Fractal and Fractional doi: 10.3390/fractalfract10110798

Authors: Jun Lang Xiaolei Wang Meng Wang Yun Lei Luwei Qiu Yutong Han

Quantitative characterization of pore-structure heterogeneity is an important problem in tight sandstone reservoirs, and because pore systems depend on scale, a single fractal dimension cannot fully describe multiscale pore complexity. Sixteen tight sandstone samples from the fourth member of the Xujiahe Formation (T3x4), Sichuan Basin, were analyzed by high-pressure mercury intrusion, thin-section petrography and petrophysical measurements; segmented fractal dimensions were calculated by pore-throat interval, samples were classified by displacement pressure (Pcd), and multifractal analysis was performed on the same intrusion curves. (1) Pore heterogeneity is scale-dependent, and no single parameter set captures its different aspects. D(1) and D(2) do not correlate significantly with the multifractal parameters, and only macropore D(3) is negatively correlated with the low-measure-end (low-q) parameters. (2) Dmax − D0 and ΔD differ in their petrophysical associations. Dmax − D0 is positively associated with porosity (r = 0.56) and permeability (r = 0.77) and negatively associated with displacement pressure (r = −0.68); ΔD is associated mainly with permeability and not with the sorting coefficient. An increase in Dmax − D0 corresponds to a larger contrast between the high-measure and low-measure regions; ΔD mainly reflects the pore-throat combinations linked to permeability. (3) Quartz and carbonate cement show the clearest associations with Dmax − D0, in opposite directions; rock fragments and matrix show no significant associations. Their correlations with Dmax − D0 are r = 0.55 and −0.55, respectively, and none of these minerals correlates significantly with ΔD. (4) Carbonate cementation is associated with displacement pressure, but high displacement pressure is not limited to carbonate-rich samples, and two diagenetic pathways can be distinguished within the high-Pcd class. Samples with ≥12% carbonate cement have Pcd above 1.0 MPa; low-carbonate samples can also develop high Pcd through strong compaction, and Dmax decreases from 1.38 to 1.16 with increasing Pcd. These results show that multifractal analysis, combined with the displacement-pressure classification, can describe pore-structure heterogeneity in tight sandstones and provide a reference for comparable tight reservoirs.

Fractal and Fractional doi: 10.3390/fractalfract10110797

Authors: Anca Alexandra Pop Eva-Henrietta Dulf Cristina Alina Silaghi Horatiu Silaghi

Automating insulin delivery requires controllers that remain well-behaved despite large inter-patient variability. This paper investigates a fractional-order proportional–integral (FOPI) controller for blood glucose regulation and compares it with a classical integer-order PI controller. Six virtual patients, two from each of three age groups (children, adolescents and adults), were generated with a SimBiology/ Matlab2025 implementation of the Dalla Man glucose–insulin meal model, using patient-specific age, fasting glycemia and ingested carbohydrate load (75–110 g). Instead of estimating the parameters of the compartmental model, a simple input–output description was used for each patient. The integer-order PI controller was designed from phase-margin and cut-off-frequency specifications, whereas the FOPI controller (parameters Kp, Ki and λ) was optimized using the fractional order to reduce overshoot and settling time simultaneously and adding the iso-damping condition. The results suggest that the additional tuning degree of freedom λ yields a faster nominal response in most patients and a much less variable overshoot, at the price of a more demanding design. Additionally, a graphical user interface is described that enables patient-specific model generation and controller design, allowing clinicians and patients alike to simulate therapeutic effects and compute appropriate dosages.

Fractal and Fractional doi: 10.3390/fractalfract10110796

Authors: Najla Alghamdi Maryam Ahmad Alyami Nouf Khalid Alghefari

This paper examines a coupled multi-term Caputo fractional q-differential system with nonlocal multi-strip q-integral boundary conditions. An equivalent integral formulation allows us to establish existence through the Leray–Schauder nonlinear alternative and uniqueness through Banach’s contraction principle. We further derive Ulam–Hyers and Ulam–Hyers–Rassias stability results, along with their generalized forms. The study develops our earlier fractional model in the q-calculus setting, with a numerical example supporting the theoretical analysis.

Fractal and Fractional doi: 10.3390/fractalfract10110795

Authors: Muath Awadalla Maryam G. Alshehri

This paper develops a variational and operator-theoretic framework for a class of fractional Sturm–Liouville-type problems involving left and right Riemann–Liouville derivatives on weighted Hilbert spaces. The form domain is defined through a one-sided Riemann–Liouville graph norm, and the associated symmetric bilinear form is shown to be densely defined, closed, and coercive. The representation theorem for closed forms then yields a self-adjoint realization, whose operator domain is characterized variationally and related to the formal fractional differential expression through its distributional realization. The resulting operator has compact resolvent and a strictly positive, purely discrete spectrum consisting of eigenvalues of finite multiplicity tending to infinity, while the corresponding eigenfunctions can be chosen to form an orthonormal basis of the underlying weighted Hilbert space. The eigenvalues are further characterized by the min–max principle. As an illustrative application, a regularized fractional Legendre-type problem is shown to satisfy the hypotheses of the abstract framework. Finally, under uniform convergence of the weights to a positive limiting classical weight, the associated bilinear forms are proved to converge on the common core Cc∞(a,b) to the corresponding classical Sturm–Liouville form as α→1−. The classical-limit result is formulated at the level of forms on the common core and does not assert operator or spectral convergence.

Fractal and Fractional doi: 10.3390/fractalfract10110793

Authors: Alexandru Tudorache Rodica Luca

We investigate the existence and multiplicity of positive solutions for an ω-Riemann–Liouville fractional differential equation on an infinite interval supplemented with nonlocal boundary conditions containing Riemann–Stieltjes integrals and ω-Riemann–Liouville fractional derivatives of various orders. In the proof of our main results we use the Guo–Krasnosel’skii fixed point theorem and the Leggett–Williams fixed point theorem.

Fractal and Fractional doi: 10.3390/fractalfract10110794

Authors: Sudsakorn Inthidech Nopanom Kaewhanam Thammanun Chatwong Apichit Kampala Sitthiphat Eua-apiwatch Sivarit Sultornsanee

A fractional flow direction need not be determined by its yield boundary. We study a componentwise left Riemann–Liouville operator with fixed coordinates, scales, terminals and internal variables. Proposition 2 constructs two globally strictly convex polynomial functions. They define the same closed ellipse and have identical boundary derivatives through any prescribed finite order. Their normalized fractional directions nevertheless differ for every order between zero and one. A half-order circle example gives an exact separation of 30.008768 degrees. Thus, neither strict convexity nor finite boundary data restore representation invariance. Polynomial identities, adaptive quadrature and independent Grünwald–Letnikov refinement verify the construction. A published modified Cam-clay surface provides a further operator benchmark. A coupled AJOP-T diagnostic, Lode scans and a map over lower terminal, fractional order and relabeling coefficient then illustrate possible mechanical consequences under stated admissibility conditions. The main result identifies constitutive information that cannot be recovered from finite boundary data alone. These synthetic tests verify the equations and implementation. They do not calibrate or validate a material model.

Fractal and Fractional doi: 10.3390/fractalfract10110792

Authors: Jianhui Li Zhen Wang Xin Liu Xiaojie He

We study a consumption-adjusted fractional Black–Scholes terminal-value equation with deterministic coefficients. An absolutely continuous adjustment yields explicit put and call formulas through cumulative drift, variance, and discount factors. For deterministic time-varying volatility, the variance coefficient retains the historical covariance of the fractional integral. The representation follows a standard Gaussian reduction; the analysis identifies the joint coefficient changes induced by the adjustment and their Brownian limit. Cash and stock consistency tests show that the adjusted operator generally does not preserve the prices of the original market. A deterministic scaling relates its solution to European valuation in an auxiliary diffusion market, without identifying that market with the fractional asset process. Reproducible quadrature and finite-difference experiments verify the formulas, while classical option benchmarks illustrate the distinction from American optimal stopping.

Fractal and Fractional doi: 10.3390/fractalfract10110791

Authors: Hye-Rim So Gang-Gyoo Jin Gun-Baek So

The box-counting method is a fundamental tool for estimating fractal dimensions from 2D binary images, but its accuracy depends heavily on the sampling strategy used to select box sizes. Traditional approaches—such as geometric and arithmetic stepping—either produce sparse data at large scales or bias toward over-sampling at small scales. More recent strategies, including modified geometric and coverage-ratio stepping, improve pixel utilization but still suffer from low utilization rates and uneven data distribution on log–log plots. We propose an enhanced sampling strategy called the group competition step (GCS) method. GCS partitions the sampling range into subgroups and applies a localized competition mechanism based on coverage ratio values to produce a more balanced distribution of step sizes. The proposed method was evaluated alongside five existing sampling schemes across a dataset of 120 deterministic fractal images encompassing both geometric and non-geometric scales paired with fractional box-counting. On geometric benchmark images, GCS achieved superior performance with the lowest mean absolute error (MAE = 0.0062) and mean absolute percentage error (MAPE = 0.3561%), securing the lowest root mean square error (RMSE) in five out of eight benchmark geometries. Across the full 120-image test suite incorporating fractional remainder boundary counting, GCS maintained its performance lead (MAE = 0.0077, MAPE = 0.4569%) and recorded a Wilcoxon signed-rank test p-value of 0.5469, confirming statistical unbiasedness. These results demonstrate that GCS provides a more reliable framework for precise fractal-dimension estimation.

Fractal and Fractional doi: 10.3390/fractalfract10110790

Authors: Sunzhuo Yang Xiaolin Liu Chenyang Han Yuxin Shen Siyuan Ju

To address the complex pore structure of porous media and the overprediction of flow rate by the baseline fractal viscous-flow model under high pressure differences, a gas mass-flow-rate model was developed for gas-permeable steel. The pore fractal dimension and tortuosity fractal dimension were determined from CT-derived pore-size statistics and a porosity constraint. Pore-size distribution, flow-path tortuosity, and gas compressibility were incorporated into the model, while the nonlinear resistance correction coefficient was introduced through an assumed structure-informed closure based on the relative contribution of pore-structure characteristics. For the four back-pressure conditions investigated, the correction reduced the mean relative error from 13.0–14.4% to 2.3–7.0%. Without relying on experimental fitting of pressure-difference–flow-rate data, the proposed model effectively improves the prediction of non-Darcy gas flow under high pressure differences.

Fractal and Fractional doi: 10.3390/fractalfract10110789

Authors: Rattiya Meesa Ratinan Boonklurb Nontawat Bunchak Phiraphat Sutthimat

The question is whether the reset-time distribution alone determines long-time first-passage behavior when intrinsic waiting times are heavy-tailed. This paper studies complete resetting in a two-state semi-Markov chain motivated by spatial variable-order diffusion. For the spatial diffusion, the long-time result is conditional on an assumed expansion of the survival probability within one renewal cycle. Exact renewal equations connect long-time survival in the chain to the joint distribution of reset and intrinsic waiting times. For independent power-law resetting in the slow state, a critical reset-time tail exponent separates slow-state and fast-state dominance. For identical Mittag–Leffler reset-time distributions, independent physical-time resetting yields faster survival decay, whereas operational-time resetting through the same stable subordinator as the intrinsic waiting time preserves the slow-state exponent. For Poisson resetting in both states with a deterministic reset delay, an explicit mean first-passage formula reduces optimization over two reset rates to one dimension. A unique finite global minimizer exists under the stated assumptions. Monte Carlo simulations and numerical Laplace inversion for the chain support the analytical results. The findings identify dependence between reset and waiting times as essential to first-passage behavior in the two-state model.

Fractal and Fractional doi: 10.3390/fractalfract10110788

Authors: Burak Aggul Amir Seyyedabbasi Erdal Alimovski Kaan Arik

The performance of an optimization algorithm can be limited when the same search behavior is maintained throughout the optimization process. Fractional Newton-Derived Optimization (FNDO) uses a fixed fractional order and does not adapt its main search mechanisms during the run. To address this limitation, this study introduces SA-FNDO, an adaptive extension of FNDO that retains the original fractional order while incorporating archive-assisted guidance based on JADE, covariance-based crossover derived from CoBiDE, and CMA-ES refinement around the current best solution. SA-FNDO is evaluated on the CEC2017 and CEC2022 benchmark suites and on three static 3D UAV path-planning environments. The ablation study shows that archive guidance and CMA-ES refinement provide the clearest contributions, while the effect of covariance crossover is more problem-dependent. Across the CEC test conditions, SA-FNDO obtains the best overall mean rank among the nine compared methods, ranking first on CEC2017 and CEC2022 at D=10 and second on CEC2022 at D=20. In the UAV case study, SA-FNDO obtains feasible paths in all 90 runs and achieves the lowest feasible-run median geometric cost in all three environments. These results support SA-FNDO as an effective extension of the publicly available FNDO implementation under the tested conditions.

Fractal and Fractional doi: 10.3390/fractalfract10110787

Authors: Igor Pantic Svetlana Valjarevic Jelena Cumic Snezana Babac Jovana Paunovic Pantic

Fractal analysis has become a frequently used technique in sensory physiology for the evaluation of signal complexity, with potential applications ranging from the analysis of retinal vascular architecture to the quantification of changes in electroencephalographic recordings. Features such as fractal dimension and lacunarity may be able to detect discrete changes in both one-dimensional and two-dimensional signals associated with numerous physiological processes and disease development. However, their interpretation and translational potential are constrained by various limitations. In this concise review, we critically explore current applications of fractal analysis across major sensory domains and focus on the challenges that limit reproducibility and biological interpretation. We particularly focus on the biological ambiguity of fractal dimension and related measures, the non-equivalence of commonly used fractal metrics, acquisition-related variability, small datasets, and insufficient technical validation. Finally, we discuss issues related to the integration of fractal metrics into contemporary machine learning models, including challenges related to generalizability, overfitting, and poor explainability.

Fractal and Fractional doi: 10.3390/fractalfract10110786

Authors: Bernardo Pincheira Werner Kristjanpoller Benjamin Miranda Tabak

This paper studies the scale-dependent co-movements of monthly changes in U.S. unemployment, inflation, and the federal funds rate from July 1954 to May 2026 using contemporaneous, lagged, and asymmetric Multifractal Detrended Cross-Correlation Analysis (MF-DCCA). Significance is assessed against 200 Iterative Amplitude Adjusted Fourier Transform (IAAFT) surrogates for every pair, lag, and trend regime, and the lead–lag patterns are compared with a linear VAR benchmark. The generalized cross-Hurst exponent Hxy(q) varies with the moment order q: the cross-scaling of small co-fluctuations (q<0) is largely compatible with the surrogate null, whereas that of moderate and large co-fluctuations departs from it at horizons of up to 12 months. The lead–lag cross-persistence between policy-rate changes and unemployment, and from policy-rate changes to inflation, is approximately invariant across horizons of 1 to 12 months, whereas for unemployment and inflation, and from inflation to policy-rate changes, the cross-persistence of large co-fluctuations increases with the horizon, most robustly from inflation to unemployment. The VAR finds linear predictability only in the first group, most robustly from policy-rate changes to inflation. The asymmetric analysis shows that joint scaling depends on the phase of the guiding series, with markedly higher cross-persistence of inflation with unemployment in high-inflation phases, a pattern that is robust to the detrending order, whereas the asymmetries involving the policy rate are less robust and the asymmetric estimates at extreme moment orders are sensitive to the estimation settings. These results are descriptive rather than structural and complement conventional macroeconometric evidence.

Fractal and Fractional doi: 10.3390/fractalfract10110785

Authors: Seth Kermausuor Eze R. Nwaeze

Mathematical inequalities occupy a fundamental position in mathematical analysis, providing essential tools for estimating functions, establishing bounds for operators, and investigating the qualitative behavior of mathematical models [...]

Fractal and Fractional doi: 10.3390/fractalfract10110784

Authors: Lianting Sun Chuanzhi Cui Xiaopeng Cao Chuanfei Wang Dong Zhang Yingzhu Guan Zhongwei Wu

Accurate prediction of CO2–brine relative permeability is essential for evaluating CO2 migration and storage performance in saline aquifers. However, conventional empirical models rely heavily on fitted parameters, while existing fractal models generally provide limited descriptions of the effects of pore pressure and displacement rate on two-phase flow. To address these limitations, a pore-scale-based fractal relative permeability model is developed by coupling two-phase flow theory, fractal pore-size distribution, and numerical integration. The main innovation lies in incorporating pressure-dependent CO2 viscosity and a variable immobile water-film thickness governed by pore radius, driving pressure gradient, and brine viscosity into the effective flow radius. Compared with conventional models employing fixed water-film thickness or neglecting injection-condition effects, the proposed formulation provides a physically based representation of the responses of CO2–brine relative permeability to pore pressure and displacement rate. The model is evaluated using six displacement experiments conducted on four sandstone cores. The coefficients of determination (R2) range from 0.957 to 0.998 for the brine phase and from 0.869 to 0.955 for the CO2 phase, demonstrating good agreement between predictions and measurements. Sensitivity analyses indicate that increasing absolute permeability generally enhances the relative permeabilities of both phases. Increasing pore pressure from 10 to 55 MPa and displacement rate from 1 to 5.5 mL·min−1 generally increases two-phase relative permeability and maximum CO2 saturation while reducing residual brine saturation. The proposed model provides a physically informed approach for characterizing CO2–brine two-phase flow under varying injection conditions and supports relative permeability prediction for CO2 storage in water-wet sandstone saline aquifers.

