Fractal Fract., Volume 10, Issue 2 (February 2026) – 61 articles

In this article, the $\alpha$-fractal interpolation function $f^{\alpha }$ corresponding to any function f belonging to the weighted Sobolev space ${\mathcal{W}}_{\rho }^{r,2}\left(I\right)$ is defined. The convergence of sequences of $\alpha$-fractal interpolation functions corresponding to mappings in ${\mathcal{W}}_{\rho }^{r,2}\left(I\right)$ with respect to the uniform norm as well as the weighted Sobolev norm is discussed. It is proved that the Riemann–Liouville fractional order integral of an $\alpha$-fractal interpolation function of any map $f\in {\mathcal{W}}_{\rho }^{r,2}\left(I\right)$ is also a self-referential function interpolating a specific data set. Some aspects of the convergence of the Riemann–Liouville integral of $\alpha$-fractal functions when the original mappings converge are also analyzed. In short, by imposing certain conditions on the base function and the scale vector of a specific iterated function system, fractal perturbations of functions from weighted Sobolev spaces are defined. It is also proved that, under suitable hypotheses, the Riemann–Liouville fractional integral of these fractal mappings on Sobolev spaces is a fractal function of the same kind. Full article

Generative Adversarial Networks (GANs) perform well on natural images but often fail in domains governed by strict geometric or symbolic constraints. This work focuses on convolutional GANs and studies how their inductive biases interact with two contrasting types of synthetic image data: fractal patterns, characterized by self-similarity and scale-invariant local structure, and Euclidean shapes, defined by simple geometric primitives and rigid global constraints. Using multiple convolutional GAN architectures (DCGAN, WGAN-GP, and SNGAN), two resolutions (64 × 64 and 128 × 128), and a suite of evaluation metrics, we compare adversarial training behavior on these datasets under tightly controlled conditions. Fractal datasets yield stable training dynamics and perceptually plausible generations, whereas Euclidean shape datasets consistently exhibit structural failure modes that persist under higher resolution, smoother shape representations, and architectural stabilization. Geometry-aware metrics reveal severe violations of global shape consistency in Euclidean outputs that are not reliably captured by standard perceptual or distributional measures such as FID, SSIM, or LPIPS. We argue that these findings reflect a fundamental inductive bias of convolutional generative models toward a locally rich, scale-repeating structure rather than globally constrained geometry. Rather than indicating that fractals are intrinsically easier to model, our results show that Euclidean geometry exposes limitations of adversarial generative learning that remain hidden under conventional evaluation. From this perspective, fractal datasets serve as informative diagnostic benchmarks for probing how adversarially trained convolutional generators handle scale-invariant structure versus globally constrained geometry, and our results highlight the need for domain-aware metrics and alternative architectural biases when applying generative models to structured or symbolic data. Full article

Low-maturity shale is a prime target for in situ conversion (ICP), yet heating window selection remains largely empirical because pore evolution and hydrocarbon generation are rarely quantified in tandem. Nenjiang Formation shale from the Songliao Basin (TOC = 8.91%; R_o,max = 0.54%) was subjected to closed-system pyrolysis at 300–500 °C (20 °C h⁻¹; 72 h per step). Released oil and gas and residual chloroform-extractable bitumen (“A”) were quantified, and pore evolution was characterized using 2D low-field NMR, SEM, micro-CT, and low-pressure N₂ adsorption. Fractal dimensions (D_s and D_p) were derived from Frenkel–Halsey–Hill (FHH) fitting. Oil yield and bitumen “A” increased sharply above 350 °C and peaked at 375 °C, whereas gas generation accelerated above 400 °C and continued to increase to 500 °C. NMR indicates a temperature-dependent shift in retained hydrocarbons toward weaker confinement and higher mobility, with enhanced expulsion/mobility signals near 375 °C. At 375 °C, BJH pore volume and average pore diameter reached maxima (0.0675 cm³ g⁻¹ and 15.36 nm), while D_s and D_p reached minima (2.343 and 2.444). The coincidence of peak oil expulsion with minimum fractal complexity suggests that FHH-based fractal indices provide a quantitative metric for comparing ICP heating windows in low-maturity shale. Full article

