Fractal Fract., Volume 10, Issue 8 (August 2026) – 88 articles

Geochemical gold datasets from shear zone-hosted systems frequently contain extreme outliers that distort statistical structure, shift population boundaries, and undermine the reliability of concentration–area (C–A) fractal modeling. Conventional treatments such as discarding anomalous samples or applying fixed Winsorization thresholds often fail to preserve the multivariate relationships that control geochemical dispersion. In this research, an intelligent random forest (RF)-based model was developed to reconstruct an extreme Au outlier in stream sediment samples from the Saqqez shear zone belt by leveraging available multielement geochemical information. This study introduces a hybrid correction framework based on a machine learning algorithm and targeted Winsorization (MLA–TW) that integrates TW with RF regression to reconstruct a realistic and geochemically plausible value for a highly influential Au outlier. Three scenarios were examined: (1) modeling with the original dataset containing a 739 ppb outlier, (2) modeling after removing the outlier, and (3) modeling with a reconstructed value obtained from the MLA–TW approach. The RF model showed reliable predictive capacity (R² = 0.85), and the reconstructed value preserved both geological plausibility and nonlinear multivariate structure. Application of the C–A fractal model demonstrated that the MLA–TW scenario yielded the most stable population breaks, the most robust anomaly thresholds, and the highest spatial fidelity, successfully identifying verified gold prospects and deposits in the region. Overall, the MLA–TW framework stabilizes the C–A model and improves its robustness by reducing the statistical leverage of extreme values while preserving the nonlinear geochemical patterns essential for anomaly detection. The results confirm that this intelligent hybrid approach provides an objective and geologically meaningful methodology for refining Au threshold determination and delineating shear zone-related gold targets with improved accuracy. Full article

Fractures exhibit complex spatial structures and multiscale geometric characteristics, and their accurate characterization is fundamental to reservoir evaluation, fluid migration analysis, and rock mechanics. To address the limitations of single-scale local fractal methods in simultaneously capturing fracture details and global structures, as well as the dependence of supervised learning on labeled data, this study proposes an unsupervised fracture identification method integrating Complexity-Adaptive Box-Height Differential Box-Counting (CABH-DBC) with a self-organizing map (SOM). Local fractal features are extracted using fixed multiscale windows, while the box height along the gray-level dimension is adaptively refined according to the local grayscale standard deviation. The multiscale features are then fed into the SOM for clustering, with grayscale information assisting in fracture-cluster determination. Experiments on borehole image logs from ten depth intervals of the CCSD main borehole yield mean F1 and IoU values of 0.659 and 0.493, respectively. Compared with DBC-Kmeans, the proposed method improves F1 and IoU by 39.0% and 58.0%, respectively; compared with DBC-SOM, the strongest baseline in this study, the improvements are 16.6% and 24.8%. Ablation experiments further demonstrate the complementary contributions of complexity-adaptive box-height refinement, fixed multiscale fractal features, and SOM clustering. Full article

A fractional-order image enhancement and fractal complexity-guided fusion method was developed to improve weak fault representation and introduce complexity-aware priors into dissolved gas analysis (DGA)-based diagnosis of oil-immersed power transformers. Gas-ratio features derived from five characteristic gases were combined into an extended DGA feature sequence and converted into two-dimensional representations using the Markov transition field (MTF), recurrence plot (RP), and Gramian angular field (GAF). A fractional-order difference operator then strengthened texture details and local variations, while fractal complexity features quantified structural irregularities across fault conditions. Based on these features, a fractal complexity-guided multi-image attention fusion module was designed to adaptively integrate the three image representations. An improved RIME optimization algorithm was further employed to jointly optimize the fractional order, imaging parameters, and network hyperparameters. On the public DGA dataset, the proposed model achieved precision, recall, accuracy, and F1-score values of 97.68%, 97.51%, 97.82%, and 97.71%, respectively. External validation on a self-collected DGA dataset further demonstrated its robust cross-condition generalization capability. Full article

Active Infrared Thermography is widely used to characterize laser–tissue interactions, yet quantitative extraction of non-Fourier transport parameters often relies on lock-in demodulation or computationally intensive inverse procedures. Here we propose Modulated-Laser Active Infrared Thermography (ML-AIT), a simplified protocol that exploits the central region of the laser spot, where lateral diffusion is minimized, enabling a one-dimensional (1D) spatial analysis of the temperature profile in ex vivo. porcine adipose tissue under periodic excitation (30–200 Hz). The measured spatial profiles are interpreted with generalized Cattaneo-type bioheat formulations, including fractional-order extensions that account for memory and subdiffusive heat spread in heterogeneous media. By fitting the frequency-dependent stationary spatial distributions, we compare classical Fourier, hyperbolic, and generalized Cattaneo descriptions within a unified analytical framework. The results show that, under the investigated conditions, the measured spatial decay is predominantly governed by optical attenuation, enabling the extraction of an effective optical attenuation coefficient from the thermographic profiles. ML-AIT thus provides a straightforward experimental and analytical approach for studying modulated laser-induced temperature profiles in ex vivoporcine adipose tissue. Full article