Fractal and Fractional doi: 10.3390/fractalfract10110783

Authors: Jinrong Cao Jiang He Linming Dou Qiang Wang Xiaoliang Hu Xinyuan Tian

Quantifying crack evolution in ultra-thick hard strata is important for understanding mining-induced failure and energy release. Taking the 6306 working face at the Dongtan Coal Mine as a case study, physical similarity experiments and UDEC-Trigon simulations were combined with box-counting analysis to investigate crack fractal evolution and the influence of rock compressive strength. The numerical results were used to examine the correspondence between crack fractal dimensions and model released energy. The results show that localized cracks progressively propagate upward and laterally and coalesce into an extensive network, forming a failure region narrower at the top and wider at the base. The physical-model crack fractal dimension increases from 1.2386 to a maximum of 1.3910 and subsequently fluctuates slightly, reaching 1.3798. Tensile crack fractal dimensions increase progressively before exhibiting small fluctuations, whereas shear crack fractal dimensions follow a sequence of slow growth, rapid increase, pronounced decrease, and renewed increase. During the principal period of crack development, the maximum increases in tensile and shear crack fractal dimensions relative to the beginning of this period are 0.1861 and 0.1637, respectively. The corresponding increase in calculated released energy is 1.7126 × 1010 J, accounting for 60.32% of its total recorded net increase. Higher calibrated compressive strength suppresses early crack formation and restricts spatial propagation and coalescence, resulting in lower fractal dimensions. As failure progresses, tensile crack fractal dimensions under different strength conditions gradually converge, whereas shear crack fractal dimensions retain a greater dependence on strength. These findings characterize the distinct evolution of tensile and shear crack networks and their observed association with model energy release.

Fractal and Fractional doi: 10.3390/fractalfract10110782

Authors: Feng Xu Xiao-Jun Yang

This paper develops closed-form prices for binary call options in a time-inhomogeneous Gaussian diffusion whose effective volatility and covariance coefficients are motivated by a sub-mixed fractional Vasicek model with H∈(3/4,1). We derive the pricing PDE by delta hedging and establish the equivalent martingale measure, market completeness, and the Feynman–Kac representation within this diffusion. Applying the Mellin transform—with an explicit treatment of the admissible transform strips and of the regularisation of the discontinuous payoffs—we obtain explicit analytical formulae for both cash-or-nothing call (CONC) and asset-or-nothing call (AONC) options. The closed-form formulae are validated independently by a finite-difference solution of the pricing PDE and by Monte Carlo simulation of the equivalent diffusion: the relative errors are below 0.1% with 106 simulated paths, and degenerate cases recover the classical Black–Scholes binary option prices up to machine precision. Within this diffusion, sensitivity analysis shows that the fractional parameter H has a state-dependent but quantitatively modest impact on prices (within about ±1.2% in our benchmark experiments), the effect being strongest for out-of-the-money options.

Fractal and Fractional doi: 10.3390/fractalfract10110780

Authors: Sebastian Raubitzek Georg Goldenits Sebastian Schrittwieser Philip König Kevin Mallinger

Fractional optimization methods and fractal activation functions are two independent directions for improving neural network training. Fractional optimizers extend first-order optimization through fractional derivatives and memory effects, whereas fractal activations introduce multi-scale nonlinear representations based on self-similar Weierstrass- and Blancmange-type functions. Here, we investigate their interaction within a unified experimental framework. We evaluate fractional optimizer families on Ackley and Himmelblau benchmark surfaces, in standard form and with additive Weierstrass-type perturbations, and then in feed-forward neural networks with conventional and fractal activations on ten classification datasets. The comparison includes standard methods, Herrera-type local scaling optimizers, explicit and adaptive memory-based fractional optimizers, and other representative literature methods. Fractional optimization and fractal activations show useful but selective pairings. Herrera-type local fractional scaling performs well with selected fractal activations in network training, while explicit Grünwald–Letnikov memory is competitive on the controlled surfaces without being preferentially favoured by the perturbation: the unmodified baselines rank first on both additive variants. Adaptive memory improves plain memory substitution in several cases, supporting controlled fractional memory as a promising direction rather than a universal replacement.

Fractal and Fractional doi: 10.3390/fractalfract10110781

Authors: Chen Zhang Zhihan Shi Guangming Zhang

This paper investigates resilient leader-following consensus of singular fractional-order multi-agent systems subject to partial denial-of-service (DoS), sensor and actuator deception, measurement noise, and external disturbances. The descriptor structure requires regularity, impulse-freeness, and algebraic consistency in addition to fractional consensus stability. A descriptor-compatible augmented observer is first constructed to estimate the dynamic state and deception signals. A distributed compensation protocol is then synthesized over a finite family of normal and partial DoS topologies. To avoid reset-type switching arguments that are problematic for global-memory Caputo dynamics, the Caputo lower terminal is kept fixed and a common-storage condition is imposed for every admissible topology. After semi-explicit descriptor reduction, real-lifted fractional bounded-real inequalities provide a convex observer synthesis condition, a modal consensus-gain synthesis condition, and a post-design attenuation certificate for the observer-error channel. The resulting theorem guarantees admissibility and a prescribed network H∞ bound under arbitrary switching among the certified topologies. Simulations on a five-follower singular network include zero-initial-condition H∞ verification, attack estimation, modal-sector checks, parameter sensitivity, and structure-adapted comparisons with representative recent methods.

Fractal and Fractional doi: 10.3390/fractalfract10110779

Authors: Mohamed Aibout Khelifa Daoudi Abdeldjabar Bourega Jose Savarimuthu Prabakaran Raghavendran Khidir Shaib Mohamed

This research investigates the controllability of Hilfer fractional dynamical systems with state-dependent delays in Rn, including systems with a linear control component and nonlinear delayed dynamics. Using Mittag–Leffler matrix functions, a mild solution representation is established. For the system with a state-dependent delay and a linear control component, controllability is studied through the associated controllability Gramian and its invertibility. The Gramian’s invertibility alone does not establish controllability of the full system; the sufficient conditions also account for the state-dependent delayed term. For the nonlinear system, a sufficient condition for relative controllability is established using the Schauder fixed-point theorem, provided that the associated operator is well-defined and continuous; maps a suitable nonempty, closed, bounded, and convex set into itself; and has a relatively compact image, as assumed in Theorem 2. The theoretical results are illustrated through a nonlinear numerical example involving fractional dynamics, nonlinear delayed interactions, and a state-dependent delay. Furthermore, an artificial neural network framework is developed to approximate the resulting numerical state trajectories. Its performance is evaluated using independent test data, trajectory comparisons, absolute-error analysis, and regression metrics. The root mean squared error on the test data is 0.009669, while the coefficients of determination are 0.999961 for the first state component and 0.999121 for the second state component. These results demonstrate the effectiveness of the proposed analytical framework for investigating controllability and the artificial neural network approach for approximating computed trajectories of nonlinear Hilfer fractional systems with memory- and state-dependent delay effects.

Fractal and Fractional doi: 10.3390/fractalfract10110778

Authors: Guiyao Ke Jun Pan Haijun Wang

Although scholars have uncovered various dynamics of the integer-order four-dimensional Faraday disk dynamo, such as effects due to induced azimuthal eddy currents, unstable periodic orbits, and zero–zero–Hopf bifurcation, the global stability and attractive sets of this system, and especially those of its fractional-order counterpart, have received little attention. In this study, applying Lyapunov functions and the comparison principle, we establish the global stability of nontrivial equilibria and the existence of global Mittag–Leffler attractive sets (GMLASs) of a Caputo fractional-order four-dimensional Faraday disk dynamo, which are validated by several numerical examples.

Fractal and Fractional doi: 10.3390/fractalfract10110777

Authors: Bandar Bin-Mohsin

This article is devoted to the derivation of new inequalities within the theory of fractal–fractional calculus. Its principal contribution is the construction of a new auxiliary identity of fractal–fractional type. By exploiting this identity, we establish a variety of Bullen–Simpson-type inequalities associated with functions whose local fractional derivatives possess generalized harmonic convexity. Moreover, by employing the generalized Hölder and power mean inequalities, we derive supplementary bounds that further enhance the scope of the analysis. The accuracy of the theoretical developments is illustrated through a representative example, together with graphical visualizations that substantiate the consistency of the proposed results. Finally, an application to certain special fractal means is presented in order to demonstrate the effectiveness and practical relevance of the obtained inequalities.

Fractal and Fractional doi: 10.3390/fractalfract10110776

Authors: Nan Rao Zhixuan Jia Yuxiao Yang

Detrended fluctuation analysis (DFA) and its multifractal extension (MF-DFA) estimate scaling exponents from a fixed segmentation of the data. On a nonstationary series, however, a segment that crosses a transition of the local Hurst function H(t) mixes two scaling regimes and biases the estimate. We propose a local Hurst-guided segmentation for MF-DFA that leaves the classical fluctuation function and polynomial detrending unchanged and adapts only the grouping of the observations to H(t): a rolling DFA curve is estimated, its transitions are located by CUSUM or PELT, and the multifractal spectrum is estimated separately within each detected regime, with a boundary inset that discards observations of uncertain regime membership. When H(t) is constant the procedure reduces to standard MF-DFA. We prove that regime mixing induces a spurious multifractal width in global MF-DFA, and we bound the error that a misplaced transition propagates into the state-level h(q) and show that a boundary inset of the order of the detection error removes it. Simulations use multifractional Brownian motion with stepwise and continuously varying Hurst functions. The segmented estimators recover the state spectra and remove the spurious multifractal width that global MF-DFA attributes to the series as a whole. An application to the daily S&P 500 index over 1999–2026 completes the study.

Fractal and Fractional doi: 10.3390/fractalfract10110775

Authors: Xiaoqian Yuchi Xuanhong Du Yao Zhao Pan Shu Wenhua Yang

To address the challenge of intense pressure manifestation induced by remaining coal pillar from gently inclined multi-coal seam mining, a physical similarity simulation experiment was conducted using microseismic, thermal infrared, and borehole television monitoring. The overlying strata movement process, energy release characteristics, and fracture fractal evolution patterns were systematically investigated. The 3DEC numerical simulation was employed to evaluate the effectiveness of roof cutting measures. The results indicate that the overburden movement can be divided into three stages: formation of remaining coal pillar, suspension and fracturing of interlayer rocks, and subsidence and migration of overburden. The suspension of rock strata and the load transfer from uncollapsed strata are the key factors inducing dynamic disasters. Furthermore, the occurrence of disaster-causing energy events exhibited concentration and suddenness; only two IV high-energy events were recorded, accounting for 3.62% of the total energy, corresponding to the sudden fracturing of rock strata and the collapse of the entire overburden. A threshold of 250 J can serve as a criterion for identifying high-energy disaster events in similar models. Combined with the fractal dimension evolution process of fractures, it followed a pattern of growth, stabilization and renewed growth. After the upper-seam and W1121 working face were mined, the fractal dimension increased to 1.324. It remained within 1.372–1.377 during interburden suspension and reached 1.417 after interburden failure. These changes indicate that repeated multi-seam mining promotes upward fracture-zone propagation, potentially affecting the height of the water-conducting fracture zone. Finally, roof cutting was simulated in potentially hazardous coal-rock zones to evaluate pressure relief. Following pressure relief, suspension and hinging of fractured rock beams were largely eliminated. Overburden subsidence became more uniform, while lateral compression on one side of the panel coal pillar decreased markedly. Compaction within the goaf also became more uniform. These findings provide a theoretical basis for safe and efficient multi-seam mining and dynamic hazard control in mines with similar conditions.

Fractal and Fractional doi: 10.3390/fractalfract10110774

Authors: Tao Li Xuyuan Gao Zhoncheng Jiang Wei Liu Dengke Li Xuancheng Yuan Bowen Yao

Wheel polygonization is a typical out-of-roundness fault in high-speed trains, and multi-order roughness spectra can quantitatively describe its fault condition. Variations in operating speed, wheel diameter, and reprofiling state cause differences in axle-box vibration responses, making the diagnosis of multi-order roughness amplitudes difficult. A fractional-order-enhanced transform-domain multi-instance learning method with FiLM modulation is proposed for quantitative fault diagnosis of wheel polygonization. First, axle-box vibration acceleration, operating speed, and measured roughness data are jointly processed to construct multi-instance samples containing angular-domain time series, order-domain spectra, vibration statistics, and operating-condition information. Second, a model integrating transform-domain multi-instance learning and FiLM modulation is constructed, combining a state-adaptive multi-regression mechanism with order-aware correlation mapping to achieve quantitative diagnosis of roughness amplitudes across orders 1–32. Finally, the Grünwald–Letnikov fractional-order difference is introduced into the loss function to enhance the model’s ability to learn inter-order correlations and overall spectral-shape features, thereby reducing diagnostic errors in multi-order roughness amplitudes. Experimental results show that the proposed method achieves an RMSE of 1.157 dB, 60.5% lower than that of CNN-LSTM, the best-performing comparison method, providing a quantitative diagnostic method for high-speed train wheel condition assessment and reprofiling evaluation.

Fractal and Fractional doi: 10.3390/fractalfract10110773

Authors: Fikret Gölgeleyen Seher Seda Sarıköse Bayram Masahiro Yamamoto

In this article, we consider an initial-value problem for a linear ordinary differential equation with multi-term time fractional derivatives with relaxation factors, which make the current work differently from existing studies in the literature. We establish the unique existence and the stability of the solution to the problem within the framework of suitable Sobolev spaces. Moreover, we present an improved regularity result for the solution. Our treatment is based on the operator theory in Sobolev spaces.

Fractal and Fractional doi: 10.3390/fractalfract10110772

Authors: Hongzhu Li Wei Zheng Junzhe Jiang Shaobin Cai Weiyao Zhu Debin Kong

Waterflooding intensity strongly affects oil–water migration in deep-water sandstone reservoirs with bottom edge water, yet oil mobilization across different oil-occupied pore scales and the evolution of residual oil occurrence spaces under different waterflooding modes remain unclear. Constant high-rate and rate-increasing waterflooding experiments were conducted on deep-water sandstone cores. Online nuclear magnetic resonance (NMR) monitoring and fractal analysis were used to distinguish oil-occupied pore scale intervals from oil-phase T2 signals and quantify the evolution of oil saturation and fractal dimensions under different waterflooding intensities. In 0–100 pore volumes (PV), rate-increasing waterflooding yielded slightly lower recovery than constant high-rate waterflooding. During 100–1000 PV, however, its recovery increment reached 8.68–12.39 percentage points versus 2.91–6.01 percentage points under constant high-rate waterflooding, increasing final recovery by 5.25–6.17 percentage points. Oil mobilization was dominated by macropore and mesopore intervals, with limited micropore mobilization. In 100–1000 PV, rate-increasing waterflooding caused larger oil saturation reductions in macropore and mesopore intervals (4.43–10.60 and 1.75–3.99 percentage points) than constant high-rate waterflooding (3.05–4.42 and 0.01–1.30 percentage points). Fractal dimensions increased during waterflooding, with the largest changes occurring in the micropore interval. Fractal dimension increments in mesopore and macropore intervals were 0.09–0.12 and 0.04–0.06 under rate-increasing waterflooding versus 0.04–0.07 and 0.03–0.05 under constant high-rate waterflooding. The results provide core scale insight into the relationship between injection history, late-stage oil depletion, and the T2 distribution of remaining oil in deep-water sandstone.

Fractal and Fractional doi: 10.3390/fractalfract10110771

Authors: Golib Berdiev Shahzoda Anarova Navruz Khujakulov

This study develops a hybrid fractal–parametric modeling (HFPM) workflow integrating recursive affine iterated-function systems, finite-scale box-counting, fixed-thickness prism extrusion, and NSGA-II to explore trade-offs between geometric material volume and geometric complexity in generative architectural façades. A common 729 × 729 raster and four box-counting scales (3, 9, 27, and 81) are applied to both investigated recursion depths (r = 3 and r = 4). Across five independent NSGA-II runs, the pooled final non-dominated reference set yielded geometric material-volume reductions of 34.58% and 44.57% within 1% and 2% deviations, respectively, from the baseline finite-scale fractal dimension (Df). Mean hypervolume was 0.26182 ± 0.00113, evaluated in the original objective space against a common reference point (1.000000, 0.323566); mean inverted generational distance was 0.01732 ± 0.00244, evaluated using range-normalized distances to the empirical pooled reference front. Independent fixed-recursion optimization, simple branch-pruning controls, and density-seed, grid-origin, raster-resolution, and scale-window sensitivity tests further delimit these findings. Branch removal contributes substantially to the volume reductions, while the measured trade-offs and tolerance-feasible solutions depend on the chosen box-counting window. The results therefore establish a computational, protocol-dependent geometric trade-off for the investigated fixed-thickness prefactal prisms, not structural efficiency, equivalent façade functionality, fabrication feasibility, cost savings, or environmental benefits.