Beamforming plays a central role in enhancing the performance of communication systems; however, suppressing sidelobes in planar antenna arrays (PAAs) while maintaining a compact aperture remains a challenging nonlinear optimization problem. This article presents a two-dimensional (2D) beamforming synthesis framework for PAAs based on the Fractional-Order African Vulture Optimization Algorithm (FO-AVOA), with the objective of minimizing the peak sidelobe level (PSLL) through the joint optimization of amplitude excitations and element placements. The proposed method is benchmarked against established metaheuristic optimizers, including Particle Swarm Optimization (PSO), the Gravitational Search Algorithm (GSA), hybrid PSO–GSA (PSOGSA), the Runge–Kutta Optimizer (RUN), the Slime Mould Algorithm (SMA), Harris Hawks Optimization (HHO), and the baseline African Vulture Optimization Algorithm (AVOA). Simulation results demonstrate that the FO-AVOA, coupled with the proposed 2D formulation, yields superior sidelobe suppression relative to the competing approaches, achieving a lower PSLL with fewer radiating elements, thereby reducing array complexity and overall implementation cost. The obtained results validate the suitability of the FO-AVOA for solving PAA in the context of BFA beamforming and suggest the potential utility of the FO-AVOA for pattern synthesis for other array shapes in various communication systems. Full article

This paper studies the issues of human balancing and stability in the sagittal plane using fractional and integer order time delay feedback control. The neural-mechanical model of human balance is represented as an inverted pendulum controlled by torque. We present a finite-time stability (FTS) analysis for closed-loop neutral time delay systems (NFOTDSs) with fractional order $1<\beta <\alpha \le 2$. By employing a generalized Gronwall inequality, we derive new FTS criteria for these systems in terms of the Mittag-Leffler function. Finally, a suitable numerical example is presented to show the effectiveness of the proposed method. Full article

Accurately determining the T₂ cutoff value is critical for evaluating fluid mobility in deep tight reservoirs, yet strong pore structure heterogeneity challenges traditional methods. This study proposes a non-destructive prediction method based on multifractal singularity spectrum analysis of nuclear magnetic resonance T₂ spectra. Using 10 tight sandstone cores from the Denglouku Formation (Songliao Basin), we quantify the intrinsic relationship between multifractal parameters and T₂ cutoff values. Results indicate that the minimum generalized dimension (D_min) and singularity spectrum width (Δα) are not merely mathematical fits but reveal the physical mechanisms controlling fluid binding in micro-throats. A multivariate regression model based on these parameters significantly outperforms traditional methods in accuracy (R² > 0.85). This approach provides a robust, non-destructive tool for identifying reservoir ‘sweet spots’ without compromising core integrity. Full article

The primary objective of this research article is to investigate the concept of extended $\mathcal{F}$-metric spaces and to establish a series of fixed point theorems for generalized contractions within this framework. We further introduce and analyze the notion of interpolative Kannan-type cyclic contractions in extended $\mathcal{F}$-metric spaces, deriving several novel fixed point results associated with these mappings. In addition, we obtain common fixed point theorems for rational contractions, thereby extending and unifying a variety of existing results available in the literature. To highlight the novelty and effectiveness of the proposed results, several illustrative examples are provided. Moreover, the theoretical findings are successfully applied to the solution of Atangana–Baleanu fractional integral equations as well as Volterra integral equation of Hammerstein type, demonstrating their practical significance and wide-ranging applicability. Full article

