensitivity diagnostics—we investigate the transitions between quasi-periodicity, chaos, and stochastic disorder. The study reveals that quasi-periodic attractors exhibit robust topological structure under moderate noise but progressively disintegrate as stochastic intensity increases, leading to high-dimensional chaotic-like behavior. Recurrence quantification and Lyapunov spectra validate the transition from coherent dynamics to noise-dominated regimes. Poincaré maps and sensitivity analysis expose multistability and intricate basin geometries, while the Fokker–Planck formalism uncovers non-equilibrium steady states characterized by circulating probability currents. Together, these results provide a unified framework for understanding the geometry, statistics, and stability of noisy nonlinear systems. The findings have broad implications for systems ranging from mechanical oscillators to biological rhythms and offer a roadmap for future investigations into fractional dynamics, topological analysis, and data-driven modeling. Full article

25 pages, 7807 KiB

Open AccessArticle

Effects of Scale Parameters and Counting Origins on Box-Counting Fractal Dimension and Engineering Application in Concrete Beam Crack Analysis

by Junfeng Wang, Gan Yang, Yangguang Yuan, Jianpeng Sun and Guangning Pu

Fractal Fract. 2025, 9(8), 549; https://doi.org/10.3390/fractalfract9080549 - 21 Aug 2025

Abstract

Fractal theory provides a powerful tool for quantifying complex geometric patterns such as concrete cracks. The box-counting method is widely employed for fractal dimension (FD) calculation due to its intuitive principles and compatibility with image data. However, two critical limitations persist [...] Read more.

Fractal theory provides a powerful tool for quantifying complex geometric patterns such as concrete cracks. The box-counting method is widely employed for fractal dimension (FD) calculation due to its intuitive principles and compatibility with image data. However, two critical limitations persist in existing studies: (1) the selection of scale parameters (including minimum measurement scale and cutoff scale) lacks systematization and exhibits significant arbitrariness; (2) insufficient attention to the sensitivity of counting origins compromises the stability and comparability of FDs, severely limiting reliable engineering application. To address these limitations, this study first employs classical fractal images and crack samples to systematically analyze the impact of four minimum measurement scales (2,

\sqrt{2}

, 3,

\sqrt{3}

) and three cutoff scale coefficients (cutoff-to-minimum image side ratios: 1, 1/2, 1/3) on computational accuracy. Subsequently, the farthest point sampling (FPS) method is adopted to select counting origins, comparing two optimization strategies—Count-FD-Mean (mean of fits from multiple origins) and Count-Min-FD (fit using minimal box counts across scales). Finally, the optimized approach is validated through static loading tests on concrete beams. Key findings demonstrate that: the optimal scale combination (minimum scale:

\sqrt{2}

; cutoff coefficient: 1) yields a mere 0.5% average error from theoretical FDs; the Count-Min-FD strategy delivers the highest stability and closest alignment with theoretical values; FDs of beam cracks increase continuously with loading, exhibiting an exponential correlation with midspan deflection that effectively captures crack evolution; uncalibrated scale parameters and counting strategies may induce >40% errors in inferred mechanical parameters; results stabilize with 40–45 counting origins across three tested fractal patterns. This work advances standardization in fractal analysis, enhances reliability in concrete crack assessment, and provides critical support for the practical application of fractal theory in structural health monitoring and damage evaluation. Full article

(This article belongs to the Special Issue Fractal and Fractional in Construction Materials)

15 pages, 3280 KiB

Open AccessArticle

Fractal Scaling of Storage Capacity Fluctuations in Well Logs from Southeastern Mexican Reservoirs

by Sergio Matias-Gutierres, Edgar Israel García-Otamendi, Hugo David Sánchez-Chávez, Leonardo David Cruz-Diosdado and Roberto Cifuentes-Villafuerte

Fractal Fract. 2025, 9(8), 548; https://doi.org/10.3390/fractalfract9080548 - 21 Aug 2025

Abstract

This study focuses on a hydrocarbon reservoir located in southeastern Mexico. The analysis uses well log data derived from petrophysical evaluations of storage capacity. The structural complexity of the reservoir and observed heterogeneity in Cretaceous units motivate a fractal-based characterization of spatial fluctuations. [...] Read more.

