Conventional Rate Transient Analysis (RTA) models, based on homogeneous fracture assumptions, are inadequate for characterizing flow in complex fracture networks of heterogeneous unconventional reservoirs. This study develops a fractal-based RTA (FD-RTA) workflow integrating lithofacies analysis, microseismic fracture interpretation, and post-fracturing production data from the Yingxiongling shale oil field in the Q’aidam Basin. The workflow is applied to eight horizontal wells completed in layered and laminated dolomites. Results show that the two lithofacies exhibit distinct fractal flow behaviors. Layered dolomite tends to develop preferential flow pathways, characterized by rapid initial depletion followed by declining supply capacity, with the half-flow dimension (δ) decreasing from 0.299 to 0.074 during production. Laminated dolomite displays stronger fracture-matrix interaction and sustained production performance, with δ increasing from 0.469 to 0.678 as multi-scale fractures are progressively activated. The FD-RTA workflow effectively links fracture complexity with production behavior, providing a dynamic characterization tool for evaluating hydraulic fracturing effectiveness in shale oil reservoirs.
A fractional-order mathematical model is derived for mpox transmission dynamics that explicitly incorporates the influence of public awareness through a new awareness-dependent transmission function. The proposed incidence function models the reduction in disease transmission as the level of awareness increases, thereby capturing the dynamic interaction between epidemic progression and behavioral response. The model further accounts for asymptomatic infection, hospitalization, recovery, and awareness evolution using Caputo fractional derivatives to incorporate memory effects. Basic mathematical properties of the model, including positivity, boundedness, and existence of solutions, are established. The basic reproduction number, R0, is derived using the next-generation matrix approach, and a sensitivity analysis is performed to identify the epidemiological parameters that most strongly influence disease transmission. Local and global stability analyses demonstrate that the disease-free equilibrium is locally and globally asymptotically stable whenever R0<1, while an endemic equilibrium exists and is globally asymptotically stable when R0>1 and a forward transcritical bifurcation occurs at R0=1. Numerical simulations validate the analytical findings and illustrate the significant role of sustained public awareness in reducing disease transmission and mitigating epidemic outbreaks. The results obtained from the proposed model suggest that combining behavioral awareness strategies with conventional public health interventions can substantially improve the long-term control of mpox.
Volcanic gas and isotope anomalies can retain delayed effects of shallow hydrothermal storage, transfer and selective chemical removal before they are measured at the surface. A single Caputo order provides a parsimonious memory law, but finite cutoffs, multiple relaxation slopes and discrete exchange domains may require broader descriptions. This theoretical study formulates hydrothermal memory first as a causal Volterra process and distinguishes positive-spectral and Sonine-compatible refinements of that class without assuming that either implies the other. Exact initialized diffusive realizations, positive finite-reservoir approximations, a quadratic storage identity and finite-window kernel-to-observable error bounds are developed. Synthetic experiments compare memory classes through scalar, gas-ratio and isotope responses. The principal statistical benchmark is deliberately restricted to the nested Caputo and exponentially tail-tempered families, thereby testing whether one additional cutoff parameter is supported. In a key negative-control experiment, omission of a slow inherited mode caused AICc to prefer tail tempering in 72.0% of replicates even though the post-observation kernel remained Caputo. This demonstrates that incorrect memory initialization can masquerade as a constitutive cutoff. The study has not yet been validated against real volcanic gas or isotope records, and no site calibration is claimed.
Geological carbon sequestration is an important technology for mitigating climate change by reducing atmospheric CO2 emissions. The geological storage process is governed by complex multiphase flow behaviors in porous media. Classical two-phase flow models are generally formulated using integer-order derivatives, which may not adequately capture the memory effects caused by complex porous media structures. In this work, a pressure–saturation coupled two-phase flow model with a fractional saturation equation is established. The normalized quasi-static pressure problem is first analyzed, while the fractional saturation problem is studied through a conditional fixed-point argument. Under explicitly assumed compatibility, uniform pressure regularity, L2-valued well-definedness and local Lipschitz continuity of the saturation operator, and truncation regularity, an invariant-region estimate ensures that the gas saturation remains in the physical interval [0,1]. These stronger operator properties are imposed as structural hypotheses rather than derived from the basic L2 saturation and H1 pressure spaces. The pressure and saturation solution operators are then composed, and a contraction argument on a sufficiently small time interval establishes conditional local existence and uniqueness within the prescribed admissible class KR(T)×DT, where DT⊂Xp(T). This work provides a conditional analytical framework for the fractional pressure–saturation system under explicit structural hypotheses.