Clarifying the complex interaction between climate risks and the new energy industry chain is of key significance to advancing the energy transition and strengthening industrial chain robustness. This research pairwise-matches the climate physical risk and the climate transition risk with the entire range of the new energy industry chain segments, comprehensively examining the pairwise interactive relationships. By applying the MF-ADCCA series of methods, it was revealed that there are prevalent asymmetric cross-correlated multifractal characteristics between climate risks and the new energy industry. The long-term memory under the upward trend of the market is distinctly stronger than that under the downward trend. Given that this correlation can indirectly reflect market efficiency differences, this paper constructs the Hurst Volatility Sensitivity Index (HVI) and the Hurst Asymmetry Index (HAI) and further proposes the Unified Market Efficiency Index (UMEI). Its innovative advantage resides in the balanced integration of volatility efficiency and structural symmetry, in turn enabling a comprehensive assessment of the new energy market efficiency under climate risk perturbations. Static analysis reveals that the overall market efficiency of the new energy industry under the climate transition risk is generally higher than that under the climate physical risk, and the market efficiency of mature upstream and midstream new energy segments is significantly superior to that of the downstream. Dynamic evolution characteristics indicate that market efficiency has typical time-varying traits, the evolution of which is often driven by significant policies or extreme events. The climate transition risk tends to trigger aperiodic structural adjustments, while the climate physical risk mostly induces periodic efficiency fluctuations. This study furnishes solid evidence for the new energy market in coping with climate risks. Full article

Motivated by the close relationship between the matrix square root and the matrix sign function, this paper develops a new high-order iterative framework for computing the principal square root of a matrix and its inverse. The proposed approach is derived from a rational fixed-point iteration associated with a scalar nonlinear equation and is extended consistently to the matrix setting. The method is shown to be globally convergent and to achieve fourth-order convergence. Numerical experiments demonstrate that the new scheme outperforms several classical and Padé-based methods. Full article

This paper develops a unified analytical approach for pricing a broad class of volatility-linked financial derivatives under the sub-mixed fractional geometric Brownian motion model. The proposed framework captures key empirical features of financial markets, including correlated non-stationary Gaussian increments and long-memory dependence, while preserving the semimartingale property required for arbitrage-free pricing. We present the exact distribution of the realized variance as a quadratic form of correlated non-stationary Gaussian increments, which leads to a closed-form expression for the cumulative distribution function via a Laguerre-series expansion. These distributional results enable analytical pricing formulas for an extensive family of volatility-linked derivatives. Monte Carlo simulations confirm the accuracy and computational efficiency of the proposed formulas, while numerical investigations illustrate the significant impact of non-stationarity, long-memory effects, and the Hurst parameter on derivative values. These results contribute to a deeper theoretical understanding and more effective computational methods for pricing nonlinear volatility derivatives in markets characterized by persistent temporal dependence and non-stationary stochastic dynamics. Full article

Fractional calculus approach is used to analyze the model of surfactant transport by anomalous diffusion and its adsorption on an interface in a liquid-liquid system. The anomalous diffusion is modeled by time-fractional partial differential equations in the bulk phases. The adsorption of surfactant is described by the corresponding time-fractional Neumann boundary conditions at the interface. The adsorption process is considered under mixed barrier-diffusion control, described by first-order ordinary differential equation, which relates the subsurface concentration with that on the interface. A second relation between these concentrations is derived in terms of a fractional equation by application of Laplace transform technique. By combining both relations the subsurface concentration is eliminated and a single multi-term fractional ordinary differential equation for the surfactant concentration on the interface is derived. Different adsorption kinetic models are considered. In the case of Henry adsorption isotherm the model is linear and possesses analytical solution in terms of multinomial Mittag-Leffler functions. In the cases of Volmer and van der Waals adsorption isotherms nonlinear differential equations of fractional order are obtained. They are reformulated in equivalent integral form, which is used for computer simulation of the process of adsorption. Numerical results are presented and compared with analytical asymptotic predictions. Full article

Person re-identification (Re-ID) using infrared surveillance cameras has attracted increasing attention due to its robustness under low-light conditions. However, infrared images generally suffer from a low spatial resolution, which degrades Re-ID performance. To address this issue, this study proposes a part attention and contrastive loss-based super-resolution reconstruction network (PCSR-Net) and a unified infrared-only Re-ID framework. The proposed PCSR-Net consists of a correlation-based super-resolution reconstruction network (CoSR-Net), a feature extractor for Re-ID, and a part attention mechanism that estimates the importance of different body regions. In addition, contrastive loss and part-aware reconstruction loss are incorporated to guide the super-resolution process toward identity-discriminative representations. Experimental results on DBPerson-Recog-DB1 and SYSU-MM01 demonstrate that the proposed method outperforms state-of-the-art approaches in terms of the equal error rate (EER), mean average precision (mAP), and rank-1 accuracy, validating its effectiveness for infrared-based person Re-ID. Full article

