Fractal Fract., Volume 10, Issue 3 (March 2026) – 75 articles

In this work, we provide a convolution type operator ${\mathsf{\Lambda }}_{\nu ,b}$ that is produced by the generalized Marcum Q-function and examine how it maps to various Janowski-type subclasses of harmonic univalent functions. Since the Marcum Q-function has an integral form via the lower incomplete gamma function, the convolution operator ${\mathsf{\Lambda }}_{\nu ,b}$ can be understood as a fractional type integral operator operating on the coefficients of harmonic mappings. Applying ${\mathsf{\Lambda }}_{\nu ,b}$ to harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D}$, we derive coefficient inequalities, and inclusion relations for various subclasses of harmonic and analytic univalent functions. In particular, we give sufficient conditions for ${\mathsf{\Lambda }}_{\nu ,b}\left(f\right)$ to belong to Janowski-starlike families such as ${\mathcal{S}}_H^{\ast }(F,G)$, ${\mathcal{K}}_H^0$, and ${\mathcal{R}}_H(F,G)$. Closure properties of the Janowski class under the proposed operator are also established. Numerical tables and examples confirm the inclusion results, and graphical plots illustrate how the operator reshapes the image domains for different parameter pairs $(\nu ,b)$. Numerical illustrations are provided to visualize the geometric steering effect induced by the Marcum Q-function and its fractional-order damping behavior. Full article

Despite its importance in modeling subdiffusion in fractal and heterogeneous media, a rigorous and computational scheme for solving the fractional diffusion equation on generalized comb structures over unbounded domains has remained elusive, mainly due to the nonlocal memory effect and slow spatial decay of solutions. To the best of our knowledge, we address this long-standing gap by presenting a fully integrated framework that simultaneously resolves both challenges. We derive the governing equation from constitutive relations and establish exact absorbing boundary conditions (ABCs) for the multi-skeleton comb model, a result absent in prior work. A transparent Dirichlet-to-Neumann (DtN) map, constructed via Laplace analysis, rigorously handles skeletal Dirac delta singularities and eliminates spurious reflections without empirical parameters. Furthermore, we propose a novel structure-preserving finite difference scheme that applies the sum-of-exponentials (SOE) approximation not only to the interior Caputo derivative but also to the convolution kernels arising from the ABCs. This yields a dramatic reduction in computational complexity, from quadratic $O\left(N_t^2\right)$ to quasi-linear $O(N_{t}log{N}_t)$, while preserving the physics of anomalous transport. We prove the well-posedness, unconditional stability, and convergence of the method. Numerical results confirm theoretical error estimates and show excellent agreement between simulated particle distributions, mean square displacement profiles, and exact asymptotics, validating both accuracy and robustness. The speedup (CPU time ratio Direct/Fast) is about $1.00\times$–$1.23\times$ for $N_t=5000$ in our tests. Our approach sets a new benchmark for simulating anomalous dynamics in fractal-inspired media. Full article

This paper investigates the fractal dynamical behavior of a discrete Caputo fractional-order hepatitis C virus model. First, we analyze the stability of the system by using spectral radius and design the fractional-order controller based on coordinate transformation. Then, a nonlinear coupling controller is constructed to achieve synchronization between two fractional-order models with different parameters and different fractional orders, and the synchronization is supported by rigorous mathematical proof. Numerical simulations are used to verify the effectiveness of control and synchronization. Full article

This paper addresses the stabilization problem for a class of fractional-order memristive reaction–diffusion Cohen–Grossberg neural networks with time-varying delays under an intermittent boundary control framework. Two scenarios are considered: systems without parametric uncertainties, for which asymptotic stability is established, and systems with uncertainties, for which robust asymptotic stability is ensured. By constructing appropriate Lyapunov functionals and employing Wirtinger-type inequalities, the fractional Razumikhin approach, and key properties of the Mittag–Leffler function, sufficient stability conditions are derived in terms of linear matrix inequalities with reduced conservatism. Furthermore, the effects of time-varying delays and control activation intervals on the stabilization performance are systematically investigated. The effectiveness and advantages of the proposed control methodology are validated through numerical simulations. Full article