Closed-form Caputo gradients in critical-state stress-fractional plasticity have relied on the polynomial Modified Cam-Clay surface. We extend them to the non-polynomial teardrop bounding surface by defining the operator as a Caputo derivative in the logarithm of normalized pressure, orientation-corrected for that decreasing map. On this axis, the surface becomes power–exponential, its critical-state terminal falls at exactly t* = 1/Ψ independently of Ω, and the fractional gradient reduces to incomplete-Beta–Kummer and Humbert-Φ₁ closed forms, verified against singularity-aware quadrature over 1240 cases to relative errors below 10⁻¹¹. The flow rule recovers associated flow as α → 1 and is exactly associated at the critical state. This is a computational study of operator semantics on inherited calibrations, not an experimental validation. Substituting it for the fixed-window Grünwald–Letnikov flow of a companion factorial collapses the dominant flow main effect from −60% to below 0.2% for both clays at every overconsolidation ratio tested, across drained and approximately undrained paths. The collapse is conditional: it is a property of the critical-state dwell, and at a laboratory-scale budget the operators still differ by about 15% of baseline. Window semantics, not fractionality alone, decide where flow influence resides in a factorial design. Full article

Underground coal mining can induce deformation and failure of overlying strata and ground surface, which seriously endangers the safety of human life and property. During mining, the immediate roof rock successively experiences initial caving (fixed support on four sides) and periodic caving (fixed support on three sides and free on one side). Different boundary conditions alter the unloading and deformation processes such as cracking and fracturing of immediate roof rock, thereby affecting its subsequent mechanical behavior of compaction and deformation, and resulting in differences in the movement law of overlying strata. In this paper, the numerical simulation method is adopted to investigate the variation laws of unloading and fracturing characteristics of immediate roof rock under initial caving and periodic caving with thickness-width ratio (t/w), length-width ratio (l/w), unloading stress (σ_u) and specimen strength (σ_c), and the corresponding action mechanism is revealed. The fractal evolution law of fractured immediate roof rock obtained from this study can quantitatively evaluate the compaction characteristics of caved rock, provide refined parameter support for surface subsidence prediction and possess guiding significance for stope surrounding rock control engineering. The results show that the fragments formed after the failure of immediate roof rock are mainly block-strip shaped under both first caving and periodic caving conditions. With the increase in the thickness-width ratio, the flexural rigidity of immediate roof rock increases and crack propagation is restrained, so that the particle-size–mass fractal dimension of fragments increases first and then decreases for the two caving modes. The increase in length-width ratio weakens the propagation of secondary fractures and raises the particle size of fragments, while the overall variation in particle-size–mass fractal dimension is small under the two working conditions. As the unloading stress continuously rises, the coupled tension-shear effect inside the rock gradually intensifies, and the failure mode changes from tension-shear failure to global shear failure. Accordingly, both the particle-size–mass fractal dimension and fractal dimension of crack distribution increase first and then decrease under first caving and periodic caving conditions. The increase in the strength of immediate roof rock raises the energy consumption during rock failure, and large-size fragments are more likely to be generated, which reduces the particle-size–mass fractal dimension and increases the particle size of fragments under both caving modes. Meanwhile, internal micro-fractures continuously initiate and propagate with the growth of rock strength. For specimens with relatively high strength, crack propagation is inhibited and the development of secondary fractures is weakened, leading to an evolution trend that the fractal dimension of crack distribution increases first and then decreases. Under identical parameter conditions, the particle-size distribution and crack complexity for first caving are mainly affected by geometric parameters; the particle size of fragments is primarily controlled by specimen strength; and the unloading stress threshold governs the transition of failure mode. For periodic caving, the crack-initiation location is first determined by asymmetric boundary constraints. The thickness-width ratio dominates the particle-size distribution of fragments, and unloading stress as well as specimen strength further regulate the complexity of cracks. Full article

Ultrasonic attenuation in particle–cementitious heterogeneous media results from the coupled effects of matrix memory dissipation and particle-induced random heterogeneity, which cannot be readily distinguished using conventional homogeneous-medium models. This study develops a unified stochastic fractional wave framework that couples Caputo fractional dissipation with a random-potential representation of spatial heterogeneity. The main contribution is an analytically tractable amplitude–phase formulation that separates the leading-order roles of the two mechanisms: fractional dissipation primarily governs exponential amplitude attenuation, with $\kappa (\omega )\propto {\omega }^{\alpha -1}$, whereas the random potential mainly modulates local phase propagation and introduces finite scattering-type amplitude corrections. By transforming the governing equation into a frequency-domain Helmholtz form and applying Wentzel–Kramers–Brillouin (WKB) asymptotic analysis, explicit scaling relations are obtained for both attenuation and phase fluctuations. Two-dimensional Helmholtz simulations support the predicted attenuation law and show that the relative $L_2$ error of the WKB phase prediction decreases from 25.23% to 5.36%, while the covariance-based fixed-receiver ensemble phase-variance prediction lies within the 95% confidence interval of 30 independent realizations. Single-frequency ultrasonic transmission experiments provide complementary trend-level evidence, showing reduced tail retention and increased descriptive tail attenuation with increasing particle volume fraction. The proposed framework provides a mechanistically interpretable basis for distinguishing dissipation-dominated and heterogeneity-induced ultrasonic responses. Full article