This article examines the behavior of seismic noise fields over the Japanese islands recorded by the F-net seismic network for 1997–2025. This paper uses nonlinear noise statistics: the entropy of the wavelet coefficient distribution, the Donoho–Johnston (DJ) wavelet index, and the multifractal singularity spectrum support width. These parameters were chosen because their changes reflect the complication or simplification of the noise structure. Changes in the structure of seismic noise properties are analyzed in comparison with a sequence of strong earthquakes. Using a model of the intensity of interacting point processes, the effect of the leading of local noise property extrema relative to the seismic event times is estimated. Using the Hilbert–Huang decomposition, the synchronization of the amplitudes of the envelopes of noise property time series for different IMF levels is estimated. A sequence of weighted probability density maps of extreme values of noise properties is analyzed in comparison with the mega-earthquake of 11 March 2011 and the preparation of another possible strong seismic event. Full article

This article is devoted to constructing the fractional powers of operators and their matrix approximations. A key feature of this study is the use of a spectral approach that remains applicable even when the base operator does not generate a semigroup. Our main results include the convergence rate of matrix approximation, derived from resolvent estimates, and a practical algorithm for constructing matrix approximations. The theory is supported by examples. Full article

The Kuramoto–Sivashinsky (KS) equation and its fractional form (FKS) are widely used across scientific fields, including fluid dynamics, plasma physics, and chemical processes, to model nonlinear phenomena such as shock waves. It is worth emphasizing that this contribution is part (II) of a larger, systematic research program aimed at modeling, for the first time, completely nonintegrable, nonplanar, and fractional nonplanar evolutionary wave equations. This work focuses on the nonplanar KS framework and its applications to dust–acoustic shock waves in a complex plasma composed of inertial dust grains and inertialess nonextensive ions. This study analyzes both the nonplanar integer KS and nonplanar FKS equations, accounting for geometric effects. This is because the nonplanar model is most suitable for analyzing various nonlinear phenomena (e.g., shock waves) that arise and propagate in plasma physics, fluids, and other physical and engineering systems. Since the nonplanar KS equation is a fully non-integrable problem, its analysis poses a significant challenge for studying the properties of nonplanar shock waves in plasma physics. Therefore, the primary objective of this study is to analyze the nonplanar KS equation using the Ansatz method, thereby deriving semi-analytical solutions that simulate the propagation mechanism of nonplanar shock waves in various physical systems. Following this, we investigate the effect of the fractional factor on the profiles of nonplanar dust–acoustic shock waves to elucidate their propagation mechanism and assess the impact of the memory factor on their behavior. To achieve the second goal, we face a significant challenge because the model under study does not support exact solutions and is more complex than simpler physical models. Thus, the Tantawy technique is employed to overcome this challenge and to analyze this model for generating highly accurate analytical approximations suitable for modeling nonplanar fractional shock waves in various plasma models and in other physical and engineering systems. Full article

We propose a fast L2-1_σ finite element method for solving the time fractional Keller–Segel equations with a Caputo fractional derivative of $\alpha \in (0,1)$. Firstly, the fast L2-1_σ scheme on the graded mesh is used to discretize the time fractional derivative. This approach relies on the sum of exponentials (SOE) skill to speed up the convolution kernel. Thus, we overcome the computational cost caused by the nonlocality of fractional derivatives. Then, by combining finite element discretization in spatial direction, a fully implicit numerical scheme is derived. Subsequently, we establish the stability and an $\alpha$-robust error analysis of the fully discrete scheme. Finally, we present some numerical examples to demonstrate the correctness of our theoretical results. Full article