Fractal and Fractional doi: 10.3390/fractalfract10110770

Authors: Yiting Lin Xiyuan Cheng Yaokang Li Yunan Wei Wenbin Cheng

This paper presents a systematic structural cryptanalysis of a color-image cryptosystem based on finite-time projective synchronization in fractional-order inertial memristive neural networks. By examining the algebraic structure and plaintext–ciphertext dependencies of its encryption pipeline, several attack-relevant structural properties are identified, including plaintext-independent sequence reuse, algebraically redundant XOR operations, shared encryption material across RGB channels, and the absence of interpixel diffusion. Under a fixed secret configuration and a bijective spatial disruption map, the complete encryption process is shown to reduce to Cc(i)=Pc(π(i))⊕K(i), where π denotes a reusable spatial permutation and K is an equivalent XOR mask obtained by combining the two nominal diffusion sequences. At the image layer, this representation shows that encryption is determined by the induced permutation and equivalent mask. Exploiting these properties, a chosen-plaintext attack is developed to recover the equivalent mask using an all-zero image and reconstruct the spatial permutation through base-256 position encoding. The implemented two-query construction supports up to 224 pixels, whereas an optimal two-query design supports up to 240 pixels; one query suffices for at most 216 pixels. A differential chosen-plaintext attack further recovers the equivalent key by eliminating the mask through related plaintext differences. In addition, a known-plaintext attack reconstructs the permutation using mask-invariant positional fingerprints under explicit uniqueness conditions, while chosen-ciphertext attacks enable direct plaintext recovery with one nonchallenge query or equivalent-key recovery with two queries. The two-query CPA is also evaluated through an encryption-oracle-only interface. Experimental results on 512×512 RGB images confirm exact equivalent-key recovery and lossless plaintext reconstruction. All eight tested image cases yield exact recovery under the fixed context, with seven repeated runs per case used to summarize timing, and six heterogeneous known plaintext–ciphertext pairs suffice for exact recovery in the principal experiment. In general, known-plaintext recovery identifies the permutation when the positional fingerprints are unique. Finally, an invertible, module-wise proof-of-concept design is specified for context-dependent sequence generation, cross-channel coupling, bijective spatial disruption, and independent bidirectional feedback diffusion; its relation to the attacks analyzed in this paper is assessed structurally.

Fractal and Fractional doi: 10.3390/fractalfract10110769

Authors: Leyang Li Xingping Lai Yuchen Hou Pengfei Shan Huicong Xu Kai Li Zhaohang Li Cheng Lu Yonghang Cai Huien Li

Microseismic monitoring is a crucial technical approach for deep underground engineering and the early warning of dynamic disasters. However, the high-precision automatic picking of microseismic P-wave first arrivals under strong non-stationary background noise remains a critical challenge in this field. To address the limitations of conventional time-frequency denoising methods, in which hard thresholding may oversmooth the onset edges of first arrivals and consequently cause STA/LTA or global AIC algorithms to be trapped by noise-induced local extrema, this paper proposes a novel automatic P-wave picking method. This method relies on the synchrosqueezing transform (SST) and local box-counting dimension for denoising, integrating permutation entropy with the Akaike Information Criterion (AIC). During the denoising stage, a dual-discrimination mechanism combines SST-based time-frequency component selection with local box-counting dimension analysis. Initially, candidate high-energy time-frequency points are selected from the SST representation using adaptive thresholds, while the remaining components are attenuated to reduce diffuse background noise. Subsequently, the local box-counting dimension of the continuous time-frequency plane is calculated to characterize the differences in the time-frequency fractal structure of the signal. Based on this, an asymmetric fractal gain operator is constructed to suppress noise disturbances while providing directional energy compensation for the first-arrival jump edges, thereby mitigating the smoothing distortion of weak waveforms during the denoising process. In the picking stage, the permutation entropy and variance features of the local waveforms are extracted to formulate a joint feature function for locating candidate onset regions. This is combined with a sliding AIC to perform a localized fine search, overcoming the vulnerability of traditional global searches to noise extrema interference and achieving high-precision localization of P-wave first arrivals. Verification using real mine data demonstrates that the proposed method significantly enhances signal quality under complex background noise conditions while effectively preserving the initial onset characteristics of the seismic phases. Under simulated input SNRs of 20, 10, and 0 dB, the proposed method achieves SNR gains of 17.22, 15.98, and 10.54 dB, respectively, corresponding to a mean gain of 14.58. The method also yields the lowest RMSE among the compared denoising methods at input SNRs of 10 and 0 dB. The research results indicate that this method can provide reliable technical support for the accurate picking and source localization of microseismic events in highly noisy environments.

Fractal and Fractional doi: 10.3390/fractalfract10110768

Authors: Nikolay Hinov Plamen Stanchev

This study evaluates whether fractional-order shaping provides a meaningful performance advantage for robust current control of a frequency-controlled parallel-resonant inverter when compared under identical conditions with strongly tuned integer-order controllers. A nested fractional extension of an integral lead–lag regulator is implemented using an explicit integrator and a finite-band Oustaloup approximation. Three controller parameterisations are optimised using matched differential-evolution searches with five independent starts and 6000 objective evaluations per start. An exact rational two-plant certificate establishes a necessary lower bound of 0.02885438 for the inherited continuous-time objective. The re-tuned integer controller achieves a worst observed gain of 0.02885823, whereas the selected fractional controller reaches 0.02890450 at λ = 1.000448. The selected order therefore lies effectively at the integer limit, and the benchmark leaves very limited numerical headroom for improvement through fractional shaping. All five evaluated controllers settle successfully across 46 branch-specific cycle-synchronous simulations using a common actual-cycle zero-order-hold implementation. With two additional command-delay cycles, two local augmented maps become unstable for both the selected fractional controller and the re-tuned integer controller. Approximation-order analysis and controller-inclusive Jacobians further distinguish measured-output settling from slowly decaying internal memory. The results therefore identify benchmark-specific limits of fractional shaping rather than a general limitation of fractional-order control.

Fractal and Fractional doi: 10.3390/fractalfract10110767

Authors: Adel Emam S. M. Khaled A. El-Dali Zaki Mrzog Alaofi

A two-dimensional causal benchmark is developed for a normally incident finite-width Gaussian beam that jointly generates Beer–Lambert carrier and heat sources in a fractional nonlocal photo-thermoelastic semiconductor. Colored amplitude noise and beam-centre jitter are treated as correlated properties of the same optical field. Fourier transformation of the physical transverse profile generates the continuum of transverse wavenumbers, while unilateral Laplace transformation retains the frequency dependence of the fractional heat law. The two-input spectral matrix is propagated through a seven-output transfer operator for T, N, u, v, σxx, σxy, and σyy. Independent time-domain verification uses correlated Ornstein–Uhlenbeck paths generated by their exact discrete transition and filtered through the full frequency-dependent transfer, rather than sampled from the analytical output covariance. This test exposed and corrected a conjugation-order error in the correlated cross-spectral projection. After correction, 256 paths containing 12,582,912 retained samples reproduce every field variance within 1.274% and every selected cross-field correlation within 0.00365. A 180 × 101 frequency-wavenumber grid differs from a 260 × 161 reference by at most 0.944% in variance and 0.00226 in correlation. Exact signed attribution shows constructive amplitude-pointing interaction for five fields but destructive interaction for v and σxy. The innovation-correlation sweep is affine to machine precision. The work is positioned as a dimensionless verification and mechanism study, not a calibrated device reliability prediction.

Fractal and Fractional doi: 10.3390/fractalfract10110766

Authors: Xuan Cui Shengli Yang Chunyu Duan Kai Wang Huiying Zhou Aoxiang Wang

Rock–coal–rock (RCR) composite structures in inclined coal seams exhibit asymmetric deformation and inclination-dependent failure. This study investigates how coal–rock interface inclination affects mechanical response and fracture development using uniaxial compression tests, CT imaging, three-dimensional reconstruction, and multifractal analysis. As the inclination increased from 0° to 40°, load-bearing capacity decreased and the dominant failure mode changed from splitting through mixed tensile–shear failure to interface-controlled shear failure. Crack volume and surface area decreased with inclination; the crack-to-rock/coal volume ratio was 0.04 at 10°, while the crack specific surface area was 4.345 and reached a maximum of 4.432 at 40°. A pseudo-color enhancement workflow improved the identification of fine cracks that were difficult to capture by conventional binary processing. Multifractal parameters quantified crack-network complexity and revealed pronounced anisotropy among specimens and spatial heterogeneity within each specimen. Coal-bearing regions generally sustained greater damage, whereas rock regions exhibited more heterogeneous crack distributions. Correlation analysis linked the mechanical response to fracture geometry and multifractal characteristics, showing that interface inclination progressively promotes weak-plane-controlled instability. These findings clarify the fracture mechanism of inclined RCR composites and provide a basis for stability assessment in inclined coal-seam mining.

Fractal and Fractional doi: 10.3390/fractalfract10110765

Authors: Tarek Aboelenen Ahmed S. Almohaimeed Tarek M. Abed-Elhameed

This paper investigates stability and complete synchronization of distributed-order fractional nonlinear dynamical systems (DONDSs) with a constant time delay. For the continuous Caputo distributed-order model, a Laplace-transform resolvent approach gives the sufficient incremental small-gain condition qw=(L+M)∥Kw∥L1<1 under stated well-posedness and resolvent assumptions. The resolvent requirements are verified for scalar damping A=aI, for which the criterion reduces to L+M<a, and for symmetric positive-definite A, for which L+M<λmin(A) is sufficient; well-posedness remains a separate hypothesis. For the finite multi-term model, the corresponding condition is q=(L+M)∥Km∥L1<1. A conditional history-space generalized Wright Lyapunov criterion is also established, with decay described through a scalar multi-term relaxation function and its generalized Wright series representation subject to additional inversion conditions. We develop an Adams product integration predictor–corrector method, examine contraction of its endpoint corrector, and compare it with backward-Euler convolution quadrature and a fully implicit Adams endpoint. Finite multi-term Liu and financial systems illustrate the framework, accompanied by numerical studies of order distributions, delays, control gains, perturbations, and parameter interactions. Finally, an active control construction cancels nonlinear and delayed mismatch terms under ideal full state and exact model assumptions, leaving a stable linear distributed-order synchronization error system.

Fractal and Fractional doi: 10.3390/fractalfract10110764

Authors: Luyun Sun Benda Zhou

Regional inequality is commonly measured with indices (e.g., the Gini and Theil indices) that depend only on the multiset of values and are therefore permutation-invariant: they cannot distinguish configurations differing solely in how the same values are ordered. This paper formalises that limitation as an information boundary and shows that the multifractal spectrum width (Δα) carries qualitatively distinct information. Four propositions are established: (1) with group membership fixed, the Gini, Theil, generalised entropy and Dagum Gini indices are permutation-invariant; (2) Δα is not, and can move in either direction under permutation; (3) Δα still varies appreciably when permutation-invariant indices are held constant, so the two classes are not informationally equivalent; and (4) controlled synthetic fields serve as a proof of concept linking Δα to multiscale heterogeneity. Permutation invariance secures the objectivity and comparability of inequality indices, but it also delineates their information boundary: Δα carries structural information that these indices cannot, in principle, capture, making the two complementary rather than substitutable. A finite-sample baseline and the Multifractal Magnitude Index (MMI) yield an operational diagnostic protocol; because Δα is defined only relative to the ordering rule and the MF-DFA parameterisation, all comparisons are made within a single parameterisation.

Fractal and Fractional doi: 10.3390/fractalfract10110763

Authors: Bibo Zheng Yunqiu Xu Wei Dai

This paper is mainly concentrated on finite-time synchronization (FTS) for discontinuous fractional-order multi-layer coupled neural networks (FMCNNs) under aperiodic intermittent control. The discontinuous activation function is introduced to capture abrupt transitions and switching phenomena widely observed in practical neural dynamics. However, such nonsmooth behaviors significantly increase the complexity of controller design and synchronization analysis. To cope with these problems, an aperiodic intermittent controller with a simple structure is developed within the Filippov differential inclusion framework, ensuring effective handling of nonsmooth neural dynamics while reducing control cost. More importantly, a piecewise fractional-order differential inequality is established to characterize the evolution of the Lyapunov function over the control and rest intervals under aperiodic switching. This inequality serves as a fundamental analytical tool for deriving sufficient conditions for FTS and estimating the settling time. Compared with the existing results, the proposed framework relaxes the restrictions on the power exponent and provides a more general analytical tool for investigating the FTS of discontinuous FMLCNNs. In the end, a simulation example is provided to show the validity of the proposed method.

Fractal and Fractional doi: 10.3390/fractalfract10110762

Authors: Tingting Dai Haiying Wang Yuan Zhou Qingyuan Wang Shaohui Wang

We analyze a campaign-renewed Caputo model of HIV transmission on heterogeneous contact networks, with risk-triggered screening and antiretroviral-treatment linkage. The model has a spectral next-generation threshold and an explicit positive matrix Mittag–Leffler jump–flow operator. Positivity, global extinction under a positive linear resolvent comparison, and conditional robust hybrid-contraction theorems are established without assuming that the nonlinear infected–treated subsystem is cooperative. A common weighted norm and watchdog-bounded intervention intervals are required; these assertions do not apply unmodified to a fixed-history Caputo system. An independent reconstruction of the complete 80-dimensional cycle matrix gives spectral radius 1.0688 for the stated nominal baseline (α=0.85, one-year cycle), correcting an earlier noncontractive setting that had been described as stabilizing. A separate six-mode intervention family admits a common weighted-norm bound below 0.9981. Newly executed nonlinear 20-year experiments, including mandatory watchdog events, report actual campaign counts, ART transfers and burden under single-factor and nominal-budget-matched comparisons; a periodic strategy can outperform risk-triggered intervention at the same total nominal pulse budget. Three seeded 100-graph synthetic ensembles clarify structural variability but do not certify arbitrary switching. Beijing surveillance counts from 2005 to 2020 are independently redrawn with one missing AIDS observation marked explicitly; these data do not empirically identify the fractional order, contact network or controller.

Fractal and Fractional doi: 10.3390/fractalfract10110761

Authors: Xinzhu Ma Tengyu Ma Longsuo Li

The propagation of Rayleigh waves on the liquid-saturated poroviscoelastic half-space, considering fractional viscoelasticity and squirt-flow (BISQ) effects, is investigated using the surface stiffness matrix method. First, the state transfer equation of a Rayleigh wave on the fractional liquid-saturated poroviscoelastic material is derived from the governing equations and constitutive relations that account for fractional viscoelasticity and squirt-flow effects. Then, the transfer matrix with fractional and squirt-flow factors and the stiffness matrix of the state vector are obtained by solving the state transfer equation of a Rayleigh wave, and, subsequently, the surface stiffness matrix of a Rayleigh wave on the fractional BISQ poroviscoelastic half-space is derived. Finally, the application of mechanically traction-free conditions, as well as permeable and impermeable conditions, yields the frequency dispersive relation of the Rayleigh wave. The dependences of the Rayleigh wave speeds on the characteristic squirt-flow length and the fractional viscoelasticity of the solid and liquid phases are numerically discussed. It is observed that the Rayleigh wave speed is influenced not only by the viscoelasticity but also by the squirt-flow length. This work provides a stable numerical implementation for fractional BISQ Rayleigh wave problems, which suffer from high-frequency instability when solved by conventional transfer matrix approaches.

Fractal and Fractional doi: 10.3390/fractalfract10110760

Authors: Ines Ben Omrane Abdulmajeed Alqasem Mourad Ben Slimane Hamdi Zorgati

We investigate pointwise rectangular anisotropic and directional densities for Borel probability measures with compact support. Contrary to traditional densities, these new notions have the advantage to weight the side lengths of local boxes differently and take into account long thin boxes. We employ similarities with statistical physics to extract interesting multifractal information using the Hausdorff dimension with respect to the coverings by rectangles with small enough Lebesgue measure (instead of requiring all side lengths to be small). We apply our analysis for invariant measures for some iterated function systems.

Fractal and Fractional doi: 10.3390/fractalfract10110759

Authors: Shiyou Lin Junying Chen Aijie Yang

We study the variable-order fractional abstract Cauchy problem D0α(t)Su(t)=Au(t)+f(t), u(0)=u0, in a Banach space X, where A is sectorial and α(t):[0,∞)→(0,1) is a Scarpi transition function. Using the Sonine-pair structure of the Scarpi kernels, we convert the problem into a Volterra integral equation and, adapting the resolvent-family framework of the constant-order theory to this variable-order setting, introduce a variable-order resolvent family through Laplace transform techniques. In contrast to the constant-order case, the variable-order resolvent family is strongly continuous and exponentially bounded, but no analyticity in t is claimed or used. For exponentially bounded transitions α(t)=α2+(α1−α2)e−ct, the factorization sq(s)=sα1·s(α2−α1)c/(s+c) links the variable-order resolvent to the constant-order α1-resolvent. We prove the existence of this resolvent family, establish the existence of mild solutions by direct construction via the variable-order resolvent family and Laplace transform methods, prove the uniqueness in the class of strong solutions of the Volterra integral equation via a duality argument, and derive continuous dependence on the data. To the best of our knowledge, this provides the first complete well-posedness theory for inhomogeneous Scarpi-type variable-order abstract Cauchy problems in general Banach spaces. Two examples (A=−λI and A=Δ) illustrate the theory; the latter yields well-posed variable-order subdiffusion equations, of which a numerical illustration is provided at the end of the paper.