This paper investigates the projective synchronization problem for a class of non-autonomous neural networks with Caputo fractional derivatives and mixed delays. First, by simultaneously introducing discrete and distributed delays into the model, the drive and response systems and the corresponding error system are constructed, and a synchronization controller is designed. Second, by constructing an appropriate quadratic Lyapunov function and estimating the nonlinear and mixed-delay terms in a unified manner, a fractional Halanay-type delay inequality is derived. Based on this inequality, sufficient conditions are obtained to guarantee global asymptotic projective synchronization for this class of neural networks. Finally, numerical simulations are presented to verify the theoretical results and the numerical consistency of the adopted discretization scheme.
We present a unified framework for two nonlinear rational difference equations with arbitrary delay. A residue-class decomposition reduces each delayed equation to independent second-order recurrences. Product and quotient transformations then reduce these recurrences to first-order Möbius maps. This structure yields explicit solutions through powers of 2×2 matrices and supports a systematic analysis of admissibility, singular cases, equilibria, periodic solutions, multiplier spectra, and asymptotic behavior. A spectral lifting relation connects the multipliers of each residue map to those of the full delayed system and identifies the neutral directions introduced by reconstruction. We then define a Caputo fractional difference extension of the reduced Möbius dynamics. At order one, the ordinary reduced map is recovered exactly and the admissible fixed points are unchanged. For 0<ν<1, the extension is studied only at the reduced-map level and is not asserted to reconstruct a solution of the original delayed equations. For fractional orders below one, stability depends on the order. In particular, a negative attracting Möbius multiplier produces an explicit critical-order threshold, and locally stable fractional solutions converge algebraically rather than geometrically. Numerical examples verify the exact reconstruction, spectral lifting, ordinary asymptotic regimes, and fractional-order threshold.
In this paper, a fractional-order active disturbance rejection repetitive control (FADRRC) scheme is proposed for fractional-order time-delay systems. The unified control framework integrates a bandwidth-parameterized fractional extended state observer (FESO), modified repetitive control and a filtered Smith predictor. The FESO online estimates system states and lumped disturbances, the filtered Smith predictor compensates time-delay deviation to weaken sensitivity to model mismatch, and repetitive control eliminates steady-state error for periodic reference signals. By fractional frequency-domain stability theory and the small-gain theorem, two BIBO stability criteria for nominal and mismatched cases are derived, reformulating the controller design as a pole placement problem of fractional-order transfer functions. Numerical simulations on a fractional-order time-delay PMSM servo system verify that the proposed FADRRC possesses superior periodic tracking accuracy and disturbance rejection capability compared with existing ESO-RC and DOB-RC methods, which confirms the validity of the presented control strategy.
The paper investigates the asymptotic stability for the control of systems described by multi-order fractional differential equations. By utilizing generalized Lyapunov functions and the Kalman–Yakubovich–Popov lemma, frequency-domain criteria are derived to evaluate asymptotic stability. The formulated criteria are similar to the ‘Popov Circle Criterion,’ but the circle parameters are determined by the system’s fractional order and the control parameters. Additionally, the asymptotic stability condition requires that all polar plots associated with the multi-fractional-order system lie inside the circle defining the criterion. Human–Robot System applications highlight the investigation techniques and the particularities of the presented criteria.
This paper presents a novel fractional-order hybrid observer framework for robust sensorless control of brushless DC (BLDC) motor drives in unmanned aerial vehicle (UAV) propulsion systems, addressing the fundamental limitations of conventional integer-order observers through the lens of fractional calculus. The proposed architecture synergistically integrates high-frequency square-wave signal injection for zero/low-speed operation with an adaptive fractional-order extended Kalman filter (AFEKF) augmented by online stator resistance and flux linkage estimation, capitalizing on the memory and hereditary properties inherent to fractional-order systems. A minimum-order current observer enables accurate three-phase current reconstruction using a single DC-link sensor, substantially reducing hardware complexity and cost. The complete algorithm is implemented on an STM32H7 microcontroller and experimentally validated on a 1.5 kW drone propulsion testbench and in-flight platform. Results demonstrate reliable startup under 50% rated load, stable operation from standstill to 5000 RPM on the UAV motor (and validated up to 22,000 RPM on a high-speed test motor, <4° electrical position error at 5 kRPM, and strong robustness against 35% stator resistance variation. In-flight tests confirm improved thrust smoothness and hover stability compared to conventional sensorless strategies. The proposed fractional-order architecture offers a practical, resilient, and computationally feasible solution for next-generation autonomous aerial systems, establishing a new paradigm for observer design in electric propulsion.
