Fractal and Fractional

The fractal characteristics of coal pore–fracture networks and their evolution under compression are essential for predicting rock mass failure and fluid transport. This study combines micro-CT scanning with fractal theory and seepage mechanics to investigate the structural evolution of coal under uniaxial compression and its impact on fluid transport. CT scans were performed at four characteristic stages (initial, elastic, plastic, and failure) to reconstruct three-dimensional fracture networks. Quantitative analysis reveals that fracture porosity increases sequentially from 0.44% to 5.01%, with the failure stage reaching 11.4 times the initial value. Fracture length and aperture distributions follow power-law scaling, and their fractal dimensions exhibit distinct evolution patterns: length dimension increases from 2.43 to a peak of 2.56 in the plastic stage and then drops to 2.47 at failure, while aperture dimension decreases from 2.29 to a trough of 2.12 before rebounding to 2.26. These patterns reflect a dynamic adjustment of network complexity, transitioning from primary fractures to micro-fracture dominance and finally to main fracture coalescence. Based on the Knudsen number, three diffusion regimes of Fick, transition and Knudsen are identified. A fractal permeability model is developed by idealizing the pore space as tortuous capillaries, showing that permeability scales with the fourth power of the maximum pore diameter and is positively influenced by the fractal dimension and the number of large pores. Furthermore, a coupled seepage–stress model is derived, incorporating pressure transmission, shear transmission, and crack opening coefficients. The damage variable is expressed as a function of stress level and fractal dimension. These findings provide theoretical support for predicting gas transport and failure behavior in coal under coupled hydro-mechanical conditions. Full article

Traditional human motion prediction methods attempt to discover the relationship between observed and future motion sequences. However, due to the dynamic complexity of human motion, existing methods cannot fully capture the interrelationships among motion sequences, and their performance remains unsatisfactory. In this work, we propose a novel Two-stage Refinement (TSR) framework for human motion prediction. It consists of two branches: (i) a traditional motion prediction branch for preliminary prediction, and (ii) an auxiliary refinement branch designed to estimate and compensate for the preliminary prediction errors. In this way, we can obtain better prediction performance than with traditional one-stage methods. To further bridge the gap between predicted results and groundtruth, we introduce a novel fractional-order differential loss function in this work. Existing methods use only integer-order differences to capture instantaneous state changes, often failing to account for the long-range temporal dependencies in human motion. By contrast, the inherent memory effect of the fractional-order differential loss function can account for long-term dependencies and enable precise tuning of high-order trajectory derivatives, thus yielding more physically realistic motion sequences with minimal error accumulation. Comparative experiments demonstrate that our proposed Fractional Optimization-based Two-stage Refinement Framework (FOTSR) outperforms most existing works on three benchmarks (including Human3.6M, CMU-Mocap, and 3DPW). Full article

The high numerical computing cost of time-fractional diffusion equation (tFDE) models over long time periods is a major obstacle to their real-world applications. Therefore, this study presents a rapid adaptive finite difference method, which uses the sum-of-exponentials (SOE) technique to quickly evaluate the kernel function and adopts the trial-and-error (T&E) method to select optimal time steps. For a uniform number of time steps N_T with T >> 1, the cumulative computational cost of the approximate fractional derivative can be reduced from O($N_T^2$) for the T&E method to O(N_T log N_T). To evaluate the accuracy and computational efficiency of the proposed method, a comprehensive comparison is conducted based on three numerical examples. Numerical results show that the SOE-T&E technique provides more accurate results with fewer grid points, compared with uniform mesh method. Moreover, the SOE-T&E technique reduces the computation time by 88.98% compared to the T&E method for the same error level in our numerical examples. Full article