This study focuses on a hydrocarbon reservoir located in southeastern Mexico. The analysis uses well log data derived from petrophysical evaluations of storage capacity. The structural complexity of the reservoir and observed heterogeneity in Cretaceous units motivate a fractal-based characterization of spatial fluctuations. The objective is to assess the fractal scaling of storage capacity fluctuations using the dynamic Family–Vicsek framework. Critical exponents

α

(roughness),

β

(growth), and z (dynamic) are obtained through structure function metrics. Data collapse techniques and local Hurst exponent distributions are used to explore long-range memory and spatial heterogeneity across wells. This study aims to classify storage capacity fluctuation records based on Euclidean or fractal geometries. This analysis allows a novel characterization of storage trends in the reservoir. The analysis reveals persistent scaling behavior, indicating long-range correlations in the storage capacity fluctuations. Multiscale patterns and variations in local Hurst exponents highlight the presence of multifractality and regional heterogeneity. Specifically, the spatial distribution of local Hurst exponents obtained in this study enables the inference of statistical properties in synthetic wells, providing key input for the structural and functional characterization of the reservoir’s geological model. This approach aims to identify preferential subsurface flow pathways for hydrocarbons and gas. Full article

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

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19 pages, 295 KiB

Open AccessArticle

On Some Inequalities with Higher Fractional Orders

by Lakhdar Ragoub

Fractal Fract. 2025, 9(8), 547; https://doi.org/10.3390/fractalfract9080547 - 19 Aug 2025

Abstract

The novelty herein pertains to a class of fractional differential equations involving the Hadamard fractional derivative of higher order. Our investigation encompasses the fractional integral operator of a logarithmic function. The mathematical tools utilized in this study are derived from an important function, [...] Read more.

The novelty herein pertains to a class of fractional differential equations involving the Hadamard fractional derivative of higher order. Our investigation encompasses the fractional integral operator of a logarithmic function. The mathematical tools utilized in this study are derived from an important function, wherein its behavior in terms of maximum value facilitates the establishment of bounds necessary for proving the existence of solutions, specifically through Green’s function. Based on this, we endeavor to estimate the bounds of Green’s function as well as analyze its properties within the considered interval. This approach enables us to establish the Hartman–Wintner- and Lyapunov-type inequalities for a class of fractional Hadamard problems. Furthermore, we introduce a novel technique to determine the maximum value of Green’s function. Finally, we illustrate these findings through two applications. Full article

30 pages, 2186 KiB

Open AccessArticle

Dynamic Analysis of a Fractional-Order SINPR Rumor Propagation Model with Emotional Mechanisms

by Yuze Li, Ying Liu and Jianke Zhang

Fractal Fract. 2025, 9(8), 546; https://doi.org/10.3390/fractalfract9080546 - 19 Aug 2025

Abstract

The inherent randomness and concealment of rumors in social networks exacerbate their spread, leading to significant societal instability. To explore the mechanisms of rumor propagation for more effective control and mitigation of harm, we propose a novel fractional-order Susceptible-Infected-Negative-Positive-Removed (SINPR) rumor propagation model, [...] Read more.

The inherent randomness and concealment of rumors in social networks exacerbate their spread, leading to significant societal instability. To explore the mechanisms of rumor propagation for more effective control and mitigation of harm, we propose a novel fractional-order Susceptible-Infected-Negative-Positive-Removed (SINPR) rumor propagation model, which simultaneously incorporates emotional mechanisms by distinguishing between positive and negative emotion spreaders, as well as memory effects through fractional-order derivatives. The proposed model extends traditional frameworks by jointly capturing the bidirectional influence of emotions and the anomalous, history-dependent dynamics often overlooked by integer-order models. First, we calculate the equilibrium points and thresholds of the model, and analyze the stability of the equilibrium, along with the sensitivity and transcritical bifurcation associated with the basic reproduction number. Next, we validate the theoretical results through numerical simulations and analyze the individual effects of fractional-order derivatives and emotional mechanisms. Finally, we predict the rumor propagation process using real datasets. Comparative experiments with other models demonstrate that the fractional-order SINPR model achieves R-squared values of 0.9712 and 0.9801 on two different real datasets, underscoring its effectiveness in predicting trends in rumor propagation. Full article

(This article belongs to the Topic Modeling, Stability, and Control of Dynamic Systems and Their Applications)

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29 pages, 7018 KiB

Open AccessArticle

Real-Time Efficiency Prediction in Nonlinear Fractional-Order Systems via Multimodal Fusion

by Biao Ma and Shimin Dong

Fractal Fract. 2025, 9(8), 545; https://doi.org/10.3390/fractalfract9080545 - 19 Aug 2025

Abstract

Rod pump systems are complex nonlinear processes, and conventional efficiency prediction methods for such systems typically rely on high-order fractional partial differential equations, which severely constrain real-time inference. Motivated by the increasing availability of measured electrical power data, this paper introduces a series [...] Read more.