Conventional integer-order parameter designs of grid-forming inverters provide limited degrees of freedom for impedance adjustment, motivating the exploration of additional approaches for flexible impedance reshaping across different frequency ranges. This paper establishes a full fractional-order grid-forming inverter (FFO-GFMI) by incorporating fractional-order inductor-capacitor (LC) filters, corresponding decoupling control, and fractional-order multi-loop controllers into a conventional grid-forming inverter. Based on the harmonic linearization method, positive- and negative-sequence impedance models of the FFO-GFMI are developed to characterize its broadband impedance characteristics. The developed models are validated through impedance scanning, and the effects of fractional-order parameters on broadband impedance characteristics are systematically investigated. The results reveal that fractional-order LC filters mainly regulate medium- and high-frequency resonance characteristics, while fractional-order control loops provide effective low- and medium-frequency impedance reshaping. Furthermore, load-step simulations demonstrate that the selected fractional-order configuration improves dynamic performance, reducing the active power settling time from 0.3121 s to 0.1974 s and the active power overshoot from 19.75% to 4.51%.
Nonlinear electromagnetic wave propagation in complex plasma environments has attracted considerable attention due to its important applications in nonlinear optics, plasma physics, space science, and communication technologies. In the present study, a time-fractional Drinfel’d–Sokolov–Wilson equation (DSWE) is investigated under the influence of electromagnetic wave perturbations. The fractional-order formulation incorporates memory and hereditary effects, providing a more realistic description of wave propagation in nonlinear dispersive media. By employing an appropriate fractional traveling-wave transformation, the governing nonlinear fractional partial differential equation is reduced to a nonlinear ordinary differential equation. Exact solitary wave solutions are subsequently constructed using the G′G2-expansion technique. Furthermore, the nonlinear dynamical behavior of the reduced system is examined through phase portraits, bifurcation diagrams, Lyapunov exponents, sensitivity analysis, and multistability investigations. Particular attention is devoted to understanding the emergence of chaotic dynamics induced by electromagnetic wave effects and fractional-order interactions. The obtained results reveal that the fractional-order parameter significantly influences the stability, propagation characteristics, and dynamical evolution of nonlinear wave structures. The coexistence of multiple attractors, transitions between stable states, and chaotic regimes is identified for various parameter configurations. These findings provide deeper insight into the complex dynamics governed by the time-fractional DSWE and contribute to the understanding of nonlinear electromagnetic wave propagation in plasma and other nonlinear dispersive media.
In this paper, the new fractal modified Zakharov–Kuznetsov equation (fmZKe) defined on Cantor sets is investigated. The fmZKe is a non-differentiable model that arises naturally in mathematical physics, nonlinear wave theory, and plasma physics. The extended rational sine–cosine method is utilized to construct new optical soliton solutions of the model. The fmZKe is reduced to a non-differentiable ordinary differential equation by applying a non-differentiable wave transformation defined on Cantor sets. This reduction leads to a system of linear algebraic equations, which upon solving yields several exact solutions of the model. Furthermore, to establish a clear understanding of the model’s behavior, non-smooth graphical representations of the solutions are presented for various parameter values. The stability analysis of the newly obtained solutions in a classical sense is examined using stability theory, and the real-life applications of the results are highlighted.
Time delays are inherently present in active control systems as a consequence of sensor acquisition, communication lags, and actuator dynamics, and their impact on system behavior cannot be overlooked. This paper examines the effect of delayed displacement and speed feedback gains on the nonlinear lateral and vertical vibrational behavior of a MAGLEV vehicle subjected to aerodynamic and centrifugal forces. A delayed nonlinear dynamic model incorporating a fractional-order PD controller under aerodynamic excitation is first established for the MAGLEV system. Subsequently, the method of multiple scales is employed to derive the frequency response relationships, while the corresponding steady-state solutions are analyzed to determine system stability. The investigation further explores how the delays alter the nonlinear dynamic response. It also considers the impact of changing the value of the fractional-order parameter α on the vehicle’s dynamics. The results showed that increasing controller delays reduces the stability region, with displacement-feedback delays having a more pronounced effect than speed-feedback delays, while fractional-order derivatives (0<α<1) further degrade stability; consequently, the integer-order case (α=1) is recommended to achieve lower vibration levels and improved dynamic stability. The outcomes of this work provide valuable understanding of vibration phenomena encountered in MAGLEV systems and contribute to the development of improved control and optimization strategies for safer, smoother, and more reliable vehicle performance.