This paper investigates the output synchronization of fractional-order cyber-physical systems (FOCPSs) under communication constraints. To address limited bandwidth and high transmission costs, an event-triggered encoding-decoding sampled-data iterative learning control (ET-EDSDILC) protocol is proposed. The control law integrates a quantized sampling framework with an encoding–decoding mechanism to reconstruct control signals and address communication constraints. Furthermore, an event-triggered mechanism based on error energy attenuation (EEA) is developed to adjust communication frequency by monitoring error trends, thereby reducing unnecessary data transmissions. By applying fractional-order calculus and the contraction mapping principle, sufficient conditions for output synchronization are derived. Numerical simulations show that the proposed ET-EDSDILC framework reduces communication overhead and data redundancy while maintaining tracking performance, offering a solution for FOCPSs under communication constraints. Full article

Transitional marine–continental shale reservoirs are typified by intricate pore architectures and pronounced heterogeneity; accurate characterization of their pore network and fluid mobility underpins reservoir appraisal and sweet-spot forecasting. Focusing on the Longtan Formation transitional shales in the Sichuan Basin, this study integrates NMR T₂ spectrometry, geochemical–mineralogical assays and multifractal analysis to elucidate multi-scale heterogeneity of the pore framework and its governing mechanisms. Results reveal that the investigated shales are characterized by low porosity (0.46–7.43%) and high bound fluid saturation (66.77–97.28%). Multifractal spectral width (Δα) and degree of multifractality (ΔD) serve as robust metrics of pore heterogeneity, correlating closely with rock composition (e.g., TOC and clay content). By combining multifractal indices, mineralogical assemblage and fluid movability, the samples are classified into three reservoir archetypes, with Type I (weakly heterogeneous—high quality) identified as the prospective developmental sweet spot. This work provides a theoretical and methodological backbone for quality assessment and play-ranking of transitional marine–continental shale reservoirs. Full article

Using 48 shale samples from the lower member of the Longmaxi Formation in Well YL, in the Middle Yangtze region, we investigate the fractal characteristics of pore structures across different shale lithofacies based on total organic carbon (TOC), X-ray diffraction (XRD), and low-pressure N₂ adsorption analyses. The shale succession is dominated by three lithofacies—clayey shale, mixed shale, and felsic shale—with mesopores and micropores forming the principal pore systems. N₂ adsorption–desorption isotherms exhibit pronounced hysteresis loops, and ln(V) versus ln(ln(P₀/P)) plots show distinct two-segment behaviour, indicating dual fractal dimensions within the pore network. The fractal dimension of small pores (Df₁ = 2.75–2.87) is consistently higher than that of large pores (Df₂ = 2.01–2.39), suggesting stronger structural heterogeneity in micropore–mesopore systems. Felsic shale exhibits the highest fractal dimensions, followed by mixed shale, whereas clayey shale shows the lowest values. Fractal dimensions correlate positively with TOC, clay minerals, and pyrite content, but negatively with quartz, feldspar, and carbonate minerals. Lithofacies therefore exert a first-order control on pore fractal characteristics through their influence on mineralogical composition and organic matter abundance. These results demonstrate that fractal dimensions provide a robust quantitative metric for evaluating reservoir heterogeneity in Longmaxi Formation shales. Unlike previous studies that examined pore complexity at the bulk-rock scale, this study adopts a lithofacies-resolved dual-fractal framework to quantify multiscale pore heterogeneity and explicitly elucidate the roles of mineralogy and organic matter in controlling pore complexity. Full article

The thermal stability of the shot-peened gradient microstructure in LPBF AlSi10Mg during annealing at 500 °C for up to 8 h was investigated. EBSD boundary maps were analyzed using the box-counting method to determine fractal dimension, D, as a quantitative descriptor of grain-boundary geometrical complexity. It was found that D decreased with depth from the dynamically recrystallized surface layer (0–10 µm; D = 1.73; ECD = 0.8 ± 0.2 µm) through the transition layer (10–40 µm; D = 1.56; ECD = 2.4 ± 0.7 µm) to the matrix (>40 µm; D = 1.16; ECD = 3.0 ± 0.7 µm). After 5 min, the surface network simplified (D = 1.63; ECD = 2.4 µm), whereas the transition layer exhibited increased complexity (D = 1.72; ECD = 3.7 µm), suggesting a strong contribution of near-surface particle pinning and extensive recovery/polygonization within the subsurface. The matrix showed a transient increase in D (1.16 → 1.71), associated with fragmentation of the cellular Si network. Continued annealing reduced D in the surface and transition layers to 1.50 and 1.64 after 1 h due to progressive boundary smoothing and consumption of deformation substructure. Prolonged exposure triggered sparse discontinuous recrystallization exclusively within the transition layer, producing abnormally large grains that migrated bi-directionally into both the pinned surface layer and the bulk matrix. After 8 h, the gradient microstructure collapsed and the boundary trace became disconnected, yielding an apparent exponent D ≈ 0.97 at the map scale. Full article