Econophysics is an interdisciplinary field that applies physics concepts to economic and financial systems. By utilizing tools such as statistical physics, including fractal analysis and entropy measures, econophysics helps model the complex and non-linear dynamics of equity markets. This paper examines the intrinsic dynamics and regularity in information content in green finance markets (carbon, clean energy, and sustainability markets) by means of range scale analysis (R/S), detrended fluctuation analysis (DFA), fractionally integrated generalized auto-regressive conditionally heteroskedastic (FIGARCH) process, and Shannon entropy (SE). The empirical results can be summarized as follows. First, prices in all markets are persistent; however, returns are likely random as estimated Hurst exponents are close to 0.5. Second, the FIGARCH process shows that volatility series in carbon and sustainability markets are persistent, whilst volatility in clean energy is anti-persistent. Third, in carbon and sustainability markets, entropy is high in prices compared to returns and volatility series. On the contrary, the clean energy market shows lower entropy for prices than for returns and volatility. In sum, it is concluded that price and volatility series are predictable, whilst return series are not. Finally, based on a rolling window framework, it is concluded that the COVID-19 pandemic and the Russia–Ukraine war have altered long memory and randomness in all three green finance markets. Full article

Prolonged water injection in unconsolidated sandstone reservoirs can induce pore rearrangement and modify flow pathways, thereby affecting reservoir performance. However, quantitative characterization of pore evolution in both temporal and spatial dimensions remains limited. This study investigates the mechanisms of pore-structure evolution during extended injection through a series of multi-scale experiments. Scanning electron microscopy and X-ray diffraction analyses were employed to compare mineral composition and microstructural characteristics before and after injection, while in situ nuclear magnetic resonance (NMR) monitoring captured the dynamic evolution process, enabling pore-size classification from T₂ spectra and fractal assessment of structural complexity. Segmented NMR measurements at different distances further resolved spatial heterogeneity. The results show that prolonged water injection reduced permeability by 10.4–32.1%, whereas porosity exhibited only minor variation, indicating that the decline in flow capacity is primarily controlled by pore–throat structural adjustment rather than pore volume loss. Mineralogical redistribution and fine-particle migration decreased the median pore radius by 21.5–51.8% and the micropore fractal dimension by 23.8–76.5%, with stronger responses observed at higher permeabilities, while meso- and macropore fractal dimensions remained nearly unchanged, indicating preferential modification of micropores with preservation of the main connected flow framework. Consistently, NMR responses reveal pronounced spatial heterogeneity along the flow direction. The NMR signal changes at the injection end were 11.2–18.4% and 7.7–21.7% during the early and intermediate stages, respectively, both exceeding those at the distal end (2.9–12.4% and 1.9–17.1%). These results indicate a downstream-attenuating structural modification gradient. The findings provide new insights into pore-structure evolution during prolonged water injection and offer a scientific basis for optimizing water-injection strategies in unconsolidated sandstone reservoirs. Full article

In this article, we investigate the existence and approximate controllability of a class of Sobolev-type fractional stochastic differential equations of order $1<\delta <2$ with infinite delay. The analysis is carried out in an abstract Hilbert space framework, incorporating fractional dynamics together with stochastic perturbations. By employing techniques from fractional calculus, semigroup theory, and fixed point theory, particularly the Banach contraction principle along with compactness arguments, we establish the existence of mild solutions for the proposed system. Subsequently, sufficient conditions for approximate controllability are derived by combining operator-theoretic methods with stochastic analysis. The novelty of this work lies in extending controllability results to Sobolev-type fractional stochastic systems of order $1<\delta <2$, where both the higher-order fractional structure and stochastic effects are treated simultaneously within a unified framework. This generalizes and complements several existing results in the literature that mainly address deterministic systems or fractional differential equations of order $0<\delta \le 1$. Finally, an illustrative example is presented to demonstrate the applicability and effectiveness of the theoretical findings. Full article

The fractional Fourier transform (FRFT) serves as an effective tool for linear frequency modulated (LFM) signal parameter estimation, whose performance depends on the search efficiency for the optimal transform order. To address the issues of fixed inertia weight in the standard particle swarm optimization (PSO) algorithm, which tends to fall into local optima and suffers from insufficient convergence accuracy, this paper introduces a proportional-integral-derivative (PID) control strategy and proposes a PID-PSO-FRFT-based LFM signal parameter estimation method. This approach introduces a PID controller, which takes the deviation between the particle’s current position and the global best position as input and dynamically adjusts the inertia weight through proportional, integral, and derivative regulation, thereby achieving an adaptive balance between global exploration and local exploitation capabilities of the particles. Simulation results demonstrate that, compared with the basic PSO-FRFT algorithm, the proposed method significantly improves the estimation accuracy of the center frequency and chirp rate of LFM signals under SNR conditions ranging from −9 dB to −7 dB, while considerably reducing computation time, exhibiting superior noise resistance, and exhibiting superior robustness. Full article