In this paper, motivated by the modified Tremblay fractional differential operator and the normalized generalized Bessel–Maitland function, we introduce the generalized Bessel–Maitland–Tremblay function via the Hadamard product transformation. This construction established an integrated framework that connects fractional calculus with geometric function theory. By utilizing estimates for the gamma and digamma functions together with monotonicity properties of the associated coefficient sequences, we establish sufficient conditions under which the proposed function belongs to various important subclasses of normalized analytic functions. In particular, criteria are obtained for uniform convexity, starlikeness and convexity of order $\delta$, $\nu$-uniform starlikeness, and $\nu$-uniform convexity, as well as exponential starlikeness, exponential convexity, and lemniscate-type conditions. Several special cases are shown to recover previously known results for the normalized generalized Bessel–Maitland function and the identity mapping. Graphical analysis and numerical examples are presented to show the geometric behavior of the proposed function and to verify the theoretical findings. Full article

Permeability models are essential for quantitatively describing coal permeability evolution and predicting gas migration during deep mining. Deep coal subjected to mining disturbance commonly exhibits pronounced nonlinear changes in permeability, limiting the applicability of conventional models. In this study, coal is idealized as a dual-component medium comprising the matrix and fractures, and deformation of both components induced by mining-related stress changes and gas adsorption is incorporated into the model. The conventional Weibull statistical damage variable is generalized to a fractional-order form using the Caputo derivative, yielding a Mittag–Leffler-type damage evolution law. By coupling this formulation with matrix–fracture deformation and an exponential damage–permeability term, a fractional-order damage-based permeability model is established to describe the complete evolution from elastic deformation through pre-peak damage to post-peak failure. The model parameters are calibrated separately using published datasets for protective-seam mining, top-coal caving, no-pillar mining, and a full-process loading case. The calibrated model yields coefficient of determination (R²) values of 0.9374, 0.9625, 0.9875, and 0.9980, respectively. The identified fractional order is λ = 1 for the three mining-disturbance datasets, whereas the full-process dataset yields λ = 0.7734. For the full-process dataset, the fractional-order model reduces root mean square error (RMSE) and mean absolute error (MAE) by approximately 31.4% and 34.7%, respectively, compared with its integer-order counterpart. Sensitivity analysis shows that λ, p, ε_d, and γ play distinct roles in permeability evolution. At an axial strain of 0.8%, increasing ε_d from 0.721% to 1.121% decreases k/k₀ from 2.6919 to 1.3433, whereas increasing γ from 0 to 2.543 increases k/k₀ from 1.0003 to 3.0334, indicating that ε_d and γ strongly affect the strain level and magnitude of permeability enhancement, respectively. The proposed model provides an effective tool for characterizing the nonlinear permeability evolution of deep coal under mining disturbance. Full article

This article introduces the concept of proximal enriched contractions for nonself mappings by integrating the enrichment technique of Berinde and Păcurar with the relation-theoretic approach of Alam and Imdad. The existence of best proximity points for this notion is ensured by two distinct results. The proofs are aligned with the proximal enriched contraction condition and are based on a constructive iteration scheme that generates a sequence converging to the best proximity point of T. Furthermore, the sequence generated by the constructive iteration satisfies the Cauchy property via a nonstandard inequality between consecutive elements. These theoretical results are utilized to derive an algorithm for computing an approximate best proximity point of T. Numerical examples in ${\mathbb{R}}^3$ are presented to demonstrate the effectiveness of both the main results and the proposed algorithm. Additionally, an application section is included wherein the fixed point result derived from our best proximity point theorems is employed to establish the existence of solutions for a nonlinear fractional integral equation. Full article

The paper investigates the asymptotic stability for the control of systems described by multi-order fractional differential equations. By utilizing generalized Lyapunov functions and the Kalman–Yakubovich–Popov lemma, frequency-domain criteria are derived to evaluate asymptotic stability. The formulated criteria are similar to the ‘Popov Circle Criterion,’ but the circle parameters are determined by the system’s fractional order and the control parameters. Additionally, the asymptotic stability condition requires that all polar plots associated with the multi-fractional-order system lie inside the circle defining the criterion. Human–Robot System applications highlight the investigation techniques and the particularities of the presented criteria. Full article

Nonlinear electromagnetic wave propagation in complex plasma environments has attracted considerable attention due to its important applications in nonlinear optics, plasma physics, space science, and communication technologies. In the present study, a time-fractional Drinfel’d–Sokolov–Wilson equation (DSWE) is investigated under the influence of electromagnetic wave perturbations. The fractional-order formulation incorporates memory and hereditary effects, providing a more realistic description of wave propagation in nonlinear dispersive media. By employing an appropriate fractional traveling-wave transformation, the governing nonlinear fractional partial differential equation is reduced to a nonlinear ordinary differential equation. Exact solitary wave solutions are subsequently constructed using the ${\displaystyle \frac{G^{\prime }}{G^2}}$-expansion technique. Furthermore, the nonlinear dynamical behavior of the reduced system is examined through phase portraits, bifurcation diagrams, Lyapunov exponents, sensitivity analysis, and multistability investigations. Particular attention is devoted to understanding the emergence of chaotic dynamics induced by electromagnetic wave effects and fractional-order interactions. The obtained results reveal that the fractional-order parameter significantly influences the stability, propagation characteristics, and dynamical evolution of nonlinear wave structures. The coexistence of multiple attractors, transitions between stable states, and chaotic regimes is identified for various parameter configurations. These findings provide deeper insight into the complex dynamics governed by the time-fractional DSWE and contribute to the understanding of nonlinear electromagnetic wave propagation in plasma and other nonlinear dispersive media. Full article