This paper develops a direct analytical framework for synchronizing and controlling fractional-order octonion-valued fuzzy bidirectional associative memory neural networks (FOOVFBAMNNs). Octonion algebra is neither commutative nor associative, which limits the application of standard analytical tools. To address this challenge, we first propose a generalized Cauchy–Schwarz inequality tailored to the octonionic domain, which operates directly without relying on system decomposition. This inequality lays the groundwork for a Lyapunov-based stability analysis that retains the system’s inherent geometric structure to avoid decomposition into real-valued components. Based on this framework, we derive concise 2-norm inequality criteria, which are sufficient to guarantee Mittag-Leffler synchronization of the proposed model. We also employ a Particle Swarm Optimization (PSO) algorithm to systematically optimize the flexible parameters in the generalized inequality, enhancing the practical performance of the synchronization scheme. To validate the effectiveness of the proposed method, we apply it to a multi-domain image restoration task. Numerical experiments verify the performance of our method. In terms of Peak Signal-to-Noise Ratio (PSNR), the octonion-valued network with PSO-tuned parameters achieves better results than its non-optimized counterpart as well as models constructed in complex or quaternion domains. Full article

Landslides constitute a globally pervasive and highly destructive natural hazard. Although artificial intelligence (AI)-driven landslide susceptibility mapping has emerged as an effective tool for delineating high-risk zones, its predictive performance is frequently constrained by inherent data noise and insufficient characterization of landslide triggering factors, restricting the credibility of the mapping results. In this study, to remedy this limitation, we adopt fractal analysis to extract latent inherent information from topographic features. Specifically, the box-counting method and multifractal analysis are applied to excavate the intrinsic nonlinear characteristics embedded in eight topographic factors, and an improved K-means algorithm is utilized to perform feature selection and construct a dedicated fractal feature dataset, which is fed to advanced AI models. Our results indicate that the information dimension ($D_1$) of the slope gradient, the correlation dimension ($D_2$) of aspect, land relief, the $D_2$ of roughness, the $D_2$ of plan curvature, the multifractal spectrum width ($∆\mathsf{\alpha }$) of profile curvature, the $D_2$ of elevation, and the surface cutting depth were the most effective features, demonstrating superior performance in capturing landslide targets. Comparative performance evaluations reveal that AI models trained on fractal features demonstrate substantially superior predictive capabilities compared to AI models trained on raw features. This superiority is consistently evidenced across key evaluation metrics, including overall accuracy, kappa coefficient, F1-score, and predictive efficiency, demonstrating that the integration of fractal characteristics significantly augments model robustness and predictive efficacy. To mitigate the ‘black-box’ problem of AI modeling, Shapley additive explanations were employed to quantify individual feature contributions and elucidate the underlying predictive mechanisms. Our findings indicate that the integration of fractal analysis yields highly discriminative and robust feature representations, thereby expanding the representational capacity of the models and improving predictive accuracy. Furthermore, a joint assessment of spatial uncertainty and susceptibility maps demonstrates that these models exhibit low predictive variance and high spatial stability when delineating high-susceptibility zones. Notably, models utilizing fractal-derived features achieve superior spatial capture efficiency. The resultant topographic features characterized by fractal representation and selected via the improved K-means algorithm can significantly improve the predictive performance of trained AI models in landslide susceptibility mapping tasks, offering a scientific and viable technical approach for future landslide prediction and prevention. Full article