Fractal and Fractional doi: 10.3390/fractalfract10110758

Authors: Alberto Ballesteros-Rodríguez Lino González-García Miguel-Angel Sicilia

Spectral width, the standard multifractal measure of market complexity, is established for actively traded sustainable assets such as carbon allowances and green bonds. Tokenized green assets trade thinly, and we calibrate how far the measure can be trusted, using KlimaDAO’s carbon tokens, whose prices we rebuild from on-chain swaps so that inactivity is observed rather than inferred. Four problems appear. Filling empty intervals with the last price widens the spectrum when the market is unchanged, and the part driven by large fluctuations less so. A test on overlapping rolling windows reports differences that are not there too often. Aggregator prices make the carbon tokens look more different than their own trades show. And the execution price of a swap, which carries the fee, narrows the spectrum of a deep pool. Under these controls the four carbon tokens have wider spectra than wrapped Bitcoin and Ether rebuilt from the same exchanges, a difference so few units cannot establish at conventional levels, and no wider than conventional tokens that trade as thinly. Among conventional tokens, width falls with the market capitalisation of the asset. Whether the pools’ quality tiers differ remains undetermined, since no two tiers were liquid over the same period.

Fractal and Fractional doi: 10.3390/fractalfract10110757

Authors: S. M. M. Rasouli

We investigate the dynamics of the early universe within an emergent fractional cosmological framework generated by a measure-deformed Newtonian action, without introducing additional propagating degrees of freedom. Considering asymptotic and dynamical requirements, we construct a minimal fractional potential that, together with the time-dependent fractional kernel, generates a regular pre-inflationary stage, quasi-de Sitter inflation, a dynamically controlled exit, and a subsequent finite duration radiation-like phase. The exact solution of the reduced early-time equation is obtained in two equivalent representations, and regularity selects a unique physical branch with finite principal curvature invariants at the initial boundary. Remarkably, the same exact trajectory evolves asymptotically toward the quasi-de Sitter regime, while an exact nonlinear analysis of homogeneous Hubble deviations establishes its asymptotic attractor character within the admissible expanding sector. Inflation ends dynamically near the zero crossing of the effective fractional force, leading to controlled background-level consistency relations among the inflationary duration, the fractional parameter, the effective equation of state parameter, and the exit scale. For the physically relevant post-inflationary regime, the model admits the characteristic radiation-like expansion over a finite post-inflationary interval. These results show that a coherent sequence of regular pre-inflationary, inflationary, and post-inflationary background dynamics can emerge from a simple and unified fractional Newtonian framework without invoking an inflaton field.

Fractal and Fractional doi: 10.3390/fractalfract10110756

Authors: Yue Yu Yifei Pan Changgang Li Shuolin Zhang Jingyi Zhang Yan Zhang Libo Zhang

Fractal geometry offers a natural language for the self-similarity and scale invariance of power networks, yet future receiving-end grids shaped by high renewable penetration, high load density and large HVDC infeed have no historical counterpart. This paper formalizes the synthesis of such grids as a Constrained Fractal Iteration System (CFIS), in which synthetic topology is generated by the scale-progressive composition of three fractal operators, each coupling a spatial clustering/topology operator with an engineering constraint-projection operator. A hybrid metaheuristic realizes the operators through dynamic-weighted K-means (DWK-Means), social-learning particle swarm optimization (SL-PSO) and constraint-embedded 2-opt (CE-2-opt). On a provincial receiving-end grid at 162, 234, and 363 GW, the generated topologies are scale-invariant over a well-defined box-size range: the box-counting dimension rises from 1.675 to 1.800 with load and shows a clear voltage-layered hierarchy; the constrained dimension remains within 1.5% of the unconstrained value, showing that engineering feasibility costs almost no fractal quality. The 500 kV station set attains a correlation dimension of 1.81 versus 1.69 for the real existing grid, consistent with the interpretation of the synthetic grid as an idealized fractal upper bound. Short-circuit levels are brought within the 63 kA limit after correction, and renewable curtailment and load shedding remain zero; N-1 screening of all elements shows that the worst post-contingency loading falls from 98.5% at 162 GW to 91.4% at 363 GW, satisfying the thermal limits under single contingencies. Against classical topology generators built on the same node set, the proposed backbone attains full degree-two redundancy and a comparable fractal dimension with about 24% less line length, and an ablation of the redundancy bound shows that it, rather than the short-circuit limit, governs steady-state security at the low load scale.

Fractal and Fractional doi: 10.3390/fractalfract10110755

Authors: Bao Chen Tianlan Xiong Juanjuan Gao Yuchao Tang

Low-light image enhancement is an important problem in image processing. In this paper, we propose a novel adaptive variational model based on fractional-order gradients for low-light image enhancement. Specifically, the low-light part of the low-light image is enhanced, that is, the contrast of the low-light image is enhanced. The core concept of the proposed technique involves partitioning the image domain into low-light and brighter regions by applying an optimal threshold value. Then, a fitting term is further used for the low-light parts, so that the pixels in the low-light area can be enhanced by adjusting the contrast and brightness parameters. The existence of a minimizer is established under the stated assumptions, and the uniqueness of the solution is proved under the corresponding convexity conditions. An efficient split Bregman method is developed to solve the proposed variational model. Experimental results demonstrate the effectiveness of the proposed method for low-light image enhancement.

Fractal and Fractional doi: 10.3390/fractalfract10110754

Authors: Emtnan E. Al-Humaidan Saleh F. Aljurbua Tarek Aboelenen

We develop a fractional pseudo-spectral collocation method for steady and time-dependent fractional differential equations on the half-line, which are based on previously introduced generalized rational Jacobi functions (GJRFs). Rational cardinal functions are constructed at mapped Jacobi–Gauss points, yielding explicit nodal differentiation matrices for left-sided Riemann–Liouville and right-sided Weyl operators, including higher-order derivatives. A boundary-adapted second-kind construction enforces the homogeneous condition at the finite endpoint. For the representative left-sided order-μ matrix with 0<μ<1, we make explicit its diagonal equivalence to a Jacobi-weighted fractional Lagrange matrix on [−1,1]; in GJRF modal coordinates, this gives a finite-N factorization, determinant identity, inverse bound, and a conditional consistency-to-solution nodal error estimate. These transferred finite-dimensional structural results do not constitute a uniform-in-N convergence theorem. The method is applied to fractional advection and diffusion equations, multi-term problems, Burgers-type equations, and Schrödinger-type equations with unilateral fractional spatial operators. A Jacobi-parameter sensitivity study shows that matching the basis parameter to the endpoint behavior of the solution is essential for rapid convergence; with this choice, the linear and nonlinear tests exhibit rapid error reduction across the tested fractional orders. Temporal refinement recovers the expected L1 convergence rates, and matched-degree-of-freedom comparisons with GJRF Petrov–Galerkin discretizations are also reported. The results demonstrate the effectiveness of the GJRF collocation framework for the half-line problems considered.

Fractal and Fractional doi: 10.3390/fractalfract10110753

Authors: Jeferson Andrés Garzón-González Oscar Martínez-Fuentes Juan Manuel Torres-Arce Omar Guillén-Fernández Esteban Tlelo-Cuautle Jose de Jesus Rangel-Magdaleno

Fractional differential equations and fractional-order chaotic systems are attractive for secure communications, but the nonlocal memory of the Caputo derivative makes their digital implementation costly, so iterative methods are preferred for developing digital implementations of physical systems. This paper characterizes seven members of the Lorenz family (Lorenz, Chen, Burke–Shaw, Liu, Shimizu–Morioka, Lü, and Yang) and implements them on an FPGA using the short-memory principle. Starting from rectangular (REC) and predictor–corrector (PC) product-integration rules, we derive incremental iterative forms with a finite memory window suited to digital hardware. Largest Lyapunov exponent maps and bifurcation diagrams over fractional order and memory duration identify chaotic regimes for the Caputo case that, to our knowledge, have not been reported; for most systems, they stabilize once the window exceeds about 2 s. On a Cyclone V FPGA with a common Q1.11.20 fixed-point format, each architecture lets you select any of the seven systems and a memory depth of 1–1000 samples without resynthesis. For a 2 s window, REC uses fewer resources (3258 ALMs, 60 DSP blocks), whereas PC, thanks to its larger time step, reaches a higher estimated state-generation rate (330,457 versus 291,839 state vectors/s). Finally, 8-bit pseudorandom sequences extracted from the Lorenz system (1 Gbit per architecture) pass NIST SP 800-22 and the TestU01 Rabbit, Alphabit, and BlockAlphabit batteries. In addition, we present numerical simulations and their digital implementation to validate the numerical schemes.

Fractal and Fractional doi: 10.3390/fractalfract10110752

Authors: Kareem T. Elgindy

Accurate evaluation of the Caputo fractional derivative (CFD) of arbitrary non-integer order is essential for modeling memory-dependent phenomena. However, the classical Shifted Gegenbauer Pseudospectral (SGPS) method attains exponential convergence only on a zero-measure set of the fractional order parameter; for all other orders, the transformed integrand exhibits reduced endpoint regularity, limiting quadrature convergence to algebraic rates. We introduce a ϕ-generalized algebraic transformation that enforces Cr endpoint regularity for any prescribed integer r>0, and establish a convergence hierarchy for the interpolation-induced truncation error (ITE) across analytic, Gevrey, C∞, and finitely smooth data. For data analytic in a disc around the evaluation point, the transformed integrand has a convergent fractional power series with geometrically decaying coefficients, which explains the fast pre-asymptotic quadrature convergence observed numerically. Precomputable weighted Fractional Shifted Gegenbauer Integration Matrices (FSGIMs), assembled by a batched formulation, reduce each online evaluation to one O(Mn) matrix–vector product whose cost does not depend on the history length, apart from the retained sample buffer. Double-precision MATLAB benchmarks give excellent accuracies for problems with known solutions and improve on several finite-difference, diffusive-representation, and adaptive-quadrature baselines by orders of magnitude. Microsecond-scale latency on a Cortex-M4-class microcontroller is projected from operation counts and desktop timings; it has not been measured on hardware. A two-cycle NASA battery case study illustrates the feasibility of CFD-based State-of-Health features.

Fractal and Fractional doi: 10.3390/fractalfract10110751

Authors: İrem Çay İrem Bağlan Ḣüseyin Budak Erman Aslan

The present work focuses on both the analytical solvability and numerical treatment of a nonlinear time-fractional pseudo-parabolic equation subject to periodic boundary conditions. Rather than imposing the standard global Lipschitz requirement on the nonlinear source term, which is prevalent in the existing literature, the existence and uniqueness results are established by assuming a novel convexity hypothesis. The periodic structure of the spatial domain is incorporated through a generalized Fourier representation, which transforms the original problem into a system of time-fractional equations for the corresponding Fourier coefficients, permitting the analysis to be carried out in an appropriate Banach space. For the numerical investigation, the Caputo derivative with respect to time is discretized by means of the L1 scheme, while a finite difference approximation is employed for the spatial derivatives. Computational experiments demonstrate that the maximum absolute errors are successfully reduced to the order of 10−6, and the calculated numerical convergence rates closely match the theoretical expectation of O(τ2−α) across different fractional orders. These robust numerical results confirm that the proposed scheme faithfully reflects the qualitative properties derived from the theoretical analysis, validating the effectiveness of the convexity-driven analytical approach.

Fractal and Fractional doi: 10.3390/fractalfract10110750

Authors: Nan Wang Rui Meng Fazhan Tao Ji Zhang

This paper investigates adaptive predefined-time and prescribed-accuracy output-feedback control for integer-order strict-feedback nonlinear systems subject to asymmetric dead-zone actuation and unavailable states. Although the controlled plant is integer order, the proposed feedback mechanism employs complementary noninteger powers below and above unity. The resulting dynamic fractional-power feedback loop assigns the convergence-time bound through a single time parameter while maintaining a prescribed residual tracking band. Instead of requiring the exact inverse of the actuator characteristic, the asymmetric dead zone is decomposed into a sign-dependent positive gain and a bounded offset that are incorporated directly into the control recursion. A neural state observer reconstructs the unavailable states, radial basis function neural networks approximate the unknown nonlinearities, and command-filtered backstepping avoids the repeated differentiation of virtual controls. Lyapunov analysis proves that all closed-loop signals remain bounded and that the tracking error enters the prescribed accuracy set within the predefined-time bound. Simulations involving a benchmark nonlinear plant and a single-link manipulator demonstrate the effectiveness of the fractional-power feedback, state-estimation, and dead-zone compensation mechanisms.

Fractal and Fractional doi: 10.3390/fractalfract10110749

Authors: Tingting Li Shuangxia Lu Youming Guo

COVID-19 and influenza may produce overlapping waves, yet an overall least-squares fit need not represent their high-activity weeks well. We compare implemented integer-order, Caputo, and Atangana–Baleanu–Caputo (ABC) discrete schemes for a common ten-compartment transition structure with seasonal transmission and external influenza input. The schemes are calibrated with ordinary mean-normalized weighted least squares (J1) and a peak-weighted objective (J2). We fit the first 66 of 83 weekly observation pairs and assess the remaining 17 weeks. ABC gives the smallest training peak root-mean-square error for both series under either objective. Within ABC, J2 reduces COVID-19 and influenza training peak errors by 9.50% and 42.98%, respectively, relative to its J1 fit. Integer order achieves the smallest ordinary loss and lower temporal validation error. The autonomous disease-free and conditional invasion thresholds characterize the underlying unforced vector field, not the fitted seasonal system. These findings support peak-weighted calibration for retrospective high-activity reconstruction in these data, but they do not establish superior future-peak prediction.

Fractal and Fractional doi: 10.3390/fractalfract10110748

Authors: Claudio Urrea

Kinematic redundancy and fractional-order sliding-mode control (FOSMC) are seldom combined in fault-tolerant manipulation. Here, they are combined on a rail-mounted SCARA arm subject to actuator loss of effectiveness (LOE) and elbow locking. The sliding surface contains a Riemann–Liouville integral of order λ∈(0,1], with λ=1 giving the integral sliding mode (ISMC). Ultimate boundedness is proved under LOE, a post-lock feasibility condition yields a fault-aware redundancy resolution rule, and the short-memory Grünwald–Letnikov implementation is shown to be an exact convolution with a truncated kernel for which the sliding dynamics are stable for any memory length. With jointly tuned gains, a common activity budget, and no added lag or delay, the accuracy-tuned FOSMC-J lowers the tracking error by 49% without faults and by 67% under LOE relative to the best ISMC (20 runs per case; 1.5–2.5 Hz disturbance). However, with an actuator lag of 2 ms, FOSMC-J failed in 9 of 10 lock runs. The moderate-gain FOSMC never failed with 2–4-ms lag, but tolerated less delay than the equal-gain ISMC. Under the lock condition, moderate-gain controllers of both orders always recovered, while high-gain controllers failed in about a third of 100 runs, with a violated margin (integer) or deep saturation (fractional). Moderate gains at reconfiguration removed almost all of the failures.

Fractal and Fractional doi: 10.3390/fractalfract10110747

Authors: Enrico De Santis Alessio Martino Antonello Rizzi

Multifractal analysis has evolved from a formalism for heterogeneous scaling into a broader framework for investigating structure, dynamics, representation, computation, and learning systems. This critical integrative review synthesizes methodological foundations, evidence criteria, complex-system applications, network methods, computational uses, and multifractal analyses of learning systems. Particular attention is given to apparent versus supported multifractality and to finite size, temporal correlations, heavy-tailed distributions, estimator choices, surrogate testing, and mechanistic interpretation. To compare heterogeneous studies, we organize the literature through a three-dimensional framework in which locus identifies where multifractality is measured, role specifies how the resulting quantities are used, and evidence characterizes the support for their interpretation. The framework clarifies how multifractal information functions across description, diagnosis, representation, adaptive computation, trainable operators, and the analysis of learning systems. We further distinguish complexity in data, models, input-conditioned hidden representations, and outputs, identifying direct multifractal analysis of hidden representations as comparatively underexplored. Across domains, multifractal claims are treated as object-dependent and tied to the support, measure, scale, and estimator used to construct the analyzed object. Future progress will depend on stronger finite-size controls, cross-method comparability, causal validation, and explicit definitions of analytical objects.

Fractal and Fractional doi: 10.3390/fractalfract10110746

Authors: Aleksei Borodulin Andrei Gantimurov Vladimir Nelyub Vadim Tynchenko Ivan Malashin

Selective machine unlearning aims to suppress targeted information while preserving retained and general knowledge. We investigate valence-aware gradient memory within low-rank adaptation (LoRA), separating retention-supporting and forgetting-related directions. Earlier experiments used Gamma-scaled exponential moving averages rather than genuine fractional dynamics; we preserve that distinction and introduce a normalized finite-history Grünwald–Letnikov (GL) fractional-integral accumulator with algebraically decaying coefficients. The study progresses from synthetic transformer optimization and pretrained text classification to causal language model experiments, including a new controlled evaluation on TOFU Forget10 using Qwen2.5-0.5B-Instruct. A SciQ- and WikiText-regularized teacher reduces reference-answer NLL on original and paraphrased TOFU questions while exhibiting a small World Facts penalty in a separate audit. From this fixed teacher, four unlearning methods were compared for 120 updates over three random seeds. Dual GL yields the largest paraphrased forget-set NLL increase (2.169±0.331), exceeding dual EMA (1.964±0.250), but incurs greater retain-set (1.043±0.207 versus 0.914±0.147) and World Facts (1.333±0.645 versus 1.153±0.567) degradation. Exact paired sign-flip tests do not establish statistical significance (p=0.25 for GL versus EMA). The stronger suppression achieved with fractional memory is accompanied by greater disruption of retained and general factual knowledge, suggesting a shift in model behavior rather than more selective forgetting. Whether the targeted information becomes genuinely inaccessible rather than merely less likely to be reproduced remains unresolved.