Conventional integer-order parameter designs of grid-forming inverters provide limited degrees of freedom for impedance adjustment, motivating the exploration of additional approaches for flexible impedance reshaping across different frequency ranges. This paper establishes a full fractional-order grid-forming inverter (FFO-GFMI) by incorporating fractional-order inductor-capacitor (LC) filters, corresponding decoupling control, and fractional-order multi-loop controllers into a conventional grid-forming inverter. Based on the harmonic linearization method, positive- and negative-sequence impedance models of the FFO-GFMI are developed to characterize its broadband impedance characteristics. The developed models are validated through impedance scanning, and the effects of fractional-order parameters on broadband impedance characteristics are systematically investigated. The results reveal that fractional-order LC filters mainly regulate medium- and high-frequency resonance characteristics, while fractional-order control loops provide effective low- and medium-frequency impedance reshaping. Furthermore, load-step simulations demonstrate that the selected fractional-order configuration improves dynamic performance, reducing the active power settling time from 0.3121 s to 0.1974 s and the active power overshoot from 19.75% to 4.51%. Full article

This paper presents a novel fractional-order hybrid observer framework for robust sensorless control of brushless DC (BLDC) motor drives in unmanned aerial vehicle (UAV) propulsion systems, addressing the fundamental limitations of conventional integer-order observers through the lens of fractional calculus. The proposed architecture synergistically integrates high-frequency square-wave signal injection for zero/low-speed operation with an adaptive fractional-order extended Kalman filter (AFEKF) augmented by online stator resistance and flux linkage estimation, capitalizing on the memory and hereditary properties inherent to fractional-order systems. A minimum-order current observer enables accurate three-phase current reconstruction using a single DC-link sensor, substantially reducing hardware complexity and cost. The complete algorithm is implemented on an STM32H7 microcontroller and experimentally validated on a 1.5 kW drone propulsion testbench and in-flight platform. Results demonstrate reliable startup under 50% rated load, stable operation from standstill to 5000 RPM on the UAV motor (and validated up to 22,000 RPM on a high-speed test motor, <4° electrical position error at 5 kRPM, and strong robustness against 35% stator resistance variation. In-flight tests confirm improved thrust smoothness and hover stability compared to conventional sensorless strategies. The proposed fractional-order architecture offers a practical, resilient, and computationally feasible solution for next-generation autonomous aerial systems, establishing a new paradigm for observer design in electric propulsion. Full article

Let $\mathcal{G}$ be a homogeneous Lie group of dimension $\varkappa$. In this article, we investigate the mapping properties of the Wolff-type potential ${\mathcal{W}}_{\vartheta ,2}$ on $\mathcal{G}$, associated with a homogeneous quasi-norm, and show that it coincides, up to an explicit dimensional constant, with the Riesz potential of doubled order $2\vartheta$ on $\mathcal{G}$. Using this identity together with Hedberg’s trick and a dyadic decomposition of the convolution kernel, we establish single-weight and two-weight boundedness inequalities for ${\mathcal{W}}_{\vartheta ,2}$ in the global Morrey spaces $M_p^{\mu }\left(\mathcal{G}\right)$. In contrast to the linear Riesz and Bessel–Riesz potentials, the Wolff potential belongs to a broader family of operators associated with quasilinear elliptic equations of p-Laplace type. We believe that these boundedness properties and the associated inequalities play an important role in the further advancement of nonlinear potential theory. Full article

In this paper, the new fractal modified Zakharov–Kuznetsov equation (fmZKe) defined on Cantor sets is investigated. The fmZKe is a non-differentiable model that arises naturally in mathematical physics, nonlinear wave theory, and plasma physics. The extended rational sine–cosine method is utilized to construct new optical soliton solutions of the model. The fmZKe is reduced to a non-differentiable ordinary differential equation by applying a non-differentiable wave transformation defined on Cantor sets. This reduction leads to a system of linear algebraic equations, which upon solving yields several exact solutions of the model. Furthermore, to establish a clear understanding of the model’s behavior, non-smooth graphical representations of the solutions are presented for various parameter values. The stability analysis of the newly obtained solutions in a classical sense is examined using stability theory, and the real-life applications of the results are highlighted. Full article