The Dirac operator links harmonic analysis, physics and hypercomplex signal representations. However, most Dirac-based imaging methods remain integer order and lack spectral adaptability. In this paper, we propose a novel fractional Dirac framework for edge detection. Some fundamental properties are obtained, including square factorization, Liouville-type properties, and uncertainty principles with sharpened constants in a limiting case. Then, a numerically stable discrete realization is developed based on the quaternion Fourier transform. This realization yields an edge detector for both grayscale and RGB images. Experiments on benchmark datasets show that the proposed method produces coherent contours and competitive boundary-detection performance compared with classical gradient methods and recent transform-based detectors. Full article

Based on the Infinite State Representation (ISR) of the Riemann–Liouville integral, the energy stored in a fractional-order integrator is revisited, together with the energy dissipated through Joule losses. Using an idealized ultracapacitor model based on the Constant Phase Element (CPE), i.e., a fractional-order capacitor, theoretical expressions for the stored and dissipated energies during current charging of the CPE are derived. Numerical simulation of the fractional integrator over a frequency interval {ω_min, ω_max} validates a realistic CPE model, in which low-frequency modes correspond to energy storage, while high-frequency modes account for self-discharge and the origin of dissipated energy. This theoretical study leads to the definition of a new ultracapacitor model composed of an internal resistor and the previous realistic CPE, whose frequency-distributed representation enables prediction of the internal state variables and, consequently, the State of Charge. Full article

Multi-UAV formation control in low-altitude urban environments faces critical challenges from atmospheric boundary layer (ABL) disturbances, including turbulence, wind gusts, and communication inefficiency in resource-constrained swarms. This paper proposes an adaptive dynamic event-triggered formation control (ADETFC) strategy integrated with a finite-time disturbance observer (FTDO) for multi-agent hexarotor UAV formations operating under ABL conditions. The novelty of the proposed ADETFC lies in employing dual adaptive parameters to simultaneously account for tracking error magnitude and inter-agent formation geometry, dynamically adjusting communication frequency. A nonsingular terminal sliding mode manifold ensures rapid transient convergence and robustness against nonlinearities and inter-agent coupling. The FTDO estimates lumped disturbances with finite-time convergence to a bounded residual neighborhood, enabling reduced control gains that mitigate chattering and actuator wear. Lyapunov-based analysis establishes finite-time reachability of the sliding manifold and guarantees that the tracking error converges to a bounded residual set in finite time. The Zeno-free operation is guaranteed by a strictly positive minimum inter-event time analytically derived from system dynamics. Simulations under three ABL scenarios, including Dryden turbulence, wind gusts, and sinusoidal disturbances, demonstrate formation tracking RMSE reductions of up to 29.4%, disturbance estimation RMSE reductions of up to 54.3%, and communication-event reductions of 46.4–63.2% compared with benchmark schemes. These results confirm accurate formation tracking, efficient communication, and robust multi-agent networking under challenging wind conditions, making the framework suitable for networked UAV applications in complex environments. Full article