Fractal and Fractional doi: 10.3390/fractalfract10110745

Authors: N’Gbo N’Gbo Andrei Velichko Bruno Carpentieri

We study an equation-free screen for finite-window Mittag–Leffler compatibility, in which multi-horizon relaxation profiles are constructed out of sample by k-nearest-neighbor prediction from ensembles of scalar trajectories and compared with candidate temporal laws. The screen assesses compatibility, not the generating mechanism. In a matched, noise-free calibration, it reproduces the generating Mittag–Leffler behavior to numerical precision, while relaxation-rate heterogeneity shifts the fitted Mittag–Leffler order toward lower values. For noise-free classical distributed-rate ensembles, a Mittag–Leffler preference over a single exponential disappears when a biexponential alternative is included. Under observational noise, this recovery is incomplete: with three candidate families, the Mittag–Leffler law is still selected in 60% of runs for a classical ensemble, against 72% for a fractional reference. In the tested noise-free settings, simpler profile constructions reproduce the screening decisions, whereas kNN predicts individual trajectories far more accurately than a constant predictor. The screening interpretation of a single-record example, initially investigated as a chaotic test case, is withdrawn as evidence about the continuous system after numerical verification revealed a nonrobust simulated regime. Mittag–Leffler compatibility is therefore a comparative, condition-dependent screening outcome, not identification of fractional origin.

Fractal and Fractional doi: 10.3390/fractalfract10110744

Authors: Xiaomei Zhang Xinyu Chen Subinuer Memet Shuo Feng Xin Li Hao Chen

Maceral composition critically controls nanoscale pore structure and influences methane behavior in low-rank coal reservoirs, yet the role of individual macerals remains inadequately quantified. Therefore, vitrinite, semifusinite, and fusinite of low-rank coal were characterized by atomic force microscopy (AFM) and micro-Raman spectroscopy to reveal their surface morphology, nanoscale pore parameters, and chemical structure. Bulk coal samples enriched in different maceral assemblages were subjected to low-pressure CO2 adsorption, low-temperature N2 adsorption, and high-pressure mercury intrusion to quantify the full-scale pore size distribution. Vitrinite exhibits the highest surface fractal dimension, the largest pore number, the highest micropore proportion, and the smallest mean pore size, forming an adsorption-favorable nanostructure. Fusinite has the largest mean pore size and a pore system dominated by mesopores and macropores, favoring diffusion and desorption, while semifusinite shows transitional characteristics. Raman analysis further indicates that fusinite has higher aromatic condensation and larger structural domains, whereas vitrinite is enriched in amorphous carbon and aliphatic functional groups. At the bulk coal scale, vitrinite-rich bright coal C2 is micropore-dominated, with micropores contributing 91.79% of the total specific surface area, whereas the inertinite-rich semi-dull coal develops abundant macropores but poor connectivity. The V/(V + I) ratio decreases from bright coal to dull coal and may provide a practical parameter for reservoir evaluation. Differentiated CBM exploration and development strategies are suggested. Bright and semi-bright coals are suggested to be prioritized for sweet-spot identification. Fracturing should connect the micropore-dominated matrix to the macro-fracture network in vitrinite-rich zones and bypass mineral-clogged pore throats in inertinite-rich zones.

Fractal and Fractional doi: 10.3390/fractalfract10110743

Authors: Shengyue Li Wanrong Cao Zhaopeng Hao

Time-fractional diffusion equations with a Caputo derivative of order α∈(0,1) provide effective models for anomalous diffusion processes with memory effects, while the weakly singular fractional kernel and rough source terms generally lead to limited temporal regularity. In this paper, we develop a fully discrete scheme that combines a second-order central difference approximation in space with a spectral Petrov–Galerkin method in time. The spatial semidiscretization leads to fractional initial value problems with reaction parameters given by the eigenvalues of the spatial discretization matrix. Unlike the standard reaction-free case, the presence of these reaction parameters complicates the weighted regularity analysis. An ultraweak formulation and a bootstrapping argument are employed to establish the temporal regularity of the semidiscrete solution in weighted Sobolev spaces for source terms with regularity index r≥−α, including rough inputs. The dependence of the resulting regularity estimates on the reaction coefficients is explicitly characterized for the subsequent fully discrete error analysis. Under suitable spatial regularity of the source term, the fully discrete error is bounded by C(h2+N−m), where m=α+min{r,2α+1−ϵ} and ϵ>0 is an arbitrarily small positive constant. No coupling condition between the spatial mesh size h and the temporal polynomial degree N is required. A fast iterative solver is also developed using fast polynomial transforms and matrix–vector products. Numerical experiments support the theoretical convergence rates. For the tested scalar FIVP, the fast solver requires less computational time than the direct solver at the larger temporal polynomial degrees.

Fractal and Fractional doi: 10.3390/fractalfract10110742

Authors: Xinyu Li Sheng Zhang Bo Xu

Broer and Kaup (BK) equations that include both spatial and temporal variables were independently introduced in the early 1970s. Renowned for describing the propagation of bidirectional long waves in shallow water, the BK equations are an important classical model among nonlinear evolution equations in fluid mechanics and nonlinear science. Although the BK equations have been extensively studied, their Painlevé integrability and generalized Darboux transformations (DTs) have not been reported. Addressing these two issues constitutes the starting point of this work. First, this paper extends the BK equations to their fractional-order counterparts, the fractional-order BK (fBK) equations, and proves their Painlevé integrability. Based on the Painlevé truncation expansions, we derive the Bäcklund transformations for the fBK equations. Subsequently, we reduce the fBK equations to the fractional linear heat conduction equation and the fractional Burgers equation. Meanwhile, starting from the fractional linear heat equation, we construct fractional N-soliton solutions of the fBK equations. On the other hand, starting from the classical DTs based on their Lax pair formulation of the BK equations, we derive the generalized DTs for the BK equations. By applying a limit process to the eigenfunctions, we deduce iteration eigenfunctions within the framework of the generalized DTs. These iteration eigenfunctions, along with two forms of seed solutions of the related Lax pairs, are then used to derive first-, second-, third-, and fourth-order rational-hyperbolic function solutions of the BK equations. The obtained high-order rational-hyperbolic function solutions are new and exhibit novel spatial structures, as they combine the external explicit representations of independent variables with hyperbolic functions to form rational forms.

Fractal and Fractional doi: 10.3390/fractalfract10110741

Authors: Zengzheng Jiang Xingtao Li Zhanlei Wang Quanzhong Guan

Closely spaced lithologic contacts create scale-dependent variations in coal-measure well logs, while incomplete logging suites complicate cross-well comparison. A geology-aware multiscale stability-regularized fractal coupling graph (GeoMSR-FCG) links local analysis support to stratigraphic boundaries and represents cross-log coupling at explicit physical scales. A multilog gradient field regulates Higuchi fractal dimension (HFD) support; detrended cross-correlation analysis (DCCA) defines graph edges at 3, 6, and 12 m. Separate roof and floor contrasts capture stratigraphic asymmetry, and projection onto measured target edges enables transfer between logging suites. Evaluation combines 960 held-out synthetic realizations with a five-well proof of concept comprising 35 coal intervals in the eastern Ordos Basin. For a 25-sample embedded layer, boundary regulation reduces HFD root-mean-square error from 0.1931 to 0.1868; overall error is 0.1828, compared with 0.1823 for regularization without the boundary term. In the field wells, median effective weighted support decreases from 5.42–5.77 m to 5.28–5.49 m. At 3 m, well-balanced coal–host coupling-magnitude contrasts are −0.652 for gamma ray–deep resistivity and −0.160 for acoustic slowness–deep resistivity, while density–neutron contrasts are positive in the three wells with both logs. Transfer to the reduced Jin2 and Gao2 suites yields ROC-AUC values of 0.762 and 0.781. GeoMSR-FCG characterizes coal-measure heterogeneity through boundary-adaptive fractal support and scale-specific coupling among the measured logs.

Fractal and Fractional doi: 10.3390/fractalfract10110740

Authors: Yuhan Ma Sheng Zhang Sen Zhao Bo Xu

This paper systematically studies a space-fractional variable-coefficient Toda (sfvToda) lattice equation. The core innovation of this paper lies in incorporating both fractional power of the recursion operator and time-varying coefficients into the Toda lattice, as well as supplementing relevant theories. Firstly, based on the adjoint operator of a 2 × 2 matrix operator defined, a fractional Toda lattice hierarchy is presented. Three time-varying coefficients are introduced to combine the first three members of this hierarchy into the sfvToda lattice equation. Secondly, the orthogonality and completeness of squared eigenfunions related to the Toda lattice spectral problem (SP) are presented, and the explicit representations of the space-fractional Toda lattice hierarchy and the sfvToda lattice equation are derived. Thirdly, within the framework of inverse scattering transform (IST) theory, the linearization of the sfvToda lattice equation is achieved through scattering data, and explicit fractional variable-velocity N-soliton solutions are obtained. Finally, numerical simulations of single-soliton solutions, double-soliton solutions, and triple-soliton solutions with variable velocity are carried out by selecting appropriate coefficient functions. The results indicate that both the fractional extension and time-varying coefficients have significant regulatory effects on the spatiotemporal structure and propagation velocity of the obtained fractional soliton solutions.

Fractal and Fractional doi: 10.3390/fractalfract10110739

Authors: Lingying Huang Yifan Gu Xingtao Li Yuxiang Zhang Quanzhong Guan Yuqiang Jiang

Transitional shales are characterized by complex multiscale pore systems. A comprehensive understanding of the pore distribution across different scales and its multifractal characteristics is essential for evaluating the reservoir properties and gas-bearing potential of these shales. Based on low-temperature nitrogen adsorption (LTNA) and nuclear magnetic resonance (NMR) experiments, this study investigates the pore structure, heterogeneity and its factors associated with transitional shale from the Upper Carboniferous Taiyuan Formation on the eastern Ordos Basin. The main results are as follows: (1) LTNA results reveal Type II adsorption isotherms and H3 hysteresis loops, suggesting the widespread occurrence of slit-shaped and wedge-shaped pores. The specific surface area (SSA) and pore volume (PV) range from 0.30 to 2.16 m2/g and from 0.000235 to 0.005163 cm3/g, respectively, and are lower than those reported for representative organic-rich marine shales. (2) The NMR T2 spectra exhibit multiple peaks with variable degrees of overlap, indicating distinct relaxation-time populations and heterogeneous pore-related signals. (3) Comparison of the multifractal spectra derived from NMR and LTNA shows that the widths of the generalized fractal dimension spectrum (ΔD) and multifractal spectrum (Δα) obtained from NMR are both significantly greater than those derived from LTNA. In contrast, the information dimension (D1) and correlation dimension (D2) are significantly lower for the NMR results. These differences may reflect variations in measurement principles, characterization ranges, and sample states. (4) Exploratory correlation analysis reveals that pore structure heterogeneity is associated with organic–geochemical properties and mineral composition. TOC is positively correlated with LTNA-derived α1 (r = 0.80), while the hydrocarbon index correlates positively with LTNA-derived ΔD (r = 0.53) and NMR-derived α2 (r = 0.54). The correlations between mineral components and multifractal parameters vary among minerals, with relatively weak correlations observed for quartz.

Fractal and Fractional doi: 10.3390/fractalfract10110738

Authors: Shaoying Zhang Wei Mao Shuai Shao Yongpeng Cao Qingqing Ma Kai Zhao

Soil-water characteristic curve (SWCC) provide a fundamental description of water retention in unsaturated soils. However, conventional empirical models generally lack an explicit connection between water retention behavior and the multiscale characteristics of soil particle and pore structures. In this study, a full range SWCC model was developed based on particle-size fractal characteristics. The particle-size fractal dimension was used to describe the self-similar characteristics of the particle-size distribution, and a particle-to-pore relationship was established by incorporating the void ratio and the fractal dimension of pore tortuosity. Based on the resulting pore-size distribution, capillary and adsorptive water retention mechanisms were incorporated into a unified fractal framework. The concept of the effective specific thickness of the adsorbed water was introduced to characterize the contribution of particle surface adsorption to water retention at high matric suctions. The proposed model provides a continuous transition from capillary dominated water retention at low suctions to adsorption dominated water retention at high suctions. Its performance was evaluated using six experimental datasets covering clayey soils, silty soils, and sands over a broad range of matric suctions. Compared with the van Genuchten (VG) and Kosugi models, the proposed model achieved the lowest root mean square error (RMSE) for four of the six datasets, while its performance for the other two datasets was broadly comparable to that of the benchmark models. For Arizona silty soil #14, the RMSE was 0.0146, compared with 0.0351 for the VG model and 0.0602 for the Kosugi model, representing reductions of 58.4% and 75.7%, respectively. These comparisons indicate improved fitting performance for several soil types, particularly the silty soils and loess examined in this study. Overall, the proposed fractal framework provides a physically based link between particle-size distribution, pore structure, and soil water retention across a broad suction range.

Fractal and Fractional doi: 10.3390/fractalfract10110737

Authors: Tuoru Li Shurong Dong Weidong Chen

Structural magnetic resonance imaging supports the characterization of Alzheimer’s disease (AD)-related cognitive impairment. But existing cross-cohort AD classification methods remain limited in addressing acquisition and population shifts when source data cannot be revisited and target-side adaptation is excluded. In particular, current cross-cohort AD classification may underutilize diverse disease-related information learned from source data, while explicit characterization of regional, scale-dependent gray-matter morphology remains limited. To address these gaps, we propose FSE-CNN for source-free cross-cohort classification of AD-related cognitive status. First, the neural pathway derives decision evidence from model states retained during source training, with the inference configuration determined exclusively from source data. Second, the spatial-scale geometric evidence encoding module characterizes gray-matter morphology through subject-relative spatial organization and finite physical scales, providing complementary geometric evidence for cross-cohort classification. Finally, the resulting neural and geometric evidence are combined through protocol-specific fusion. We evaluated FSE-CNN on two cross-cohort protocols. In OASIS-2 → OASIS-1, FSE-CNN achieved 83.48% accuracy, 85.14% balanced accuracy, and 88.60% AUC, numerically exceeding Harmonizing Flows by 1.70, 1.26, and 0.39 percentage points, respectively. In ADNI → OASIS-2, it achieved 85.79% accuracy, 85.55% balanced accuracy, and 85.03% AUC, with corresponding gains of 0.71, 0.67, and 0.21 percentage points. Overall, these results demonstrate the effectiveness of FSE-CNN for cross-cohort AD-related cognitive-status classification.

Fractal and Fractional doi: 10.3390/fractalfract10110736

Authors: Yuejun Song Huijian Wen Bing Jiang Weidong Xie Haixue Wang Haiguang Wu Zhida Liu

Pore systems within different pore size intervals in shale reservoirs exhibit heterogeneity in surface roughness and structure, which can be quantitatively characterized by fractal dimensions. To quantitatively characterize the pore structure characteristics, six Longmaxi shale samples were collected from the southern Sichuan Basin. Mercury intrusion, low-pressure N2 adsorption, and low-pressure CO2 adsorption experiments were conducted to obtain the pore size distributions of macropores, mesopores, and micropores, respectively. Fractal dimensions of micropores (D1), mesopores (D2), and macropores (D3) were calculated using the volume-specific surface area (V-S), Frenkel–Halsey–Hill (FHH), and Menger models, and the controls of total organic carbon (TOC) and clay content on these fractal dimensions were investigated. The results show that surface roughness and structural irregularity decrease in the order of macropores > mesopores > micropores, with D3 (2.89–2.98) > D2 (2.777–2.812) > D1 (2.187–2.304). These differences in fractal dimensions are mainly attributed to the development of small pores within the walls of larger pores, as well as the presence of bubble-like, honeycomb-like, and beaded organic matter pores observed under scanning electron microscopy. Specifically, D1 is predominantly controlled by TOC owing to the extensive development of micropores within organic matter, whereas D2 and D3 are governed by clay content, which hosts abundant intraparticle and interparticle mesopores and macropores. These findings provide a quantitative characterization of the pore structure of the Longmaxi Formation across multiple pore size scales and may serve as a reference for future studies on pore structure characterization of organic-rich shales.

Fractal and Fractional doi: 10.3390/fractalfract10110735

Authors: Menghui Xiong Mingcheng Dai Qingyi Liu Yushun Tan

This paper is concerned with distributed online optimization with long-term inequality constraints over time-varying unbalanced directed networks. A fractional-memory-based distributed online optimization algorithm is developed to exploit both current and historical first-order information. Its search direction is constructed using normalized power-law memory weights, whose distribution is regulated by the fractional order and whose memory horizon is determined by the maximum memory lag. An auxiliary-weight mechanism is incorporated to cope with the imbalance of directed communication, while the long-term constraint formulation allows projection onto a computationally tractable superset of the original feasible region, thereby avoiding costly constrained projections at every round. Under suitable assumptions and parameter choices, sublinear static regret and cumulative absolute constraint violation bounds are established. Simulations for a distributed online ridge regression problem demonstrate the feasibility of the proposed algorithm.