Time-fractional diffusion equations describe subdiffusive transport in heterogeneous media, but their numerical treatment is complicated by the nonlocal Caputo derivative and by the weak singularity that the solution develops at the initial time. We study a global-in-time discretization that combines spectral collocation in time—on the fractional power basis ${\left\{t^{\mathsf{\ell }\alpha }\right\}}_{\mathsf{\ell }=0}^N$, evaluated at Chebyshev–Gauss–Lobatto nodes, which reproduces the leading terms of the singular expansion of the solution—with a second-order conservative finite-difference stencil in space that uses harmonic averaging of the diffusivity at the cell faces and therefore remains accurate across discontinuous media. The resulting fully discrete problem is a large, nonsymmetric, dense-in-time linear system whose two-norm condition number grows like the inverse square of the spatial mesh size, so that Krylov subspace iteration without preconditioning stalls under refinement. Exploiting the Kronecker sum structure of the discrete operator, we build a preconditioner by fast diagonalization of the spatial factor through the discrete sine transform. For constant diffusivity the preconditioner reproduces the operator exactly and yields a direct solver; for variable diffusivity it is spectrally equivalent to the operator, and we prove that the eigenvalues of the preconditioned system cluster in a disk centered at one whose radius depends only on the coefficient contrast, and not on the mesh, the number of temporal degrees of freedom, or the fractional order. Numerical experiments in one and two space dimensions confirm second-order spatial accuracy and a preconditioned iteration count that stays flat—twelve iterations from $M=32$ up to $M=1024$ in one dimension and eleven up to $M=256$ per direction in two—while the unpreconditioned count grows by more than two orders of magnitude. In time, the accuracy is spectral until round-off in the ill-conditioned Vandermonde matrix of the power basis takes over: the barrier is reached at $N=9,10,13$ for $\alpha =0.3,0.5,0.7$, where the attainable error is about ${10}^{-6}$. A benchmark against the L1 scheme on uniform and graded meshes, the Alikhanov L2-$1_{\sigma }$ scheme and Grünwald–Letnikov convolution quadrature quantifies when the global approach pays: on forced problems and on modes with $\kappa \lambda T^{\alpha }\lesssim 2$ it reaches a prescribed accuracy one to two orders of magnitude faster and with several times less memory, while for strongly damped modes the fractional power basis converges only algebraically and graded time marching is preferable below a relative error of ${10}^{-2}$. Full article

This work addresses the Ulam–Hyers stability of mild solutions of fractional noninstantaneous evolution nonlinear equations subject to noninstantaneous impulses in the Banach space through an approach based on the measure of noncompactness. Mild solutions are formulated via operators generated by a closed linear operator and a probability density function. Existence results are derived by applying the fixed point theorem for k-set contractive operators. Furthermore, Ulam–Hyers stability is established under certain hypotheses, and an illustrative example is provided to validate the derived results. Full article

Time delays are inherently present in active control systems as a consequence of sensor acquisition, communication lags, and actuator dynamics, and their impact on system behavior cannot be overlooked. This paper examines the effect of delayed displacement and speed feedback gains on the nonlinear lateral and vertical vibrational behavior of a MAGLEV vehicle subjected to aerodynamic and centrifugal forces. A delayed nonlinear dynamic model incorporating a fractional-order PD controller under aerodynamic excitation is first established for the MAGLEV system. Subsequently, the method of multiple scales is employed to derive the frequency response relationships, while the corresponding steady-state solutions are analyzed to determine system stability. The investigation further explores how the delays alter the nonlinear dynamic response. It also considers the impact of changing the value of the fractional-order parameter $\alpha$ on the vehicle’s dynamics. The results showed that increasing controller delays reduces the stability region, with displacement-feedback delays having a more pronounced effect than speed-feedback delays, while fractional-order derivatives ($0<\alpha <1$) further degrade stability; consequently, the integer-order case ($\alpha =1$) is recommended to achieve lower vibration levels and improved dynamic stability. The outcomes of this work provide valuable understanding of vibration phenomena encountered in MAGLEV systems and contribute to the development of improved control and optimization strategies for safer, smoother, and more reliable vehicle performance. Full article

Three-dimensional Riemann–Liouville time-fractional integro-differential equations provide useful prototype models for diffusion and transport processes with temporal memory and weakly singular hereditary effects. Their numerical solution is challenging because of the nonlocal fractional derivative, history-dependent fractional integral term, and large-scale discrete systems arising from three-dimensional spatial discretization. In this work, an efficient high-order compact finite difference scheme is developed for solving such problems. The Riemann–Liouville fractional derivative is approximated by the weighted and shifted Grünwald difference formula, the fractional integral term is discretized by the product trapezoidal formula, and the Laplace operator is approximated by compact difference operators. The proposed scheme achieves second-order accuracy in time and fourth-order accuracy in space. Moreover, the solvability, stability, and convergence of the fully discrete three-dimensional scheme are analyzed under suitable regularity assumptions. Numerical experiments, including examples with smooth and non-smooth solutions, verify the theoretical convergence orders and demonstrate the effectiveness of the proposed method for different fractional parameters. Full article