Rod pump systems are complex nonlinear processes, and conventional efficiency prediction methods for such systems typically rely on high-order fractional partial differential equations, which severely constrain real-time inference. Motivated by the increasing availability of measured electrical power data, this paper introduces a series of prediction models for nonlinear fractional-order PDE systems efficiency based on multimodal feature fusion. First, three single-model predictions—Asymptotic Cross-Fusion, Adaptive-Weight Late-Fusion, and Two-Stage Progressive Feature Fusion—are presented; next, two ensemble approaches—one based on a Parallel-Cascaded Ensemble strategy and the other on Data Envelopment Analysis—are developed; finally, by balancing base-learner diversity with predictive accuracy, a multi-strategy ensemble prediction model is devised for online rod pump system efficiency estimation. Comprehensive experiments and ablation studies on data from 3938 oil wells demonstrate that the proposed methods deliver high predictive accuracy while meeting real-time performance requirements. Full article

(This article belongs to the Special Issue Artificial Intelligence and Fractional Modelling for Energy Systems)

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20 pages, 434 KiB

Open AccessArticle

Large Deviation Principle for Hilfer Fractional Stochastic McKean–Vlasov Differential Equations

by Juan Chen, Haibo Gu, Yutao Yan and Lishan Liu

Fractal Fract. 2025, 9(8), 544; https://doi.org/10.3390/fractalfract9080544 - 19 Aug 2025

Abstract

This paper studies the large deviation principle (LDP) of a class of Hilfer fractional stochastic McKean–Vlasov differential equations with multiplicative noise. Firstly, by making use of the Laplace transform and its inverse transform, the solution of the equation is derived. Secondly, considering the [...] Read more.

This paper studies the large deviation principle (LDP) of a class of Hilfer fractional stochastic McKean–Vlasov differential equations with multiplicative noise. Firstly, by making use of the Laplace transform and its inverse transform, the solution of the equation is derived. Secondly, considering the equivalence between the LDP and the Laplace principle (LP), the weak convergence method is employed to prove that the equation satisfies the LDP. Finally, through specific example, it is elaborated how to utilize the LDP to analyze the behavioral characteristics of the system under small noise perturbation. Full article

(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)

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23 pages, 11219 KiB

Open AccessArticle

Texture Feature Analysis of the Microstructure of Cement-Based Materials During Hydration

by Tinghong Pan, Rongxin Guo, Yong Yan, Chaoshu Fu and Runsheng Lin

Fractal Fract. 2025, 9(8), 543; https://doi.org/10.3390/fractalfract9080543 - 19 Aug 2025

Abstract

This study presents a comprehensive grayscale texture analysis framework for investigating the microstructural evolution of cement-based materials during hydration. High-resolution X-ray computed tomography (X-CT) slice images were analyzed across five hydration ages (12 h, 1 d, 3 d, 7 d, and 31 d) [...] Read more.

This study presents a comprehensive grayscale texture analysis framework for investigating the microstructural evolution of cement-based materials during hydration. High-resolution X-ray computed tomography (X-CT) slice images were analyzed across five hydration ages (12 h, 1 d, 3 d, 7 d, and 31 d) using three complementary methods: grayscale histogram statistics, fractal dimension calculation via differential box-counting, and texture feature extraction based on the gray-level co-occurrence matrix (GLCM). The average value of the mean grayscale value of slice (Mean_{G_AVE}) shows a trend of increasing and then decreasing. Average fractal dimension values (D_{B_AVE}) decreased logarithmically from 2.48 (12 h) to 2.41 (31 d), quantifying progressive microstructural homogenization. The trend reflects pore refinement and gel network consolidation. GLCM texture parameters—including energy, entropy, contrast, and correlation—captured the directional statistical patterns and phase transitions during hydration. Energy increased with hydration time, reflecting greater spatial homogeneity and phase continuity, while entropy and contrast declined, signaling reduced structural complexity and interfacial sharpness. A quantitative evaluation of parameter performance based on intra-sample stability, inter-sample discrimination, and signal-to-noise ratio (SNR) revealed energy, entropy, and contrast as the most effective descriptors for tracking hydration-induced microstructural evolution. This work demonstrates a novel, integrative, and segmentation-free methodology for texture quantification, offering robust insights into the microstructural mechanisms of cement hydration. The findings provide a scalable basis for performance prediction, material optimization, and intelligent cementitious design. Full article

(This article belongs to the Special Issue Fractal Analysis and Its Applications in Materials Science)

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20 pages, 2993 KiB

Open AccessArticle

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 - 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)

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28 pages, 4465 KiB

Open AccessArticle

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 - 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)

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29 pages, 3058 KiB

Open AccessArticle

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 - 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

Open AccessArticle

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 - 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

Open AccessEditorial

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 - 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)

23 pages, 5400 KiB

Open AccessArticle

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 (v2.62), investigated six anthracite samples with varying degrees of metamorphism (R_o = [...] 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 (v2.62), investigated six anthracite samples with varying degrees of metamorphism (R_o = 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 R_o 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 (R_a, R_q) exhibit a negative correlation with R_o. 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 R_o, moisture (M_ad), ash content (A_ad), and volatile matter (V_daf). Among these, the fractal dimension obtained by the triangular prism measurement method has the highest correlation with R_o, A_ad, and V_daf, while the variance partition method shows the highest correlation with M_ad. 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)

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19 pages, 2887 KiB

Open AccessArticle

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

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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Fractal and Fractional

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