Tight sandstone reservoir effectiveness is governed not by pore volume alone, but by the storage and flow contributions of different pore-throat scales, their structural complexity, and their three-dimensional connectivity. This study investigates tight sandstones of the Benxi Formation deposited in a marine–continental transitional mixed siliciclastic–carbonate setting in the Gaoqiao area, southern Ordos Basin. Petrophysical measurements, red-epoxy-impregnated thin-section petrography, mercury intrusion capillary pressure (MICP), segment-specific fractal analysis of functionally defined pore-throat regimes, X-ray micro-computed tomography (micro-CT), and pore-network modeling (PNM) were integrated. The MICP responses define three pore-throat structure types and two data-derived functional boundaries at 0.708 and 0.141 μm, which separate large-pore-throat-dominated, transitional pore-throat, and fine-throat-limited intervals. Using these nominal boundaries, Type I is strongly dominated by the large-pore-throat interval, which accounts for 88.7% of total mercury intrusion, whereas Type II exhibits a mixed large-to-transitional response, and Type III is characterized by negligible large-pore-throat intrusion and pronounced fine-throat restriction. Perturbing both functional boundaries by ±5% and ±10% does not alter these principal functional distinctions, although samples close to the second boundary exhibit the expected local transitional sensitivity. Among the three segment-specific fractal parameters, the fine-throat fractal dimension, DB, shows the strongest association with median capillary pressure (r = 0.834, p < 0.001) and remains significantly related to displacement pressure, median pore-throat radius, and permeability, whereas DT shows no significant linear correlation with the tested petrophysical and MICP parameters. The fractions of the largest connected pore cluster in representative Type I–III samples are 90.26%, 72.56%, and 64.65%, while their PNM permeabilities decrease successively from 64.32 mD to 0.850 and 0.121 mD. Together, these results indicate that, for the investigated samples, reservoir effectiveness reflects the combined influence of pore-throat size configuration, segment-specific structural complexity, and three-dimensional network connectivity.
Integrated Modular Avionics (IMA) integrates safety-critical functions on shared computing and network resources, creating coupling channels through which a local fault may escalate into a catastrophic system-level scenario. This study develops a graph-based fractional-order model for cascade escalation in IMA architectures with communication delays and reconfiguration failures. The architecture is represented as a weighted directed graph of core processing modules, network switches, and remote data concentrators, where each node carries functional degradation and queue-backlog states. The proposed delayed Caputo fractional-order dynamics incorporate degradation propagation, backlog spillover, mixed-criticality priority conflict, and a state-dependent reconfiguration-failure mechanism. We establish well-posedness and positive invariance of the feasible state domain, derive a sufficient cascade threshold that separates a delay-independent, globally Mittag–Leffler stable nominal regime from a supercritical regime in which bistability and catastrophic attractors may occur, and characterize delay-induced oscillatory instability together with a memory-stabilization effect. Numerical experiments on a synthetic 22-node IMA configuration show fault absorption below the threshold, reconfiguration-contained cascades under sufficient supervisory capacity, and global escalation when reconfiguration collapses under load. The results indicate that backlog growth is an early warning signal and that maintaining the cascade threshold below unity while provisioning reconfiguration capacity above the tipping point can support safer reconfiguration-policy design in certifiable avionics.
In this study, random-order FDEs are studied with a focus on the variability of their solutions and random orders when the order of differentiation is governed by a probability distribution. Fractional differential equations (FDEs) offer a generalized modeling approach that enables the analysis of nonlocality and memory effects. However, the deterministic framework for studying FDEs neglects the random nature of real-life events. In this regard, four continuous probability distributions with bounded support (uniform, Beta, triangle and Bates) are used to analyze the variability of the solutions depending on the random order, which is defined to vary between [0.65, 0.95] according to these probability distributions with identical expected values. Monte-Carlo simulations with N=215 repetitions are performed to investigate the random-order FDEs using a predictor-corrector approach. Results show that the decrease in the deviation from the uniform distribution to the Bates distribution is reflected in the random characteristics of the solutions and the order of differentiation to almost the same extent. The findings indicate that the average solutions obtained with random orders from each distribution show almost identical results, whereas the variability changes significantly based on the distribution of the order of differentiation. This information provides useful guidance in working with probabilistic models instead of deterministic systems for uncertainty quantification or sensitivity analysis.