Rock pore–fracture systems exhibit inherent fractal characteristics, which exert a significant influence on fluid transport. In this study, coal rock is selected as the representative medium. Based on fractional calculus in physical fractal space, and by integrating operator algebra with the force–electric analogy method, a fractional order control equation is derived. To validate the proposed model, porosity measurements of coal and limestone were performed using the two-compartment Boyle’s law method and compared with conventional porosity calculation approaches. The results demonstrate that the fractional order model achieves a coefficient of determination (R²) of up to 0.99 for porosity and 0.98 for pressure, representing an improvement of approximately 0.07 over the exponential model. Moreover, the root mean square error (RMSE) of porosity is as low as 0.0008, while the RMSE of pressure is 0.0715, both significantly lower than those obtained using the exponential model. These results indicate that the fractional order model more effectively captures the non-Darcy flow behavior and the temporal evolution of porosity, providing substantially improved fitting accuracy. Further analysis reveals that the porosity–time relationship is jointly governed by fluid compressibility and pore compressibility under effective stress conditions. Comparative results across different lithologies reveal that the pore compressibility coefficient increases with porosity; for the same rock type, a higher coefficient implies a more complex pore structure and a longer equilibration time. Overall, the proposed fractional order framework provides a more accurate description of the fractal pore structures in rocks, establishing a clear link between microscale fractal geometry and macroscale fractional order response. Full article

We present a conceptual and computational investigation of vacuum memory within a discrete toy-model framework. In this phenomenological approach, we introduce an effective memory field that records virtual events and nonlocal couplings on a lattice, without claiming to derive a fundamental new field of nature. Using a discrete toy model, we simulate memory formation via virtual events, nonlocal links, and black-hole-like information sinks. The resulting dynamics exhibit long-range spatial correlations, curvature-induced accumulation, high-entropy retention zones, and distinct spectral features, indicating that the modeled memory field can store and organize information in a vacuum-like medium. Building on this foundation, we incorporate curvature-modulated vacuum memory fields into Bohmian particle dynamics. By varying the memory coupling strength λ, we demonstrate that memory gradients systematically bend particle trajectories toward curvature centers, illustrating an active role for structured memory in guiding quantum-like motion. We further show that when vacuum memory encodes the full quantum phase S(x, t) and particles are guided by the Bohmian relation $\dot{x}=m^{-1}{\partial }_{x}S$, the trajectories collapse onto a single path with machine-level precision, providing a numerical consistency check that our implementation reproduces exact pilot-wave guidance and minimal-action dynamics. Through a minimal two-site entangled-memory model, we demonstrate that coupled memory fields—without explicit particle dynamics—can spontaneously synchronize via weak informational coupling, generating robust nonlocal correlations reminiscent of entanglement. Finally, we simulate two-slit interference under vacuum memory perturbations. While random, unstructured memory preserves quantum coherence and fringe visibility, structured, phase-sensitive memory induces dephasing and suppresses interference, functioning as a phenomenological decoherence mechanism. Together, these results situate our toy model within emerging information-based views of quantum dynamics and spacetime, offering a computational platform and conceptual lens for exploring the informational dynamics of a vacuum-like medium. Full article

The fractional Langevin equation is used to analyze the erratic motion of a particle in a complex environment. Hydrodynamic backflows and restoring harmonic forces can work concurrently on the probe particle in addition to the constraints imposed by the complex environment, whose characteristics are represented in the friction force the particle encounters. The friction force, which is used for the modelling, corresponds to a damped oscillating function and it is expressed by a one-parameter Mittag-Leffler function. Analytical solutions for the relaxation functions are extracted, and through them observables like the position autocorrelation function, velocity autocorrelation function and mean square displacement are explicitly given. The competition between hydrodynamic fluctuations and frictional forces establishes a memory whose time scale is larger than the diffusional time scale, and during this time window the motion of the particle is superdiffusive. Full article