The main objective of this study is to investigate smoking dynamics, identify the most influential factors governing smoking behavior, and develop effective intervention strategies through the integration of fractional-order modeling, sensitivity analysis, optimal control theory, and artificial neural networks (ANNs). A nonlinear fractional-order compartmental model is formulated by dividing the population into potential smokers, light smokers, heavy smokers, and quit smokers. The smoking reproduction number is derived to characterize the transmission and persistence of smoking behavior within the population. To determine the impact of model parameters on smoking dynamics, both normalized forward sensitivity analysis and global sensitivity analysis based on Latin Hypercube Sampling (LHS) with Partial Rank Correlation Coefficient (PRCC) are performed. The obtained results identify the most sensitive transmission and progression parameters and demonstrate their important role in shaping smoking prevalence within the community. Furthermore, the classical integer-order model is compared with the fractional-order formulation, where the fractional model provides a more realistic description due to its ability to incorporate memory and hereditary effects associated with smoking behavior. An optimal control framework involving awareness and treatment strategies is further introduced to investigate effective smoking reduction policies. The numerical results demonstrate that awareness campaigns reduce smoking initiation, while treatment interventions increase smoking cessation, and the combined implementation of both strategies produces the most significant reduction in smoking prevalence. The consistency between the sensitivity analysis and optimal control results further supports the reliability of the proposed framework. Numerical simulations are carried out to analyze the qualitative and quantitative behavior of the system under different epidemiological scenarios. In addition, an ANN-based computational framework is employed as an efficient numerical tool to accurately approximate the complex dynamics of the proposed fractional-order smoking model with very low prediction error. Overall, the present study provides a comprehensive mathematical and computational framework for understanding, analyzing, and controlling smoking behavior within a population. Full article

This paper investigates explicit coefficient-based estimates for a class of fractional Hermite functions defined through finite power series with Gamma-function coefficients. These functions may be viewed as a fractional Hermite-type family associated with the Caputo fractional derivative of order $\alpha \in (0,1]$. An explicit representation of the fractional derivative is obtained as a finite sum of monomials with computable Gamma coefficients. This representation is used to derive a preliminary uniform estimate on bounded intervals $[0,R]$ with an explicit constant depending on $\alpha$, n, and R. Consistency with the integer-order setting is established by showing that, when $\alpha =1$, the construction reduces to a Hermite-type polynomial family and the Caputo derivative coincides with the ordinary derivative. Explicit asymptotic formulas are obtained for the associated coefficient envelope as $R\to 0^{+}$ and $R\to \infty$. Numerical experiments up to degree $n=7$ show that the ratio between the coefficient envelope and the computed supremum norm remains below approximately $1.45$ for the tested parameter range. In addition, a weighted $L^2$ estimate is derived with respect to a fractional Gaussian-type weight, yielding an explicit coefficient-based bound. The estimates obtained in this work are preliminary in nature, being based on coefficient-wise majorization, and are not claimed to be optimal. Determining sharp constants and establishing genuine norm-comparison inequalities remain open problems. The results presented here provide a rigorous starting point for the study of explicit coefficient-based estimates for fractional Hermite functions and suggest several directions for future research in fractional approximation theory. Full article

According to data from the World Federation of Societies of Anesthesiologists, numerous countries across Asia and Africa have fewer than one anaesthesiologist per 100,000 people. Upskilling nurse anaesthetists in these regions is critical to improving clinical outcomes, and interactive virtual patient simulators offer a safe environment to explore complex clinical scenarios. This paper introduces an advanced general anaesthesia patient simulator engineered to bridge the accessibility gap left by existing platforms, which often require expert programming knowledge and restrict users to manual titration. Our simulator features an intuitive graphical user interface optimised for clinical education and natively supports both manual and closed-loop anaesthesia administration. The platform includes a suite of pre-designed controllers, specifically standard PIDs and two distinct fractional-order FO-PID variants, highlighting a novel robust FO-PID framework engineered to mitigate high patient variability. The deployment of these embedded controllers is demonstrated via a Depth of Hypnosis regulation case study and validated across a diverse cohort of 19 virtual patients. Closed-loop evaluation reveals that while the standard PID achieves a lower average mean squared error during the maintenance phase, the fractional-order alternatives deliver significantly superior robustness and inter-patient consistency. Ultimately, integrating this simulator into clinical training frameworks offers a viable pathway to reduce nursing workload and enhance patient safety through optimised automated drug delivery. Full article