Fractal and Fractional doi: 10.3390/fractalfract10110734

Authors: Igor Podlubny Ivo Petráš

This paper reviews a novel class of elements for electrical circuits and the novel types of electrical circuits built from them. The common characteristic of the considered elements (fractional-order memristor; fractional-order (mem)capacitor; fractional-order (mem)inductor) is that they exhibit fractional-order behavior and include a memory effect. The survey first clarifies the distinction between constant-phase elements, linear fractional-order elements with hereditary memory, and nonlinear state-dependent memory elements (memristors, memcapacitors, meminductors), and summarizes the constraints of causality, passivity, stability, and finite bandwidth that every physical realization must satisfy. The main challenge in this field is designing and implementing such memory elements with adjustable fractional order. The developed memory components and circuits can be used in signal and image processing filters, controllers for process control and industrial automation, oscillators, and signal and image encoding/decoding. This survey compares analog and digital implementations of these devices and summarizes their reported characteristics, order adjustability, and technological maturity in a comparative table. Analog implementations reported in the literature use polymer–metal composites, supercapacitors, and SMD components, while digital implementations rely on computers, PLCs, FPGAs, microcontrollers, and embedded platforms such as Arduino or Raspberry Pi. With new fractional-order elements, researchers have investigated novel electrical circuits, including fractal-structure circuits, embedded ladder networks, and their combinations. The relevance of this field is underlined by the rapid growth in the use of electrical batteries practically everywhere, especially in vehicles, drones, and communication devices. Because batteries exhibit fractional-order behavior and memory, fractional-order modeling and fractional-order controllers can optimize charge/discharge processes, motivating the need for new fractional-order tools in control theory and the theory of chaotic systems. The paper also discusses a new type of non-uniform sampling, based on fractional-order noise, aimed at producing better fractional-order models from measurements, as well as the emerging use of fractional-order memory in neuromorphic and physical reservoir computing.

Fractal and Fractional doi: 10.3390/fractalfract10110733

Authors: Waleed Mohamed Abd-Elhameed Omar Mazen Alqubori Naher Mohammed A. Alsafri Amr Kamel Amin Mohammed Atef Abdallah Ahmed Gamal Atta

We develop a spectral collocation method based on shifted balanced polynomials for solving variable-coefficient multi-term time-fractional convection–diffusion–reaction equations. The proposed approach uses the Caputo fractional derivative and a global space–time approximation. First, explicit formulas are derived for expressing the shifted balanced polynomials in the power basis and for expressing monomials in terms of the shifted balanced polynomial basis. These formulas are then used to obtain a representation of the Caputo operator at the selected points. The approximate solution is expanded in a tensor-product basis, and the governing equation is enforced at selected interior collocation points together with the prescribed initial and boundary conditions. Theoretical results concerning the growth of the shifted balanced polynomials and the decay of the expansion coefficients are established. These results are used to prove the absolute convergence of the proposed expansion and to derive an explicit upper bound for the truncation error. Four numerical examples are presented to examine the accuracy and effectiveness of the proposed method for different polynomial degrees and fractional orders. In addition, an account of the extension of the proposed approach to problems involving nonlinear terms is presented. The numerical results show that accurate approximations can be obtained using relatively small polynomial degrees.

Fractal and Fractional doi: 10.3390/fractalfract10100732

Authors: Engin Özkan Beyza Tekeoğlu Akkaya

This study investigates the algebraic and differential properties of balancing-like polynomials, which constitute a polynomial extension of balancing-like sequences. Several fundamental properties are established, including matrix representations, addition and subtraction formulas, and polynomial analogues of the Cassini, Catalan, D’Ocagne, and Vajda identities. A product factorization is obtained, which yields an explicit description of the zeros of these polynomials. Moreover, a direct connection with the Chebyshev polynomials of the second kind is established, and the ordinary generating function of the balancing-like polynomials is derived. The derivative sequences associated with these polynomials are then systematically investigated. Explicit formulas, recurrence relations, and a self-convolution identity are obtained, together with a relation recovering the original polynomials from their neighboring derivative terms. Finally, the analysis is extended to higher-order derivatives, and a generating-function representation and a general recurrence relation for the resulting differential hierarchy are derived. The differential framework is further extended to non-integer orders by means of the Caputo fractional derivative. An explicit fractional derivative formula and a generating-function representation are obtained.

Fractal and Fractional doi: 10.3390/fractalfract10100731

Authors: Najmeddine Attia Mohammed Balegh

We study the multifractal structure of a branching random walk on the boundary of a finite-type Galton–Watson tree. For α∈R, we consider the level set EX(α)=t∈∂T:limn→∞SnX(t)n=α. The multi-type structure leads naturally to the weighted mean offspring matrix M(q) and to the deterministic pressure P˜(q)=logρ(M(q)). Using the corresponding Perron–Frobenius eigenvectors, we construct a family of multi-type Mandelbrot martingales and prove their almost sure and Lp uniform convergence on compact subsets of the admissible parameter domain. This yields the associated Mandelbrot measures and the uniform estimates required for the multifractal analysis. As a consequence, for every α in a suitable admissible set I, we obtain almost surely dimHEX(α)=P˜*(α), where P˜* denotes the Legendre transform of P˜ and dimH stands for the Hausdorff dimension. Thus the classical multifractal formalism for one-type branching random walks extends to the genuinely multi-type setting, with the scalar normalization replaced by the Perron–Frobenius spectral structure of the weighted offspring matrices.

Fractal and Fractional doi: 10.3390/fractalfract10100730

Authors: Messaoud Berkal Mohammed Bakheet Almatrafi

The exact traveling-wave solutions, equilibrium structures, stability characteristics, bifurcation phenomena, and complicated nonlinear dynamics of a space-time fractional modified Benjamin–Bona–Mahony (mBBM) equation are thoroughly discussed in this article. The governing fractional partial differential equation is converted to a nonlinear ordinary differential equation by implementing a suitable traveling-wave transformation, wherein the fractional-order parameter β serves as a spatial–temporal scaling factor in the physical coordinate frame. This ordinary differential equation is then solved using the improved modified extended tanh-function approach. Various explicit traveling-wave solutions are successfully constructed and rigorously classified into smooth localized solitary waves (such as bell-shaped and kink-type profiles), doubly periodic Jacobi elliptic functions, singular periodic waves, and rational solutions. Their explicit domains and singularities are clearly discussed. Eigenvalue investigation and linearization are employed to identify the equilibrium points and evaluate their local stability. Furthermore, changes between regular and complex patterns of behavior, such as periodic, quasi-periodic, and chaotic dynamics, are shown via bifurcation and numerical analysis by introducing an external periodic excitation into the reduced planar system. The findings reveal that the fractional parameter β modifies the wave profile geometry and propagation speed, while the chaos is mainly influenced by external forcing parameters and nonlinear system coefficients. The improved modified extended tanh-function approach, along with dynamical-systems analysis, offers a comprehensive foundation for expressing both explicit wave patterns and complicated nonlinear processes.

Fractal and Fractional doi: 10.3390/fractalfract10100729

Authors: Shuai Cao Weibo Li Xiaoqing Deng Yingting Deng Qing Li Junyi Zeng

Disturbances from multiple sources propagate through coupled motor, hydraulic, and mechanical channels in electro-hydrostatic actuators, while increasing observer bandwidth introduces a trade-off with sensitivity to high-frequency components. To address these challenges, this study proposes a cascaded control method that integrates parallel disturbance estimation with fractional-order error damping. A detailed model incorporating the instantaneous flow rate of a piston pump, leakage, fluid compressibility, and dynamic friction is developed, and a control-oriented model is derived through time-scale separation. A three-channel parallel observer is constructed according to the locations at which disturbances enter the state equations, enabling corresponding compensation within a position–pressure difference–rotational speed backstepping control cascade. A Caputo fractional derivative is introduced into the motor-level error feedback to shape the magnitude and phase characteristics within the target frequency band by adjusting the fractional order and damping gain. This term is implemented using a rational approximation over a finite frequency band. The approximation filter states are incorporated into a composite Lyapunov function. Under prescribed filter dissipation conditions, positive feedback gains, and uniformly bounded equivalent residuals, uniform ultimate boundedness of the augmented continuous-time closed-loop error dynamics is established, together with residual-dependent error bounds. Co-simulation results show that the position root-mean-square error during multilevel step tracking is 5.68 mm, representing reductions of 45.8%, 24.6%, and 12.6% relative to proportional–integral–derivative control, fractional-order linear active disturbance rejection control, and cascaded sliding-mode control with multiple-disturbance observation, respectively. In semi-physical experiments, the steady-state position error during the plateau phases remains largely within ±2 mm under loads of 5 kN and 10 kN. During motion-reversal tracking under variable loads, the position root-mean-square error is reduced by 26.8% and 14.8% relative to the first two baseline methods, respectively. These results demonstrate that, under the tested conditions, channel-specific disturbance compensation combined with fractional-order dynamic shaping improves actuator tracking performance and provides a basis for fractional-order controller design and finite-dimensional implementation analysis in coupled electrohydraulic systems.

Fractal and Fractional doi: 10.3390/fractalfract10100728

Authors: Wei Cheng Hongqing Zheng Wanqing Song Piercarlo Cattani

The reliability analysis of integrated circuit (IC) isolation structures (ISs) faces two challenges: how to accurately describe the distribution of time-to-failure (TTF) data, and how to obtain the TTF distribution under actual working conditions (AWCs). To address these issues, this paper introduces the flexible five-parameter Beta Generalized Weibull (BGW) distribution, which can represent the five characteristic failure-rate shapes (IFR, DFR, CFR, BFR and UBT). The sign of the hazard derivative is analyzed through Glaser’s eta-function theorem together with the asymptotic behavior of the hazard rate at the origin and at infinity; this yields exact, necessary-and-sufficient parameter conditions for each shape, with the origin controlled by the product aγω and the tail controlled by γ. A TTF model based on the BGW distribution is then proposed to predict the TTF distribution under AWCs, and it is proved analytically that a change of stress affects only the scale parameter δ, leaving all shape parameters unchanged. Because this invariance was previously stated as an assumption, it is verified empirically on the time-dependent dielectric breakdown (TDDB) data using per-level Weibull shape estimates with bootstrap confidence intervals, a Kruskal–Wallis test on standardized failure times, and a parametric-bootstrap likelihood-ratio test within the BGW family; all three tests support stress-invariant shapes. The reliability function is recast as a convergent infinite series, with the correct beta-integral kernel and an explicit ratio-test bound. Finally, the effectiveness and superiority of the method are verified using TDDB TTF data, and the untested 3800 V condition is predicted from the fitted linear voltage-acceleration model.

Fractal and Fractional doi: 10.3390/fractalfract10100727

Authors: Hasan Karci Oguzhan Karahan Ismet Kandilli

The growing penetration of renewable-energy sources into contemporary power networks introduces substantial difficulties in maintaining satisfactory voltage regulation and dynamic stability. Existing PID-based designs for automatic voltage regulators (AVRs) equipped with an amplifier feedback with dynamic weighting (AFDW) architecture provide limited tuning flexibility, and under the considered asymmetric saturation constraints exhibit residual steady-state error (≈0.05 p.u.) and relatively slow settling (>2.2 s), while the previously reported FOPID variant exhibits overshoot above 4%. Moreover, tilt-based structures and comprehensive robustness assessments remain relatively unexplored for this configuration. To address these limitations, a nonlinear AVR with an AFDW architecture and IEEE DC1C saturation limits is developed. The parameters of fractional-order PID (FOPID) and tilt-integral-derivative (FOTID) controllers are optimized under nonlinear saturation constraints using two mathematically inspired metaheuristic approaches: the PID-based search algorithm (PSA) and its recently proposed fractional-order extension, FoPSA. Controller tuning is carried out through a cost function that captures the transient characteristics of the closed-loop response. Extensive simulation studies indicate that FoPSA-based tuning yields more favorable transient behavior for the fractional-order controllers, with notable reductions in settling time and steady-state error relative to the reference designs. In addition, multi-domain evaluations reveal that the FoPSA-FOTID configuration maintains strong robustness against parameter variations, severe load disturbances, and changes in reference-tracking conditions. Overall, the results demonstrate the effectiveness of the proposed control framework and its potential to provide reliable and resilient voltage regulation for modern power systems.

Fractal and Fractional doi: 10.3390/fractalfract10100726

Authors: Luca Di Persio

A fractional stochastic evolution is studied for the maturity-dependent spread between green and conventional forward rates. Its Caputo resolvent links the persistence of an initial greenium shock to the local scaling of subsequent fluctuations. The nonlinear equation is globally well posed for every finite probability moment exponent at least two under globally Lipschitz coefficients. In the Gaussian benchmark, covariance asymptotics hold in the trace norm, and so apply directly to the discontinuous maturity weights defining zero-coupon ratios. At positive observation times, the local variance contains a contribution from the reweighting of the entire preceding history. The resulting temporal exponent identifies the memory order, whereas identification of the maturity order depends on information about the noise covariance. These results give a statistical interpretation of climate memory effects without equating physical-measure exponential moments with arbitrage-free prices.

Fractal and Fractional doi: 10.3390/fractalfract10100725

Authors: Hongxuan Yan Luoyi Sun

When the density at the antipode of the intrinsic Fréchet mean reaches a critical level, an interval for the mean of a long-memory circular time series that is calibrated to independent-data scaling is too short, and at Hermite rank two, it also has the wrong shape. The antipode is the cut locus, where squared distance has a kink, and the quadratic coefficient of the population objective vanishes exactly when the density there equals 1/(2π). For Yt=G(Zt), subordinating a stationary Gaussian sequence with covariances decaying at rate α∈(0,1), and contact of order 2r at the antipode, the rescaled estimator converges to a signed (2r+1)-st root. The contact order fixes the root and the Hermite rank of the score fixes what sits under it: a Gaussian at rank one, or a Rosenblatt variable at rank two under strong memory. Because the critical density is a knife-edge equality, we also give the limit along an array approaching it: when the quadratic coefficient vanishes at a rate matched to the long-memory scale of the score, the limit is a one-parameter family joining the odd root to a linear one. In hourly wind directions from fifty-three stations, the antipodal density ranges from 0.05 to 0.74 of the critical level; the records identify no subordination and do not settle the memory, so they locate series relative to the critical level rather than testing the limits.

Fractal and Fractional doi: 10.3390/fractalfract10100724

Authors: Gang Gyoo Jin Kwangcheol Shin Jongkap Ahn

The mathematical constants π and e are irrational and transcendental numbers whose infinite, non-repeating decimal expansions mimic randomness. This study investigates potential underlying structures within these sequences by treating their digits as discrete time-series signals. Using Higuchi’s algorithm, we calculate their fractal dimensions to quantify their inherent complexity. Because the efficacy of Higuchi’s method depends heavily on the maximum window size (tmax), we establish a pseudo-optimal formula for tmax calibrated against fractional Brownian motion of known dimensions. Finally, we enrich our analysis with random walk and power spectral density analysis, uncovering necessary empirical conditions for white-noise behavior and long-range dependence.

Fractal and Fractional doi: 10.3390/fractalfract10100723

Authors: Sotiris K. Ntouyas Bashir Ahmad Jessada Tariboon

Fractional calculus is concerned with integration and differentiation of arbitrary, not necessarily integer, order [...]

Fractal and Fractional doi: 10.3390/fractalfract10100722

Authors: Yufei Zhang Jijun Tian Shuguang Yang Shuo Feng Daxing Wang Xin Tong Chengjie Xu

Underground coal gasification (UCG) is an important in situ conversion technology for the clean and efficient recovery of deep and difficult-to-mine coal resources. Reactions between high-temperature char and gasifying agents such as CO2 and H2O in the reduction zone directly control the generation of valuable gases, particularly CO and H2. Calcium-bearing minerals can catalyze char gasification and participate in high-temperature ash-phase transformation. However, the differential effects of CaCO3 and CaO on coal pyrolysis–gasification behavior and ash-phase evolution under a CO2 atmosphere remain unclear. In this study, a typical low-rank Jiangna (JN) coal from the Hami region of Xinjiang, China, was subjected to sequential HCl-HF leaching to prepare demineralized coal (DC), which was subsequently loaded with 5 or 10 wt% CaCO3 or CaO. TG-FTIR-MS, high-temperature in situ XRD, SEM, N2/CO2 adsorption, and multifractal analysis were employed to investigate thermal conversion, gaseous product release, pore structure, and Ca-bearing phase evolution. Demineralization decreased the mass loss at 600–1000 °C from 50.05% to 29.14%. The CaCO3-loaded samples showed their main high-temperature mass-loss and FTIR CO-band responses above 800 °C, whereas the CaO-loaded samples showed an earlier apparent high-temperature mass-loss response at 600–800 °C. At 600 °C, the CaO-loaded samples exhibited greater mesopore development. At 900 °C, the CaCO3-loaded samples exhibited pronounced micropore and mesopore development, with micropore volume increasing by 49.1–64.9%. The nonlinear mass scaling function τ(q) and the decrease in the generalized dimension D(q) with increasing q indicated heterogeneous multiscale distributions of pore volume. At 600 °C, the micropore spectrum width (Δα) of the CaCO3-loaded samples was 0.5537–0.6375, higher than that of the CaO-loaded samples (0.4093–0.4198). In contrast, the mesopore Δα of the CaO-loaded samples was 0.6688–0.9901, markedly higher than that of the CaCO3-loaded samples (0.1660–0.2126). At 900 °C, the CaCO3 and CaO-loaded samples exhibited contrasting micropore and mesopore multifractal responses. The CaO-loaded samples showed an earlier apparent high-temperature mass-loss response under the tested conditions. However, equal-mass additive loading prevented separation of the effects of precursor form and Ca input. Although HT-XRD detected crystalline phase changes, the active Ca species and extent of Ca incorporation into ash phases remain unresolved. These results document different temperature-dependent responses to CaCO3 and CaO addition under the tested conditions. These findings provide an experimental basis for selecting Ca-based additives, regulating valuable gas generation, and evaluating high-temperature ash stability in UCG.