Complex surfaces exhibit hierarchical morphological organizations that cannot be fully described by conventional roughness parameters alone. In this study, a fractal–statistical framework is proposed to elucidate the substrate-controlled morphological evolution of 2,6-diphenyl anthracene (DPA) thin films deposited on chemically modified dielectric substrates, including hexamethyldisilazane (HMDS), octyltrimethoxysilane (OTMS), octadecyltrichlorosilane (OTS), and bare silicon dioxide (SiO₂). A multidimensional morphological descriptor vector (M_DPA) is introduced by integrating ISO 25178 areal surface parameters (H_ISO), fractal dimension (D_f), texture direction parameters (T_d), power spectral density (PSD), and scale-sensitive fractal analysis (SSFA) descriptors to quantify amplitude-based, spatial-frequency, and scale-dependent morphological information. Atomic force microscopy (AFM) topographies of 5 nm and 50 nm thick films were analyzed using complementary approaches, including ISO 25178 areal surface parameters, texture direction analysis, peak statistics, morphological envelope fractal analysis, two-dimensional Fourier analysis, power spectral density (PSD), and scale-sensitive fractal analysis (SSFA). The results demonstrate that substrate chemistry governs not only the amplitude of surface roughness but also the lateral organization, spatial frequency distribution, and scale-dependent fractal complexity of DPA morphologies. The fractal dimension analysis revealed substrate-dependent variations in surface complexity, with values ranging from 2.11 to 2.45 for 5 nm films and from 2.19 to 2.52 for 50 nm films. PSD analysis identified distinct substrate-induced modifications in spectral organization, while SSFA revealed significant changes in smooth–rough crossover scales, maximum complexity scales, and fractal surface complexity during film growth. In particular, OTMS promoted the strongest hierarchical organization for thicker films, exhibiting the highest scale-sensitive fractal complexity, whereas OTS generated highly developed but less hierarchically correlated rough structures. The integrated fractal–spectral methodology establishes quantitative relationships between substrate functionalization and multiscale surface evolution, demonstrating that morphological complexity cannot be described solely by conventional height parameters. This framework provides a robust approach for characterizing hierarchical thin-film architectures and can be extended to other organic semiconductor systems where substrate-driven morphological control is critical. Full article

Tight sandstone reservoir effectiveness is governed not by pore volume alone, but by the storage and flow contributions of different pore-throat scales, their structural complexity, and their three-dimensional connectivity. This study investigates tight sandstones of the Benxi Formation deposited in a marine–continental transitional mixed siliciclastic–carbonate setting in the Gaoqiao area, southern Ordos Basin. Petrophysical measurements, red-epoxy-impregnated thin-section petrography, mercury intrusion capillary pressure (MICP), segment-specific fractal analysis of functionally defined pore-throat regimes, X-ray micro-computed tomography (micro-CT), and pore-network modeling (PNM) were integrated. The MICP responses define three pore-throat structure types and two data-derived functional boundaries at 0.708 and 0.141 μm, which separate large-pore-throat-dominated, transitional pore-throat, and fine-throat-limited intervals. Using these nominal boundaries, Type I is strongly dominated by the large-pore-throat interval, which accounts for 88.7% of total mercury intrusion, whereas Type II exhibits a mixed large-to-transitional response, and Type III is characterized by negligible large-pore-throat intrusion and pronounced fine-throat restriction. Perturbing both functional boundaries by ±5% and ±10% does not alter these principal functional distinctions, although samples close to the second boundary exhibit the expected local transitional sensitivity. Among the three segment-specific fractal parameters, the fine-throat fractal dimension, D_B, shows the strongest association with median capillary pressure (r = 0.834, p < 0.001) and remains significantly related to displacement pressure, median pore-throat radius, and permeability, whereas D_T shows no significant linear correlation with the tested petrophysical and MICP parameters. The fractions of the largest connected pore cluster in representative Type I–III samples are 90.26%, 72.56%, and 64.65%, while their PNM permeabilities decrease successively from 64.32 mD to 0.850 and 0.121 mD. Together, these results indicate that, for the investigated samples, reservoir effectiveness reflects the combined influence of pore-throat size configuration, segment-specific structural complexity, and three-dimensional network connectivity. Full article

This paper investigates how opinion dynamics with memory effects and random communication uncertainties achieve consensus in an inhomogeneous Hegselmann–Krause (H-K) model. The Caputo fractional derivative is introduced to describe the influence of historical opinions, while multiplicative noise captures random disturbances in communication. We establish sufficient conditions under which all followers almost surely converge to the leader’s opinion. Moreover, we characterize the algebraic (power-law) convergence behavior determined by the fractional order. The analysis is based on fractional Lyapunov techniques and fractional Grönwall estimates, which enable the treatment of the nonlinear stochastic system with leadership. The results reveal how memory effects, leadership strength and stochastic perturbations jointly influence consensus formation. Numerical simulations are provided to verify the theoretical results and illustrate the effects of the model parameters on the convergence dynamics. Full article