To further tap into the vibration isolation potential of vehicle mechatronic ISD (Inerter–Spring–Damper) suspension systems, a positive real synthesis design method for vehicle ISD suspensions is proposed based on the fractional-order bridge networks. Firstly, a quarter-car dynamic model of the mechatronic ISD suspension system is established. Subsequently, based on the impedance solution method for bridge networks, a basic bridge network structure and its corresponding series-parallel network configuration are constructed for comparative analysis. Then, with the aim of improving the overall performance of the vehicle suspension system, the particle swarm optimization algorithm is employed to determine the optimal parameters for the mechatronic ISD suspension. Finally, simulation results under random road excitation at a vehicle speed of 20 m/s indicate that, compared to the traditional passive suspension system, the fractional-order bridge network-based ISD suspension system achieves reductions of 3.9%, 27.5%, and 11.8% in the root mean square (RMS) values of vehicle body acceleration, suspension working space, and dynamic tire load, respectively. In contrast, the integer-order bridge network-based ISD suspension system only achieves reductions of 3.0%, 24.2%, and 4.5% in these respective performance metrics. The results confirm that, with an equivalent number of components, the fractional-order bridge network exhibits superior vibration isolation performance relative to series-parallel networks, thereby providing novel methodological guidance for the design of vehicle mechatronic ISD suspension. Full article

This work investigates the moderate deviation principle for a class of two-time-scale Caputo fractional stochastic differential equations. The driving noise of the slow variable is fractional Brownian motion with Hurst index $H\in ({\displaystyle \frac{1}{2}},1)$. The driving noise of the fast variable is standard Brownian motion. The fractional derivative operator of the slow variable is defined by Caputo, and the derivative of the fast variable is of the integer order. The proof process is mainly based on the weak convergence method of fractional Brownian motion variational representation. We first establish the moderate deviation principle by proving the weak convergence of the single-time-scale controlled version. Subsequently, we combine Khasminskii time discretization technology to extend the theoretical framework to two-time-scale systems. Finally, a concrete computational case is offered to demonstrate the applicability of the theoretical framework. Full article

This paper studies mixed impulsive boundary value problems involving tempered $\mathsf{\Psi }$-fractional derivatives of Caputo type. By introducing exponential tempering into the fractional framework, the proposed model effectively captures systems with fading memory—an improvement over conventional power-law kernels that assume long-range dependence. The generalized tempered $\mathsf{\Psi }$-operator unifies several existing fractional derivatives, offering enhanced flexibility for modeling complex dynamical phenomena. Impulsive effects and integral boundary conditions are incorporated to describe processes subject to sudden changes and historical dependence. The problem is reformulated as a Volterra integral equation, and fixed-point theory is employed to establish analytical results. Existence and uniqueness of solutions are proven using the Banach Contraction Mapping Principle, while the Leray–Schauder nonlinear alternative ensures existence in non-contractive cases. The proposed framework provides a rigorous analytical basis for modeling phenomena characterized by both fading memory and sudden perturbations, with potential applications in physics, control theory, population dynamics, and epidemiology. A numerical example is presented to illustrate the validity and applicability of the main theoretical results. Full article

In this study, we present a unified symmetry-conservation solution analysis of a well-posed resonant nonlinear Schrödinger (NLS)-type equation incorporating spatio-temporal dispersion and inter-modal dispersion. Working within the truncated M-fractional derivative framework, we first construct exact traveling-wave solution families via the Kudryashov expansion method, together with the corresponding parameter constraints and limiting cases. We then determine the admitted Lie point symmetries and establish the associated Lie algebra, including the commutator structure, adjoint representation, and an optimal system of one-dimensional subalgebras for classification. Using the conservation theorem, we derive conserved vectors associated with the fundamental invariances of the model; in the NLS setting and under suitable conditions, these quantities can be interpreted as generalized power (mass), momentum, and energy-type invariants. Overall, the results provide explicit wave profiles and structural invariants that enhance the interpretability of the model and offer benchmark expressions useful for further qualitative, numerical, and stability investigations in nonlinear dispersive wave dynamics. Full article