Fractal and Fractional doi: 10.3390/fractalfract10100721

Authors: Yuancheng Si Yunqian Wei Saralees Nadarajah

Chinese equity indices open after foreign markets have already supplied part of the day’s information. We ask whether multiscale dependence networks improve opening forecasts conditionally on states observable before the mainland opening. Daily prices from 2010 to July 2026 support 2081 evaluation dates per index. The design distinguishes incomplete local records, stale news, index launch availability, and accumulated foreign changes during domestic closures. A fixed LightGBM model adds 57 fluctuation order dependent DCCA features to the same feasible Raw information set. Six economic state partitions, matched benchmarks, and chronological rule assessment distinguish a useful representation from unrestricted model superiority. The clearest additional evidence concerns Shanghai’s absolute prediction error. On 1036 mornings with above median foreign news disagreement, mean absolute error falls from 25.90 to 24.83 basis points, a 4.12 percent reduction; the pointwise 95 percent block interval for the 1.07-basis-point gain is [0.64, 1.52]. This secondary absolute-loss comparison has Holm adjusted p=0.0094 across 468 comparisons and p=0.0296 in a wider 1482 test audit. Its later period estimate remains positive but does not pass the corresponding multiplicity adjustment. High-minus-low gain differences also do not establish unique state dependence. Squared-error and extreme probability comparisons do not show general network dominance, and forward applied gates do not deliver an adjusted significant advantage. The evidence supports a loss- and benchmark specific conditional increment, rather than an identified multifractal mechanism or a universally preferable risk model.

Fractal and Fractional doi: 10.3390/fractalfract10100720

Authors: Hussain Gissy Mohamed Biomy Mirajul Haq Muhammad Arif Muhammad Abbas

In this paper, we use a normalized fractional q-derivative operator, constructed by combining Jackson’s q-difference operator with a fractional q-integral, to introduce two subclasses of normalized analytic functions in the open unit disk whose fractional transforms are starlike and univalent. The first subclass, Sq,λ*(ML), is defined by subordination to a normalized q-Mittag–Leffler function, whereas the second, Sq,λ*(L), is associated with a generalized limaçon domain. For both classes, we investigate sharp initial coefficient bounds and sharp estimates for the Fekete–Szegö functional. We also study inclusion and radius relations between the two classes. This comparative approach clarifies how an analytically defined Mittag–Leffler domain and a geometrically defined limaçon domain affect coefficient estimates when the underlying operator is held fixed. Relevant classical and quantum calculus results are recovered through suitable parameter choices and in the limiting case q→1−.

Fractal and Fractional doi: 10.3390/fractalfract10100719

Authors: Yeguo Sun Yinyin Li Yihong Liu Lei Liu

In contrast to finite-time and fixed-time synchronization, predefined-time synchronization allows for the convergence time to be explicitly defined as a deterministic value, making it ideal for applications that have stringent timing requirements. This paper addresses the predefined-time synchronization of fractional-order delayed memristive fuzzy cellular neural networks (FDMFCNNs). By leveraging the properties inherent in fractional calculus, we introduce a novel lemma. Based on this lemma and a discontinuous state feedback control method, we derive several innovative criteria for achieving predefined-time synchronization in FDMFCNNs by constructing an appropriate Lyapunov function. The findings presented in this study significantly enhance and broaden the existing literature. Finally, we validate the efficacy of our control strategies for predefined-time synchronization with the help of numerical simulations.

Fractal and Fractional doi: 10.3390/fractalfract10100718

Authors: Dragos Nica Călin Gheorghe Buzea Alina Stefania Dobos Dragos Ioan Rusu Adrian Timofte Diana-Corina Bostan Vlad Ghizdovat Maricel Agop Decebal Vasincu

We developed a one-dimensional Madelung-type model to test whether seismic activation dynamics improve agreement with future earthquake distributions beyond empirical baselines. The model couples activation density, activation-flow velocity, effective loading, structural heterogeneity, and a curvature-sensitive potential. Synthetic simulations examined localization, while temporally separated United States Geological Survey (USGS) Japan–western Pacific catalog tests compared the model with persistence and historical relative intensity. Principal component analysis (PCA) preprocessing excluded holdout data, and weak-evolution parameters were calibrated independently on 2022 data. On the July–December 2023 complete-catalog holdout, weak evolution underperformed persistence on most metrics, including Pearson correlation (0.4156 versus 0.4837); paired moving-block bootstrap intervals for correlation and cosine differences included zero. In January–June 2023, several continuous metrics improved over persistence, but operating-point results remained mixed. Declustering favored weak evolution over persistence on most metrics, while relative intensity retained higher Pearson and Spearman correlations. Removing the curvature term produced stronger concentration, indicating a regularizing rather than uniquely focusing role. Grid refinement revealed resolution sensitivity in strong evolution. These results support a mechanistic and numerical proof of concept whose empirical performance depends on catalog treatment, forecast window, and metric; they do not establish robust earthquake forecast superiority.

Fractal and Fractional doi: 10.3390/fractalfract10100717

Authors: Nizar Souayah

Hallucination in autoregressive language models arises when generation dynamics enter a locally stable but factually unsupported attractor. We propose FOCUS (Fractional Order Confidence for Uncertainty Signalling), a framework that replaces the fixed-geometry softmax attention kernel with the adaptive fractional graph-diffusion operator Aαl,h,t=exp(−τL^αl,h,t), where L^=I−D^−1/2P^D^−1/2 is the symmetrised normalised attention Laplacian of the causal prefix graph of each generation step (spectrum in [0,2], proved without assuming double stochasticity of P^) and αl,h,t∈(0,1) is a differentiable control variable predicted per layer l, head h, and generation step t. The key mathematical result is that for μk<1 (low-frequency modes), higher αl,h,t leads to less attenuation (more energy passes through), while for μk>1 (high-frequency modes), higher αl,h,t leads to more attenuation. By assuming that grounded evidence is low-frequency on the attention graph, high αl,h,t is thus advantageous in both spectral regimes simultaneously. The per-head specialisation of αl,h,t arises not from a static spectral tradeoff within a single head but from a training-dynamics competition between the LM loss (which rewards structural diversity across heads, including low-αl,h,t heads that pass high-frequency syntactic features) and the hallucination penalty. Because this penalty pushes αl,h,t down at hallucinated tokens, it trains αl,h,t as an indicator of hallucination rather than as a protective mechanism, and we state what this implies for the interpretation of the experiments. We prove the complete spectral theory, the causality of the construction, and the Fréchet derivative, along the admissible score-induced family of prefix Laplacians for which the zero eigenvalue stays structurally pinned, of the composite map L^↦exp(−τL^α), via the Daleckii–Kreĭn formula with a spectral-gap-dependent bound, and provide a direct Lanczos strategy for computing exp(−τL^α)v with measured error and cost. The warning mechanism is the spectral imbalance statistic It=α¯tctx/(α¯tnoise+ε), whose drop precedes hallucination onset by 6–7 tokens on hallucinating sequences in preliminary synthetic experiments. We also identify spectral localisation analysis—verifying that hallucination spans concentrate in high-frequency modes of the attention graph—as the critical empirical validation for the framework. The evidence assembled here supports FOCUS as a well-founded spectral operator and a preliminary hallucination-warning statistic; establishing it as a protective decoding mechanism, as the acronym’s original reading suggested, is left as an open empirical target.

Fractal and Fractional doi: 10.3390/fractalfract10100716

Authors: Mahammad Khuddush Saleh S. Almuthaybiri

In this paper, we study the problem D0+αu+f(t,u)=0 on (0,1), u(0)=0, D0+α−1u(1)=∫01u dA, where 1<α≤2 and the measure  dA may change sign. For a measure made of two point masses of opposite sign, we show that the function KA(s)=∫01G(τ,s) dA(τ), through which  dA enters the Green function, is nonnegative exactly when the positive weight is at least θ∗ times the size of the negative one, where θ∗ is given in closed form. Together with a simple condition Δ>0, this makes the Green function nonnegative. The threshold θ∗ decreases in α and tends to infinity as α→1+. For α<2 the Green function has no uniform Harnack estimate, so we work in a cone weighted by Ψ(t)=∫01G(t,s) ds, where Schauder’s theorem gives a positive solution with explicit bounds whenever f lies between two positive constants. For the omitted ray fixed point theorem of Avery, Henderson and Liu, we describe exactly when its key inequality holds for the two-point functional used here, and we show that the hypotheses under which we can verify it already imply the classical inward condition. Examples with α=3/2 and α=1.08 illustrate the results.

Fractal and Fractional doi: 10.3390/fractalfract10100715

Authors: Ali Khaleghi Soroush Oshnoei

False data injection attacks (FDIAs) pose a serious threat to FACTS-assisted wide-area damping controllers because they corrupt remote measurements, degrade inter-area oscillation damping, and may evade conventional detection. Data-driven detectors that ignore closed-loop physics are particularly vulnerable under multiplicative gain-ramp attacks; the problem is further exacerbated by the memory-dependent nature of fractional-order controllers, whose long-term history effects can mask attack-induced inconsistencies in the residuals unless system dynamics, actuator behavior, attack-channel constraints, and fractional-order memory are jointly embedded in the learning objective. This paper addresses that gap through a physics-informed neural network (PINN) framework that jointly estimates latent system states and reconstructs the multiplicative attack gain by incorporating plant dynamics, SVC actuator dynamics, measurement consistency, attack-channel reconstruction, and the reconstructed fractional-order command inside the SVC actuator residual. Detection is declared when a smoothed composite residual exceeds a calibrated threshold. In addition, a supervisory switching mechanism is introduced to replace corrupted wide-area control signals with reconstructed secure equivalents after confirmed detection, thereby restoring closed-loop damping performance without interrupting controller operation. Simulation results on a FACTS-based damping control system under gain-based FDIA scenarios show that the proposed detector achieves up to 99.42% accuracy, 98.86% precision, 99.18% recall, and a 99.02% F1-score, with a false alarm rate as low as 0.53%. The recovery mechanism also substantially reduces post-attack speed-deviation overshoot relative to the unmitigated case.

Fractal and Fractional doi: 10.3390/fractalfract10100714

Authors: Xueqing Bai Bangsheng Zhao Bofu Ta Siyuan Wang Jin Han Yanjun Wu Gaorun Zhong

To examine the fractal characteristics and pore architectures within marine-continental transitional shale, experimental analyses were carried out for Shanxi shale samples collected from the central Ordos Basin. Multiple test approaches including scanning electron microscopy (SEM), X-ray diffraction analysis, total organic carbon (TOC) analysis, and gas adsorption experiments were applied in this work. According to experimental measurements, the mineral composition of collected Shanxi shale samples exhibits dominant quartz and clay minerals, averaging 38.2% and 54.36%, respectively, with an average TOC content of 1.635%. CO2-accessible micropores make an important contribution to specific surface area, whereas N2-accessible mesopore range pores contribute appreciably to pore volume. The pore size distribution is characterized by two dominant peaks within the ranges of 8–20 nm and 80–85 nm. Mineral pores constitute the primary pore type within Shanxi shale, while pores associated with organic matter are infrequently observed. Micropore-specific surface area parameters show positive correlative trends relative to TOC, yet only weak correlations exist between this parameter and quartz and clay minerals. Meso-macopore-specific surface area exhibits a negative correlative tendency versus TOC content, weakly correlated with clay minerals, and no relationship with quartz content. Pore volume magnitudes are predominantly controlled by organic matter abundance and clay mineral content, with quartz content having no significant impact. Fractal dimensions computed through the Frenkel–Halsey–Hill (FHH) model indicate that D1 values for the low-pressure interval vary between 2.654 and 2.834, and D2 corresponding to the high-pressure interval are distributed from 2.638 to 2.867. The internal pore structure’s complexity and heterogeneity are more pronounced than the surface roughness of the pores. Compared with typical marine and continental shales, marine-continental transitional shale displays intermediate characteristics in both pore morphology and fractal features. The key results demonstrate that TOC content has strong positive linkages with fractal dimensions D1 and D2. By contrast, clay minerals exhibit a marked negative association against D2. Quartz content does not show a significant correlation with D1, yet faint positive linkages are detected between this indicator and D2. The present work offers quantitative constraints to assess pore structure heterogeneity and gain insights into reservoir properties of marine-continental transitional shale.

Fractal and Fractional doi: 10.3390/fractalfract10100713

Authors: Yabin Deng Pengzhou Wang Bijing Liu Yijin Wu Junqi Huang Zenan Huang

Spike-frequency adaptation (SFA) regulates neuronal excitability according to recent spiking activity, yet conventional neuromorphic models typically describe adaptation using single-timescale exponential recovery. To the best of our knowledge, most existing fractional spiking-neuron formulations introduce memory primarily through membrane or input-related dynamics rather than through the recovery of a spike-triggered adaptation current. Here, we propose fractional spike-frequency adaptation (F-SFA), in which an integer-order LIF membrane is coupled with a spike-triggered adaptation current undergoing fractional-order recovery. A compact multi-exponential embedding enables recursive implementation without explicit history convolution. Single-neuron experiments show firing rate adaptation, progressive inter-spike-interval (ISI) elongation, and post-stimulus recovery. F-SFA is further incorporated into an event-driven Lobula Giant Movement Detector 2 (LGMD2)-inspired model without feed-forward inhibition (FFI). The resulting model retains a pronounced response to dark looming while providing additional history-dependent attenuation of sustained translation and post-looming activity. These results support the feasibility of F-SFA as a compact mechanism for introducing multi-timescale spike-history-dependent adaptation into event-driven neuromorphic models.

Fractal and Fractional doi: 10.3390/fractalfract10100712

Authors: Lei Kong Tao Teng Xuan Yang Quan Wang

In coal-fired power plants, the coordinated control of boiler–turbine systems (BTSs) is required to track grid load dispatch instructions in real time while maintaining stable main steam pressure. Nevertheless, a BTS is affected by input constraints, external disturbances and model uncertainties, making it challenging for existing control schemes to stabilize the system within a fixed time. To tackle this issue, a fixed-time fractional-order sliding mode coordinated control (FTFOSMCC) strategy is put forward for the BTS. Firstly, a novel fixed-time fractional-order sliding mode disturbance observer (FTFOSMDO) is constructed by merging Riemann–Liouville fractional calculus with the fixed-time stability theory; the proposed observer is theoretically proven to drive the disturbance estimation error to zero within a fixed settling time that does not depend on the initial observation error. Secondly, the FTFOSMCC controller incorporates the Caputo fractional-order sliding surface, a novel variable exponential power reaching law, the state-of-the-art anti-windup technique, disturbance-observer-based feedforward compensation technique and fixed-time stability theorem, guaranteeing active suppression of actuator saturation and fixed-time stability of the closed-loop control system by Lyapunov’s theorem. Thirdly, simulations on a 300 MW subcritical BTS validate the exceptional tracking accuracy, disturbance rejection capability and robustness of the proposed strategy with the reduction of 33–90% in maximum tracking error, 12–98% in cumulative error, and 11–98% in time-weighted error compared to tuned PID and integer-order disturbance-observer-based sliding mode controllers.

Fractal and Fractional doi: 10.3390/fractalfract10100711

Authors: Shanshan Zheng Xiaoxu Du Min Tang Xiaowen Zhuang Gan Wang

Financial markets are complex systems characterized by heterogeneous trading horizons, nonlinear interactions, and scale-dependent dependence. From a fractal-market perspective, these features imply that risk transmission may vary across temporal scales and market states. This study examines multiscale connectedness among carbon, West Texas Intermediate (WTI) crude oil, coal, natural gas, and clean energy markets. Empirical mode decomposition (EMD) adaptively extracts intrinsic high- and low-frequency components, while time-varying parameter vector autoregression (TVP-VAR) is used to estimate connectedness across scales. Random forest, gradient boosting decision tree, extreme gradient boosting, and SHapley Additive exPlanations are employed to identify nonlinear and state-dependent macro-financial drivers. Results show pronounced scale heterogeneity: original connectedness closely tracks high-frequency dynamics, whereas low-frequency connectedness is stronger and more persistent, peaking during the 2022 energy crisis. Market roles are horizon-dependent, with WTI generally transmitting risk, carbon mainly receiving risk, and clean energy becoming an important low-frequency transmitter. Overall, the findings support a multiscale interpretation of carbon–energy risk transmission consistent with the heterogeneous-horizon view of fractal-market research.

Fractal and Fractional doi: 10.3390/fractalfract10100710

Authors: Mousa J. Huntul Abdullah A. Zaagan

This manuscript presents an efficient hybrid numerical method for solving one- and two-dimensional time-fractional Cattaneo models that incorporate the Riesz space-fractional derivative. The proposed approach combines the Crank–Nicolson scheme for temporal discretization with the classical Galerkin finite element method for spatial discretization. The stability of the semi-discrete scheme is established in the L2 norm, and a convergence analysis of the fully discrete scheme is carried out, yielding a temporal convergence order of O(Δt)3−μ and a spatial convergence order of O(R2). Finally, two numerical examples, in one and two dimensions, are provided to demonstrate the accuracy and efficiency of the proposed method and to verify the theoretical findings.