We investigate dynamic weighted fractional information-theoretic measures for linear stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H\in (1/2,1)$. Motivated by recent constructions of fractional Deng entropy and building upon explicit Gaussian solutions and closed-form fractional moments derived in previous work, we establish fully analytical expressions for the Shannon entropy, Rényi entropy, Tsallis entropy, extropy, and a continuous weighted fractional entropy $\mathbb{E}\left[X_t^p(-\log p_{X_t}\left(X_t\right))\right]$ for $p\ge 0$, expressed directly in terms of known fractional moments without density estimation. All derived measures share a universal asymptotic scaling law growing as $H\log t$, establishing a precise quantitative link between long-memory effects and information dynamics. The weighted fractional entropy further reveals remarkable structural properties as a function of the weighting order p, exposing a dual role of long memory on the system’s informational content. As a concrete application, we characterize anomalous diffusion in aging soft materials through an explicit critical time linking maximal uncertainty to the memory exponent H and the macroscopic aging rate. All results are validated through extensive Monte-Carlo simulations, demonstrating excellent agreement with the closed-form expressions across a wide range of Hurst exponents H and weighting orders p. Full article

Tight sandstone reservoirs exhibit strong pore-throat heterogeneity, which exerts important controls on reservoir quality and fluid-flow behavior. To investigate the pore-throat structure characteristics and their influence on permeability, integrated analyses of thin sections, X-ray diffraction (XRD), scanning electron microscopy (SEM), cathodoluminescence (CL) and mercury intrusion porosimetry (MIP) were conducted on the Chang 8 tight sandstones in the southern Ordos Basin (China). Results show that the Chang 8 tight sandstones are characterized by low porosity and ultra-low permeability, with average porosity and permeability of 7.5% and 0.331 mD, respectively. The pore systems mainly include intergranular, intragranular pores, intercrystalline micropores and microfractures, reflecting strong pore-throat heterogeneity. Segmented MIP analysis reveals two distinct pore-throat response intervals. The fine pore-throat segment shows valid fractal scaling, whereas the large pore-throat segment is interpreted as an early-stage intrusion response. A dimensionless MIP-derived heterogeneity index (H_MIP) was therefore used to characterize connected pore-throat heterogeneity. H_MIP ranges from 2.446 to 2.973 and shows negative associations with permeability and pore-throat radius, indicating that finer and more heterogeneous connected pore-throat systems are generally associated with lower flow efficiency. H_MIP exhibits weak to moderate associations with mineral composition, particularly with carbonate and quartz contents, whereas feldspar and clay minerals show limited relationships. Sensitivity analysis of characteristic pore-throat radii demonstrates that r₁₀ shows the strongest association with permeability within the present MIP dataset, and model performance decreases monotonically from r₁₀ to r₅₀, suggesting that early mercury-accessible coarse pore-throats are more closely related to effective fluid flow than smaller pore-throat populations in the Chang 8 tight sandstone reservoirs. These findings suggest that permeability in the Chang 8 tight sandstones is closely associated with the development of connected large pore-throats, whereas H_MIP provides empirical information on connected pore-throat heterogeneity and flow-path complexity. Full article

In this paper, we propose a new three-step iterative scheme to approximate fixed points of contraction operators in Banach spaces. Under standard Lipschitz conditions, we establish the existence, uniqueness, and strong convergence of the iterative sequence. The convergence rate and data dependence of the method are also investigated. A comparative analysis with Noor, Picard–S, Abbas–Nazir, SP, and NIP iterative methods is presented. As an application, the proposed scheme is employed to solve a fractional viscoelastic model involving a Caputo derivative of order $0<\alpha <1$, which is reformulated as a Volterra integral equation. The numerical results, including error analysis and graphical illustrations, demonstrate that the proposed method achieves faster convergence and a higher accuracy. Full article

In this work, we develop two new classes of rational contraction mappings along with the corresponding fixed point theorems for these types of contractions in suprametric spaces. Furthermore, we use the obtained results to investigate two nonlinear systems, namely, a fractional chaotic financial system and a nonlinear fractional differential equation under integral boundary conditions. Both these nonlinear problems are transformed into fixed point problems in appropriate suprametric spaces, thereby demonstrating the applicability of the developed rational contraction results to nonlinear systems with memory effects. Full article