Deep shale reservoirs in the tectonically complex margin of the southern Sichuan Basin have experienced multistage deformation, resulting in strong spatial heterogeneity of the present-day geostress field. However, the influence of stress heterogeneity on multiscale pore structure evolution and reservoir quality remains poorly constrained. Here, we integrate in-situ stress measurements, overburden porosity and permeability experiments, CO2/N2 adsorption, high-pressure mercury intrusion, SEM-MAPS (Scanning Electron Microscopy-MAPS) pore imaging, stress well profile interpretation, and multifractal analysis to quantify the controls of present-day geostress heterogeneity on pore structure evolution in deep Longmaxi Formation shale. The results show that the present-day stress regime is characterized by a strike-slip pattern (σH > σv > σh), with significant variations among different structural deformation zones. Increasing structural deformation results in enhanced differential stress, increasing by 30–80% from gentle structures to tight folds and fault-affected zones, accompanied by a 60–70° rotation of the maximum principal stress orientation. Differential stress, effective stress, differential stress coefficient, and stress structure index exhibit strong negative correlations with porosity, whereas permeability decreases nonlinearly with increasing stress, indicating progressive pore-throat compression and connectivity degradation under heterogeneous stress conditions. Multifractal analysis reveals that pore-size domains exhibit different sensitivities to stress heterogeneity. The macropore fractal dimension (DN3) shows the strongest response, followed by mesopores (DN2), whereas micropores (DN1) exhibit relatively limited variations. Fault-affected zones and strongly deformed regions display higher DN3 values (>2.8), reflecting enhanced complexity of macropore and fracture networks. In contrast, gentle structural zones characterized by curvature values <0.10 km−1 and distances >500 m from faults exhibit relatively low and stable fractal dimensions (<2.73), indicating more homogeneous pore structures. Increasing stress heterogeneity induces the transformation of organic matter pores from regular subcircular shapes to flattened and slit-like morphologies, accompanied by pore-size migration toward smaller scales (<15 nm) and enhanced pore heterogeneity (Df > 1.35). These findings reveal that present-day geostress heterogeneity governs shale pore fractal evolution and promotes the transition from micropore-dominated to heterogeneous macropore–fracture systems. This study provides quantitative insights into stress-controlled pore evolution and reservoir quality evaluation in deep shale reservoirs under complex tectonic settings.
Impacted mandibular third molar surgery is one of the most common procedures in oral and maxillofacial surgery, and an accurate preoperative assessment of surgical difficulty remains clinically important. The Pederson difficulty index is widely used for radiographic difficulty estimation, whereas fractal dimension analysis provides a quantitative assessment of trabecular bone complexity. This retrospective observational study aimed to evaluate the relationship between the Pederson difficulty index and the fractal dimension of the alveolar bone surrounding impacted mandibular third molars on panoramic radiographs. A total of 207 impacted mandibular third molars were obtained from 113 patients. Demographic data, tooth side, and Pederson difficulty categories were recorded. Each impacted tooth was classified as easy, moderate, or difficult according to the Pederson difficulty index (PDI). Fractal dimension measurements were performed in three predefined regions of interest: mesial, apical, and distal alveolar trabecular bone. The study population had a mean age of 22.63 ± 4.36 years, and the tooth-side distribution was balanced between teeth 38 and 48. No significant association was found between sex or tooth side and the Pederson difficulty category. Mesial, apical, and distal fractal values did not differ significantly according to sex or Pederson difficulty category. Panoramic radiography-based fractal dimension measurements were not significantly associated with Pederson difficulty categories for impacted mandibular third molars.
This paper investigates the qualitative and topological behavior of random solutions for a class of partial fractional random differential equations governed by the Darboux problem. Unlike classical configurations that rely on bounded or finite delays, our theoretical framework explicitly addresses systems involving unbounded infinite delay. The dynamics of the state transitions are formulated using left-sided mixed Riemann–Liouville fractional integrals and joint Caputo fractional derivatives of order ε=(ε1,ε2)∈(0,1]×(0,1]. Because of the infinite historical horizon, the underlying model is constructed and analyzed within abstract, semi-normed axiomatic phase spaces defined over topological Fréchet spaces. By avoiding restrictive compactness assumptions on the nonlinear operational bounds, we establish novel random mild existence theorems. The structural proofs are achieved through a combination of a regular, sublinear family of axiomatic measures of noncompactness and an advanced generalization of the classical Darbo fixed-point theorem tailored for Fréchet domains. Finally, a concrete mathematical example is systematically analyzed to confirm the validity, consistency, and practical applicability of the established theoretical bounds.