Fractal and Fractional doi: 10.3390/fractalfract10100709

Authors: Xiangchun Chang Hui Zhang Meng Wang Mengna Zhou Junjian Zhang

To investigate the factors of pore–fracture structure heterogeneity in middle rank coal reservoirs, middle rank coal samples collected from the deep coal seams in the northern part of the Kuqa area in the Tarim Basin are analyzed. Firstly, according to the maximum vitrinite reflectance (Ro,max), all the samples are divided into two types, Type A and B (A: Ro,max is larger than 1.2%; B: Ro,max is smaller than 1.2%). Secondly, the full-scale pore–fracture system was characterized using a combination of three experiments: high-pressure mercury intrusion testing, low-temperature CO2 adsorption testing, and low-temperature nitrogen adsorption testing. Subsequently, multifractal theory was applied to evaluate pore structure heterogeneity and investigate its governing factors. The results are as follows: (1) Type A samples exhibit superior micropore development, whereas Type B samples show a higher degree of mesopore and macropore development. Both types of samples display a “bimodal” distribution across the full-scale pore. Type A pore structure is affected by Ro,max and macerals, with mineral components having a minor impact, while Type B is regulated by all three factors. (2) The coal samples exhibit distinct multifractal characteristics. Type A pore structure heterogeneity is higher than that of Type B. This heterogeneity in Type A is dominated by the differences in the development and distribution of small-scale pores (micro- and mesopores), whereas in Type B, it is governed by the full-scale pore system distribution.

Fractal and Fractional doi: 10.3390/fractalfract10100708

Authors: Ahmadjan Muhammadhaji Yahui Niu

With the widespread adoption of internet technology globally, the propagation of rumors on social networks has attracted significant scholarly attention. This paper discusses a type of fractional-order rumor propagation model incorporating a saturation effect. In this model, it is assumed that the state of an ignorant individual will invariably change upon contact with a spreader, while the influence of time delays is also considered. Subsequently, we focus on analyzing the existence of a Hopf bifurcation arising from the time delay between ignorant individuals and spreaders, along with the dynamic behavior of the model. Stability analysis is conducted using methods such as the fractional-order stability criterion. Finally, a series of numerical simulations validate our findings and reveal that increasing the fractional order reduces the critical bifurcation value, thereby shrinking the stability region. Therefore, the stability region of the model can be regulated by adjusting the fractional order.

Fractal and Fractional doi: 10.3390/fractalfract10100707

Authors: Abdullah Alghafis Mohamed Nejlaoui

The primary objective of a parabolic distributed solar collector system (P-DSCS) is to regulate fluid heat generation so that it tracks a reference trajectory despite non-uniform solar irradiance. Fractional-order proportional-integral-derivative (FOPID) controllers offer superior dynamic flexibility to handle these complex thermal transients. However, existing tuning strategies for P-DSCS collapse frequency-domain requirements into an optimization problem solved by standard algorithms, masking critical trade-offs between competing performances and omitting closed-loop stability verification. To overcome these limitations, this paper presents an original multi-objective optimization framework for FOPID controlled P-DSCS. The tuning problem is formulated as a three-objective Optimization Problem that simultaneously minimizes phase-margin slope sensitivity for iso-damping robustness, peak sensitivity for low-frequency disturbance rejection, and complementary sensitivity for high-frequency noise attenuation. The Multi-Objective Imperialist Competitive Algorithm (MOICA) extracts a three-dimensional Pareto front, offering optimal control trade-offs. Comparative simulations against PID and literature algorithms (Multi-objective Particle Swarm Optimization (MOPSO) and Non-dominated Sorting Genetic Algorithm II (NSGA-II)) validate the superior performance of the MOICA-FOPID framework. MOICA consistently outperforms all algorithms across all metrics, improving iso-damping robustness by 38.99%. It also reduces high-frequency noise by 25.66%, while enhancing low-frequency disturbance rejection by 10.87%. Moreover, the optimal MOICA-FOPID controller maintained superior robustness despite P-DSCS gain and time-constant fluctuations.

Fractal and Fractional doi: 10.3390/fractalfract10100706

Authors: Xiaotian Hao Yuhang Tao Kangsheng Xue

Solute transport in rough fractures is governed by surface geometry, shear-induced aperture redistribution, and hydraulic conditions. This study presents a comparative three-dimensional numerical investigation of conservative solute transport in three individual self-affine fracture realizations with JRC values of 9.43, 13.18, and 17.37, five prescribed shear displacements (Us = 1–5 mm), and seven Peclet numbers (Pe = 0.1–1500), yielding 105 finite-element cases. For the selected realizations, increasing roughness and shear displacement reorganized the aperture field, enhanced streamline tortuosity, and increased flow heterogeneity, particularly in the medium- and high-roughness geometries. Under the investigated low-Reynolds-number conditions, the pressure-gradient-flow-rate relationships remained approximately linear. Mechanical aperture, hydraulic aperture, and intrinsic permeability evolved asynchronously: hydraulic transport capacity remained nearly stable in the low-roughness realization but decreased progressively in the medium- and high-roughness realizations. Pe primarily controlled the breakthrough regime. Diffusion dominated at Pe = 0.1, mixed advection-diffusion behavior occurred at Pe = 1–10, and advection dominated at Pe ≥ 100. Increasing shear displacement mainly affected the early and late portions of the breakthrough curves, whereas the central transition changed comparatively little. At high roughness and high Pe, concentration isosurfaces showed stronger preferential transport and delayed migration in low-velocity or weakly connected regions. The resulting apparent local retention denotes transient storage and delayed exchange within the fracture void space. No independent experimental validation, multilevel mesh-convergence test, stochastic ensemble analysis, systematic parameter-sensitivity analysis, or breakthrough-curve-derived residence-time, dispersivity, tailing, or transport-aperture metrics were performed. The findings should therefore be interpreted as comparative trends for the three investigated rigid-wall, isolated-fracture, steady-flow, and conservative-transport models, rather than as statistically representative JRC-class relationships or field-scale predictions.

Fractal and Fractional doi: 10.3390/fractalfract10100705

Authors: Klaudia Oleshko María de Jesús Correa López Andrey Khrennikov Qiuming Cheng José Luis Landa Ramiro Guillermo Paz Cruz Alejandro Romero Paulina Patiño Yesica Guerrero Amador

Fracture networks control fluid flow, reservoir connectivity, and production performance in carbonate systems, yet their multiscale hierarchical organization remains difficult to characterize from heterogeneous geological and geophysical datasets. Here, we introduce physics-informed visio-numerical primitives that jointly encode numerical, visual, geometric, and topological attributes of complex fracture patterns. A SYM-Fractron is a computational pipeline that fuses these patterns through the Digital Transformer (DiT) framework, enabling positional visualization and classification of Big Geodata. The multisensor, multiscale metadata constitute the symbolic–numerical representations derived directly from physical measurements. Instead of textual tokenization, the proposed framework employs an effective number metric based on Diophantine, p-adic, and multifractal approximations to transform metadata into discrete integer-based tokens. These tokens are not interpretable as new physical observables. Instead, they are introduced as computational entities designed to preserve relevant structural, spatial, and statistical positional information from the original datasets, which is likely useful for floating-point multiscale data comparison. The framework integrates two complementary computational environments: Muuk’il Kaab (MIK) for multidimensional metadata fusion and Spatial Digital Twin construction, and SYM-Fractron for hybrid binary–symbolic image analysis. Digital Twins are virtual replicas of a system that can be digitized, while Spatial Digital Twins emphasize the geospatial aspects of the whole physics-informed positional metadata. We unified coupled visual and statistical numerical representations of geological system metadata with their computational counterparts, enabling interpretable fracture classification and supporting well-trajectory optimization, using dolomitized carbonate reservoirs as an example. We achieve quantitative comparison between fracture systems and their Digital Twins through topology-preserving and position encoding. The original physical measurements are preserved and remain traceable throughout the computational workflow of signal-to-image transformations and analyses. Integer-based representations and color maps, with a user-selected chromatic scale, are used for tokenization. Validation is supported by visual and statistical consistency between the original spatially encoded datasets and their corresponding Spatial Digital Twins, while preserving the cross-scale topological consistency often lost in conventional AI workflows.

Fractal and Fractional doi: 10.3390/fractalfract10100703

Authors: Waqar Afzal Mujahid Abbas Zareen A. Khan

Fractional differential operators equipped with classical and new memory kernels have emerged as indispensable tools for modeling systems with hereditary and non-local effects, offering a flexible framework that extends far beyond the classical integer-order calculus. Within this broader landscape, pseudo-differential and Schrödinger operators play a very important role in mathematical analysis, arising naturally in the study of partial differential equations, quantum mechanics, and harmonic analysis, where they provide the fundamental tools for analyzing regularity, boundedness, and decay properties of solutions. The objective of this article is to define a new Herz norm with a variable exponent, a Morrey-type truncation, a grand modification, and an Ap(·) Muckenhoupt weight, and to prove two boundedness theorems. First, we show that every order-zero pseudo-differential operator is bounded on the resulting weighted grand variable Herz–Morrey–Hardy space HMK˙Γ,p(·)α(·),υ)(w). Second, for the Schrödinger operator L=−Δ+V with V in the reverse Hölder class RHq, q≥n/2, we replace the classical Riesz kernel with the Bessel–Riesz kernel and establish new, improved strong-type bounds for the associated Bessel–Riesz operator, from a weighted variable Lebesgue space into a weighted grand variable Herz–Morrey–Hardy–Lipschitz space built from the critical-radius geometry of L and using the same origin-centered dyadic scale as in the pseudo-differential case. Both proofs proceed via decompositions into uniform per-atom (respectively, per-ball) estimates, which are aggregated in the grand–Morrey norm under a decay condition on the Morrey parameter on the Herz–Hardy scale and under the threshold condition Γ≥1/υ on the Lipschitz-type range, together with a complementary condition on the Herz-type domain. We also show how the Bessel parameter γ enters the admissible range of integrability exponents. To further demonstrate the accuracy and consistency of these new results, we include remarks showing that, under different settings, various relevant results in the literature are recovered and improved, together with additional norm-structure and operator improvements.

Fractal and Fractional doi: 10.3390/fractalfract10100704

Authors: Nopanom Kaewhanam Thammanun Chatwong Apichit Kampala Sitthiphat Eua-apiwatch Sivarit Sultornsanee

A clay flow rule governs contraction or dilation during plastic shearing. We audit a terminal-directed scalar Caputo integral on a teardrop meridian under specified elasticity and hardening. Its zero coincides with the non-associated base-model terminal only if Ω = Ψ. Normally consolidated analysis establishes a stationary stress ratio, mean and deviatoric stresses, and surface size, with continuing plastic shear in the numerical coordinate. For inherited kaolin coefficients, the limiting pressures differ by 18.6% on a common stress locus. Both branches approach their limits asymptotically, with logarithmic divergence of the loading coordinate for the base rule and power-law divergence for the Caputo rule. Mean-stress preservation permits exact scalar composition with the Spatially Mobilised Plane transformation but supplies no physical strain mapping. Two overconsolidated substitutions, F1 and F2, defined here rather than reproduced from published models, expose hardening mismatch, exterior drift, loss of an inherited positive-modulus condition, or negative work under the declared pairing. Fujinomori stress-locus agreement cannot identify fractional order or validate deformation. These results establish compatibility requirements for the specified scalar embedding, not a complete clay model, predictive stress–strain capability, or physical dissipation proof.

Fractal and Fractional doi: 10.3390/fractalfract10100702

Authors: Agniya G. Borodina Ivan D. Solovyev Nikolay A. Kudryashov

Most studies of generalized Burgers–Huxley equations model spatial transport with a second-order diffusion term. Fourth-order spatial terms have received less attention, although they occur in long-range dispersal and regularized reaction–diffusion models and change the dominant balance of traveling-wave reductions. We consider a fourth-order generalized Burgers–Huxley-type equation with local truncated M-fractional derivatives and reaction polynomials of degrees two, three, or five. For the differentiable solutions considered here, the truncated M-operator is local, and its parameters enter through an exact nonlinear reparameterization of the independent variables rather than through a memory kernel. A derivative-enriched elliptic expansion reduces the resulting fourth-order differential equation to algebraic coefficient systems. The bounded auxiliary profiles are Jacobi sn, cn, and dn functions, with tanh and sech obtained when roots of the auxiliary quartic coalesce. Within this ansatz, the coefficient systems yield two convective quadratic, two stationary quadratic, three stationary cubic, and one stationary quintic class in centered and shifted forms; only the convective quadratic families admit nonzero speed. Reality conditions and equivalences under reflection, phase and sign changes, and singular limits identify distinct bounded profiles. These profiles may provide closed-form reference solutions for numerical studies and illustrate the effect of the local coordinate transformation.

Fractal and Fractional doi: 10.3390/fractalfract10100701

Authors: A. Alameer

In this paper, we introduce and study a normalized four-parameter Le Roy-type q-Mittag–Leffler (LR-q-ML) function defined by means of the q-Pochhammer symbol, the q-Gamma function, and the q-factorial, whose parameters directly influence the associated coefficient sequence and its growth behavior. We establish monotonicity properties of the related coefficient sequences, which are then used to derive coefficient estimates, Fekete–Szegö inequalities, upper bounds for the normalized function and its q-difference operator, and lower bounds for the real parts of the ratios involving the normalized function and its partial sums. Numerical computations together with graphical illustrations verify the theoretical estimates and show that the partial sums provide accurate approximation to the complete function as the number of terms increases. The obtained results provide a unified framework for investigating geometric properties and coefficient results of higher-parameter q-special functions in geometric function theory.

Fractal and Fractional doi: 10.3390/fractalfract10100700

Authors: Yasir Muhammad Muhammad Zeeshan Abdul Wadood Shahbaz Khan Hani Albalawi Herie Park Byung O Kang Aamir Nawaz

Reliable, economical, and secure operation of a power system depends heavily on how well its reactive power is managed. Optimal reactive power dispatch (ORPD) coordinates generator voltage magnitudes, transformer tap settings, and switched shunt compensation so that bus voltages stay within acceptable limits and transmission losses are kept low. Because these controls interact through nonlinear power-flow relations, and because tap positions and shunt banks are physically discrete while generator voltages are continuous, the problem is non-convex, multimodal, and mixed-integer. Continuous optimizers must round their solutions and therefore never search the discrete space directly, while conventional single-lag velocity updates discard the information of earlier iterations and lose swarm diversity. To address both limitations simultaneously, a sigmoid fractional binary particle swarm optimization–gravitational search algorithm (FBPSOGSA) is proposed. The velocity is updated through a truncated Grünwald–Letnikov fractional-order expansion of depth four, which retains a weighted memory of the four most recent velocities, and is mapped to a bit-switching probability through a sigmoid (S-shaped) transfer function that drives the discrete controls directly in binary space. The method is validated on the IEEE 30-bus test system for two objectives. For active power loss minimization, the FBPSOGSA attains 5.4877 MW at a fractional order of 0.2, the lowest value among all compared methods, corresponding to improvements of 19.57% over FAHLCPSO, 13.44% over OGSA, 0.96% over KHA, and 0.89% over GSA. For voltage deviation minimization, it attains 0.1059 p.u. at a fractional order of 0.1, again the lowest among all compared methods, with improvements of 15.82% over PSOGSA, 9.80% over FPSOGSA, 8.07% over FHPSO, and 64.26% over KHA. All reported control settings satisfy the system operating constraints, and the consistency of the solutions is confirmed over 25 independent runs.

Fractal and Fractional doi: 10.3390/fractalfract10100699

Authors: Nan Yang Bin Li Yang Gao Qing He Jinyang Li Jiawei Wan Jun Li

Deep excavation reduces lateral confinement and can increase tangential stress, promoting brittle failure in highly stressed rock. However, the effects of bolt number and spacing on mechanical response and damage evolution under this stress path remain insufficiently understood. In this study, parallel RFPA3D simulations were conducted for true-triaxial single-face unloading followed by continued axial loading, assuming perfect bonding at the bolt–rock interface. Two parameter series examined bolt numbers from zero to four (M0–M4) and bolt spacings from 10 to 80 mm. Stress–strain response, acoustic emission activity, failure morphology, damage area ratio (Ad), and box-counting fractal dimension (Df) were analysed. Bolt reinforcement increased post-unloading peak stress and restricted the inward extent of damage, while damage persisted near the unloaded face and within the inter-bolt regions. Peak stress increased from approximately 345 MPa in M0 to 363 MPa in M3, then decreased slightly to 358 MPa in M4, showing that reinforcement gains were not proportional to bolt number. Bolt spacing had a modest effect on peak stress but a clearer influence on damage continuity and inward extent. Among the tested spacings, the 40 mm case (D40) achieved the highest peak stress of 370.0 MPa and the lowest Ad of 23.85%, with a low Df of 1.431. In contrast, the 80 mm case (D80) exhibited more extensive and spatially complex damage (Ad = 30.78%, Df = 1.677). Ad and Df provided complementary descriptions of damage extent and spatial complexity. These numerical results provide a quantitative basis for comparing reinforcement configurations under the investigated specimen-scale conditions.