The Jurassic lacustrine oil shale in southwest China has become a primary production layer due to its high yield and substantial reserves. However, influenced by the lacustrine environment, the vertical profile of the lacustrine shale reservoir shows alternating deposits of shale and carbonate rock. This complex lithological combination results in significant heterogeneity in reservoir types, reservoir distribution, and internal structure. Currently, research on micro-pore structure and hydrocarbon storage mechanisms in lacustrine shales is insufficient, necessitating the elucidation of their micro-characteristics to support future exploration and development. This research focuses on the Da’anzhai Member of Jurassic Ziliujing Formation. Various techniques—including organic geochemical analysis, X-ray diffraction, physical property testing, gradient centrifugation, and gradient drying NMR monitoring—were employed to investigate the micro-pore structure and fluid storage mechanisms of the lacustrine shale reservoir. The following insights were gained from this research. The organic matter pores (OMP) and inorganic pores (IP) developed within the Da’anzhai lacustrine shale reservoir together create the storage space for shale oil, while micro-fractures further enhance the reservoir’s storage capacity and flow performance. Lacustrine shale oil exists in three storage states: mobile oil, bound oil, and adsorbed oil. Mobile oil is primarily located within the micro-fractures and large pores (greater than 350 nm) of the shale reservoir and is the main target for industrial extraction. Bound oil is mainly found in the meso-pores, micropores, and narrow pore structures between rock grains (30 nm to 350 nm), and, theoretically, could potentially be developed through engineering methods such as hydraulic fracturing. Adsorbed oil, due to its close binding with organic matter and clay mineral surfaces, is difficult to release effectively using conventional techniques. The OM abundance, the mineral composition of lacustrine shale, and the pore structure all influence the storage states of shale oil. While a high TOC value increases the amount of mobile oil, the strong adsorption properties of kerogen and organic matter lead to the accumulation of adsorbed oil, which inhibits oil flow. Clay minerals further restrict oil flow by enhancing adsorption, while brittle minerals facilitate the movement of mobile oil by expanding pore space. Based on fractal geometry theory and multi-scale testing results, the large pores in the Da’anzhai lacustrine shale have a high fractal dimension and exhibit complex shapes. However, as pore complexity increases, the amount of adsorbed oil rises significantly, which in turn reduces the proportion of movable oil. Full article

During coal mining, the development of joint fractures in overlying rock strata is one of the key factors that degrade the mechanical properties of rock masses, form water-conducting fracture zones, and induce safety hazards. To investigate the fracture evolution characteristics of overlying strata during coal extraction under thick and hard roof conditions, this study established a mining physical model based on similarity simulation technology, tracked the fracture evolution process, and performed quantitative analysis using fractal theory. The results show that fracture development is significantly correlated with the mining advance distance: the fractal dimension of fractures is small in the initial mining stage and gradually increases as the working face advances. When the mining width exceeds the ultimate span of the roof, local fractures expand rapidly with a sharp rise in the fractal dimension to 1.436; further increasing the mining width triggers large-scale sudden fracture expansion, resulting in severe degradation of rock mass integrity, with the maximum fractal dimension reaching 1.445. The research findings provide theoretical references for safety management and disaster prevention in coal mining under thick and hard roof conditions. Full article

In this study, the bending behavior of beams is investigated using the fractional Euler–Bernoulli beam model. This model is developed based on fractional calculus, particularly employing the Riesz–Caputo derivatives, and is capable of accurately accounting for nonlocal and size-dependent effects in structural beams. Unlike classical models that rely on integer-order derivatives, the present model uses fractional-order derivatives, which offer greater precision in analyzing beam behavior at small scales such as micro and nano levels. In this work, various beams with different boundary conditions and loading types are analyzed. To solve the governing fractional equations, a numerical algorithm based on the finite difference method is developed, which also allows for the incorporation of a variable characteristic length function along the beam. The numerical simulation results demonstrate that the order of the fractional derivative and the characteristic length have a direct impact on the amount of beam deflection. These findings indicate that the Euler–Bernoulli model based on Riesz–Caputo derivatives has high potential for realistic simulation of beam bending behavior at small scales, making it an effective tool for accurate analysis of microscale structures. Full article

In this paper, we mainly investigate the radial symmetry and monotonicity of positive solutions for a nonlinear system involving pseudo-relativistic operators and fractional derivatives of order $(0,1)$. First, we prove a more general Narrow Region Principle and a Decay at Infinity Principle, which are essential for nonlocal pseudo-relativistic operators. Then, by using the direct method of moving planes, we prove the radial symmetry and radial monotonicity of positive solutions for the nonlinear system in the bounded domain $B_1\left(0\right)$ and the whole space, respectively. Finally, we show that the positive solutions of the system are strictly monotonically increasing in a Lipschitz coercive epigraph. Full article