Let $\mathcal{A}$ denote the class of functions $g\left(z\right)=z+{\sum }_{k=2}^{\infty }{a}_k{z}^k$ analytic and normalized in $\mathbb{U}=\{z:|z|<1\}$, and let $D=\{g\in \mathcal{A}:|g^{″}\left(z\right)|\le 1,z\in \mathbb{U}\}$. We give a short, self-contained proof that D consists entirely of univalent functions by a direct line-integral estimate combined with the classical Noshiro–Warschawski theorem. We then solve, in closed form and with explicit extremal functions, two classical extremal coefficient problems for D: the Fekete–Szegő problem and the second Hankel determinant, using the exact Schur parametrization of bounded analytic functions applied to $g^{″}$. We also establish a sharp bound on a single coefficient $|a_n|$ for every $n\ge 2$, obtained by an elementary Parseval argument that does not require the full joint Schur parametrization needed for the Fekete–Szegő and Hankel problems. We then compose D with the Owa–Srivastava fractional derivative operator ${\mathsf{\Omega }}_z^{\delta }$ ($\delta \in [0,1)$), used to build fractional bi-univalent subclasses, to define the one-parameter family $D\left(\delta \right)=\{g\in \mathcal{A}:|{\left({\mathsf{\Omega }}_z^{\delta }g\right)}^{″}\left(z\right)|\le 1\}$, and extend all three results to $D\left(\delta \right)$, again with explicit extremal functions. Because the underlying weights grow at different rates in the fractional parameter $\delta$, the Fekete–Szegő problem exhibits a genuine, monotone bifurcation as $\delta$ ranges over $[0,1)$, while the Hankel and general-coefficient bounds decrease monotonically—effects that are structural consequences of the fractional operator rather than a simple rescaling of the classical case. All results are verified in two independent numerical ways: the Schur parametrization lemma itself is checked against an explicit, independently constructed rational Schur function whose Taylor coefficients are extracted numerically via the Cauchy integral formula, and the sharp bounds are checked by Monte Carlo maximization over the admissible Schur parameters. Both checks are explained in detail and agree with the closed forms to at least four decimal places. Unlike the Carathéodory class of functions with positive real part, whose coefficient body is fixed at the origin (value 1) and is classically parametrized without an extra recursive step, the Schur class $\mathcal{B}$ used here does not pin down $\phi \left(0\right)$, so its exact coefficient body genuinely requires the two-step recursive Schur parametrization applied below to $g^{″}$; identifying and exploiting this distinction is part of the technical contribution of the paper. Full article

Integrated Modular Avionics (IMA) integrates safety-critical functions on shared computing and network resources, creating coupling channels through which a local fault may escalate into a catastrophic system-level scenario. This study develops a graph-based fractional-order model for cascade escalation in IMA architectures with communication delays and reconfiguration failures. The architecture is represented as a weighted directed graph of core processing modules, network switches, and remote data concentrators, where each node carries functional degradation and queue-backlog states. The proposed delayed Caputo fractional-order dynamics incorporate degradation propagation, backlog spillover, mixed-criticality priority conflict, and a state-dependent reconfiguration-failure mechanism. We establish well-posedness and positive invariance of the feasible state domain, derive a sufficient cascade threshold that separates a delay-independent, globally Mittag–Leffler stable nominal regime from a supercritical regime in which bistability and catastrophic attractors may occur, and characterize delay-induced oscillatory instability together with a memory-stabilization effect. Numerical experiments on a synthetic 22-node IMA configuration show fault absorption below the threshold, reconfiguration-contained cascades under sufficient supervisory capacity, and global escalation when reconfiguration collapses under load. The results indicate that backlog growth is an early warning signal and that maintaining the cascade threshold below unity while provisioning reconfiguration capacity above the tipping point can support safer reconfiguration-policy design in certifiable avionics. Full article

This paper investigates the finite-time stabilization of fractional-order impulsive switched neural networks subject to actuator saturation. Such systems combine memory-dependent fractional dynamics, mode switching, impulsive updates, and bounded control inputs—features that frequently coexist in practical networks but have not previously been treated together. To address this, an impulsive control framework is developed in which the actuator saturation is handled through a convex-hull representation. Using Lyapunov stability theory, the average dwell time approach, and the Gronwall–Bellman inequality, sufficient conditions are derived to guarantee finite-time stability of the closed-loop system, and the controller gains are obtained by solving a set of linear matrix inequalities via the MATLAB (2019a) LMI toolbox. The effectiveness of the proposed strategy is validated on two-dimensional and three-dimensional numerical examples: in both cases, the synchronization error is driven below the prescribed finite-time bound ($c_2=2.5$ and $c_2=3.5$, respectively) within the finite-time horizon, while the state remains inside the admissible region despite the saturation constraint. A sensitivity analysis across fractional orders $\alpha \in \{0.85,0.90,0.95,0.99\}$ further confirms that finite-time stability is preserved throughout, with faster convergence at smaller $\alpha$. Full article

We investigate the asymptotic behavior of a Timoshenko beam system in which the free end carries a tip mass subject to a restoring spring force and a tempered Caputo fractional damping force with parameter $\eta >0$ (the case $\eta =0$ is left as an open problem). Using a diffusive state space reformulation of the fractional term, the original problem is embedded into an augmented first-order evolution system on a carefully constructed Hilbert space. Well-posedness is established via the Lumer–Phillips theorem. A spectral analysis of the governing operator, combined with the Arendt–Batty–Lyubich–Vũ theorem, shows that the associated $C_0$-semigroup is strongly asymptotically stable even when the classical equal-wave-speeds condition for the Timoshenko system is violated, provided $\eta >0$. Moreover, resorting to the Borichev–Tomilov resolvent method, we reduce the polynomial energy decay to a single resolvent exponent $\mathcal{\ell }>0$, so that the energy of every solution issued from a datum in the domain of the generator decays at least as fast as $t^{-1/\mathcal{\ell }}$ as $t\to +\infty$. We establish the estimates that control ℓ; we identify the mechanism that governs it—the inertia of the tip mass, which screens the damper at high frequency—and we measure ℓ numerically. In particular, the exponent is not dictated by the second-order character of the Timoshenko operator, contrary to what a comparison with the fourth–order beam might suggest. Full article