Multilayered geological structures are common in geotechnical engineering, where cooling shrinkage induces polygonal cracks in interlayers, compromising rock mass strength, permeability, and long-term stability. Existing thermo-mechanical studies on layered rock cracking insufficiently address the combined effects of weak interlayer geometry or interface-regulated mechanisms. To address this gap, based on meso-damage mechanics and thermodynamics, this study adopts a thermo-mechanical coupling simulation method considering rock heterogeneity, innovatively focusing on the understudied stress transfer effect at strong–weak interlayer interfaces. Systematic investigations were conducted on the initiation, propagation, and saturation of polygonal cracks in plate-like layered rocks under surface cooling, analyzing the influences of weak interlayer thickness, number, and position, and comparing surface vs. inner interlayer behaviors. Results showed that stress transfer interruption at weak–strong layer interfaces can inhibit crack propagation. Inter weak interlayers produce significantly more cracks and fragments than surface weak interlayers, with a stratified crack length distribution, and the maximum fragment area increases exponentially with weak interlayer thickness. Crack development is strongly influenced by weak interlayer thickness, with thinner layers dominated by non-penetrating cracks and thicker layers tending to develop penetrating lattice-like cracks. The inter layer stress and crack distribution exhibit fractal characteristics, with crack density decreasing layer by layer and no new cracks forming after saturation. This study clarifies the regulatory mechanism of weak interlayer features and surface cooling on crack evolution, quantifies interface effects and fractal characteristics, and provides a theoretical basis for instability prediction of layered rock structures in low-temperature geotechnical engineering. Full article

This paper proposes an iterative learning framework for a class of fractional-order nonlinear multi-agent systems operating under directed iteration-varying switching topologies. To suppress trial-to-trial fluctuations in initial states, a P-type initial condition learning mechanism is integrated into the update law, enabling each agent to actively compensate for its own startup offset in each iteration. The study designs a distributed iterative learning protocol using only local neighbor information, and this protocol can simultaneously achieve fault estimation and diagnosis. By constructing a fractional-order system model and adopting the contraction-mapping analysis method, sufficient conditions are derived in this paper, which guarantee that both the fault error and initial condition error converge asymptotically to zero as the number of iterations approaches infinity. The proposed scheme, based on iterative learning fault estimation, can effectively handle unknown nonlinearities without relying on an accurate system model. Numerical simulation results further verify the effectiveness of the designed fault observer in achieving fault estimation. Full article

The Kuramoto–Sivashinsky (KS) equation and its fractional generalizations (FKSs) arise as canonical models for a wide class of nonlinear dissipative–dispersive systems, including thin-film flows, combustion fronts, drift–wave turbulence in plasmas, and chemically reacting media, where shock-like and strongly localized structures play a central role in the dynamics. Despite their apparent simplicity, KS-type models become analytically intractable once higher-order dissipation, geometric effects, and memory (fractional) operators are incorporated, and standard perturbative or transform-based schemes often lead to cumbersome recursive structures, slow convergence, or severe restrictions on the initial data. In this work, a novel direct approximation procedure, referred to as the Tantawy Technique (TT), is developed and implemented to solve and analyze planar fractional KS-type equations and their Burgers-type reductions in a systematic manner. The central difficulty is to construct, for a given physically motivated initial profile, a rapidly convergent series in fractional time that remains stable for a broad range of the fractional order and transport coefficients, while still retaining a clear link to the underlying shock-wave physics. To overcome this, the TT combines (i) a Tanh-based exact shock solution of the planar integer-order KS equation, obtained first as a reference via the standard Tanh method, with (ii) a carefully designed fractional-time ansatz in powers of $t^{\rho }$, where the spatial coefficients are determined recursively from the governing equation in the Caputo sense. This construction yields closed-form expressions for the first few terms in the approximation hierarchy and allows one to monitor convergence through residual and absolute error measures. Full article