This paper investigates the qualitative and topological behavior of random solutions for a class of partial fractional random differential equations governed by the Darboux problem. Unlike classical configurations that rely on bounded or finite delays, our theoretical framework explicitly addresses systems involving unbounded infinite delay. The dynamics of the state transitions are formulated using left-sided mixed Riemann–Liouville fractional integrals and joint Caputo fractional derivatives of order $\epsilon =({\epsilon }_1,{\epsilon }_2)\in (0,1]\times (0,1]$. Because of the infinite historical horizon, the underlying model is constructed and analyzed within abstract, semi-normed axiomatic phase spaces defined over topological Fréchet spaces. By avoiding restrictive compactness assumptions on the nonlinear operational bounds, we establish novel random mild existence theorems. The structural proofs are achieved through a combination of a regular, sublinear family of axiomatic measures of noncompactness and an advanced generalization of the classical Darbo fixed-point theorem tailored for Fréchet domains. Finally, a concrete mathematical example is systematically analyzed to confirm the validity, consistency, and practical applicability of the established theoretical bounds. Full article

In this study, random-order FDEs are studied with a focus on the variability of their solutions and random orders when the order of differentiation is governed by a probability distribution. Fractional differential equations (FDEs) offer a generalized modeling approach that enables the analysis of nonlocality and memory effects. However, the deterministic framework for studying FDEs neglects the random nature of real-life events. In this regard, four continuous probability distributions with bounded support (uniform, Beta, triangle and Bates) are used to analyze the variability of the solutions depending on the random order, which is defined to vary between [0.65, 0.95] according to these probability distributions with identical expected values. Monte-Carlo simulations with $N=2^{15}$ repetitions are performed to investigate the random-order FDEs using a predictor-corrector approach. Results show that the decrease in the deviation from the uniform distribution to the Bates distribution is reflected in the random characteristics of the solutions and the order of differentiation to almost the same extent. The findings indicate that the average solutions obtained with random orders from each distribution show almost identical results, whereas the variability changes significantly based on the distribution of the order of differentiation. This information provides useful guidance in working with probabilistic models instead of deterministic systems for uncertainty quantification or sensitivity analysis. Full article

Deep shale reservoirs in the tectonically complex margin of the southern Sichuan Basin have experienced multistage deformation, resulting in strong spatial heterogeneity of the present-day geostress field. However, the influence of stress heterogeneity on multiscale pore structure evolution and reservoir quality remains poorly constrained. Here, we integrate in-situ stress measurements, overburden porosity and permeability experiments, CO₂/N₂ adsorption, high-pressure mercury intrusion, SEM-MAPS (Scanning Electron Microscopy-MAPS) pore imaging, stress well profile interpretation, and multifractal analysis to quantify the controls of present-day geostress heterogeneity on pore structure evolution in deep Longmaxi Formation shale. The results show that the present-day stress regime is characterized by a strike-slip pattern (σ_H > σ_v > σ_h), with significant variations among different structural deformation zones. Increasing structural deformation results in enhanced differential stress, increasing by 30–80% from gentle structures to tight folds and fault-affected zones, accompanied by a 60–70° rotation of the maximum principal stress orientation. Differential stress, effective stress, differential stress coefficient, and stress structure index exhibit strong negative correlations with porosity, whereas permeability decreases nonlinearly with increasing stress, indicating progressive pore-throat compression and connectivity degradation under heterogeneous stress conditions. Multifractal analysis reveals that pore-size domains exhibit different sensitivities to stress heterogeneity. The macropore fractal dimension (DN3) shows the strongest response, followed by mesopores (DN2), whereas micropores (DN1) exhibit relatively limited variations. Fault-affected zones and strongly deformed regions display higher DN3 values (>2.8), reflecting enhanced complexity of macropore and fracture networks. In contrast, gentle structural zones characterized by curvature values <0.10 km⁻¹ and distances >500 m from faults exhibit relatively low and stable fractal dimensions (<2.73), indicating more homogeneous pore structures. Increasing stress heterogeneity induces the transformation of organic matter pores from regular subcircular shapes to flattened and slit-like morphologies, accompanied by pore-size migration toward smaller scales (<15 nm) and enhanced pore heterogeneity (Df > 1.35). These findings reveal that present-day geostress heterogeneity governs shale pore fractal evolution and promotes the transition from micropore-dominated to heterogeneous macropore–fracture systems. This study provides quantitative insights into stress-controlled pore evolution and reservoir quality evaluation in deep shale reservoirs under complex tectonic settings. Full article