This paper investigates a class of coupled $\psi$-Hilfer fractional pantograph–Langevin equations with nonlocal integral boundary conditions. By reformulating the problem as an equivalent fixed point equation and employing Mönch’s fixed point theorem together with the Kuratowski measure of noncompactness, we establish sufficient conditions for the existence of at least one solution. Under additional Lipschitz-type assumptions, we prove Ulam–Hyers stability on a suitable closed ball and derive explicit, computable stability constants. A concrete numerical example is presented in which all hypotheses are verified and the stability constants are explicitly computed (e.g., $K_1\approx 3.811$, $K_2\approx 2.761$), illustrating the applicability of the theoretical results. The study contributes additional qualitative results to the analysis of fractional pantograph–Langevin systems within the unified framework of $\psi$-Hilfer fractional derivatives. Full article

Unmanned Aerial Vehicles (UAVs) are widely used in fields such as autonomous missions, reconnaissance, surveillance, and various industrial applications. These vehicles can perform desired tasks without human intervention in challenging environmental conditions. However, UAV control can be difficult due to environmental factors, wind disturbances, and uncertainties in system parameters. Therefore, developing reliable control strategies for UAVs is a significant challenge for researchers and engineers. This study presents a comprehensive review of rotary-wing UAVs, focusing on quadcopter and helicopter systems. Approximately 77 studies were selected from the Web of Science (WOS) database and analyzed, with an emphasis on Sliding Mode Control (SMC) and Fractional-Order SMC (FOSMC) applications in these systems. The review addresses key topics such as degrees of freedom, proposed control methods, adjustment techniques, comparative methods, fractional-order definitions, simulation tools, and explanations. The literature analysis highlights current research trends by showing the performance advantages and limitations of SMC and FOSMC methods. Furthermore, future research directions and existing knowledge gaps are discussed in detail. This review was prepared to provide the control engineering community with a comprehensive understanding of SMC and FOSMC applications in rotary wing systems and to contribute to the development of innovative and effective control strategies. Full article

Sleep disorders are prevalent in the world, and they lead to severe health issues such as cardiovascular disease and cognitive disabilities. Conventional polysomnography-based diagnosis is based on manual EEG analysis under the supervision of trained specialists, which is time-consuming and may have inter-rater variability. Although the predictions of deep learning (DL) models on the task of sleep classification of EEG have been promising, they, in many cases, do not explain the multiscale, temporal dynamics that physiological signals are characterized by. In this work, a hybrid model that is a combination of CNN and multifractal detrended fluctuation analysis (MF-DFA) was proposed to detect localized temporal features and long-term fractal-based dynamics of single-channel EEG recordings. The performance of the suggested model was tested using two separate polysomnographic datasets: the CAP Sleep Dataset of five-class sleep disorder classification (Healthy, Insomnia, Narcolepsy, PLM, and RBD) and the ISRUC Sleep Dataset on the three-class subject-independent validation. In the CAP dataset, the framework had an accuracy of 86.38%. Cross-dataset transfer to the ISRUC Sleep Dataset, where only the classification head was fine-tuned on a small labeled subset while all feature-extraction layers remained frozen from CAP training, achieved 87.50% accuracy, demonstrating that the learned representations generalize across differing recording protocols, sampling rates, and diagnostic label spaces. The experiments of ablation proved the paramount importance of the MF-DFA features, and the lack of them led to low classification rates. The findings demonstrate the clinical feasibility of applying fractal analysis in conjunction with DL to detect sleep disorders in an automated, generalizable manner, suitable for use in large-scale monitoring and resource-starved clinical environments. Full article

This paper mainly considers the fractional parameter identification algorithms of the bilinear-in-parameter autoregressive (AR-BIP) system. The data filtering technique is introduced to improve the parameter estimation accuracy of the AR-BIP system, which involves using a filter to filter the data of the identification model. The filtering-based hierarchical fractional least mean square algorithm (F-HFLMS) and the filtering-based multi-innovation hierarchical fractional least mean square algorithm (F-MHFLMS) are proposed for effective and accurate parameter estimation of the AR-BIP system. Using the multi-innovation theory and expanding the scalar innovation into the innovation vector, the F-MHFLMS could take full advantage of the input and output data information of the system. The performance of the F-MHFLMS algorithm is compared with the F-HFLMS strategy for the AR-BIP system using the values of the mean square error (MSE) and the average predicted output error. The effectiveness and accuracy of F-HFLMS and F-MHFLMS algorithms are demonstrated under the numerical experimentation based on different noise variances, fractional orders and innovation lengths. Compared with the F-HFLMS algorithm, the F-MHFLMS algorithm can acquire more accurate and robust parameter estimation. Full article

The nature of nonlinear dynamics of financial markets results in fractal geometry and chaotic behavior that can be viewed on a variety of scales in time. This paper conducts research on the fractal characteristics of the stock network and its contribution to the price of assets based on the Fractal Market Hypothesis (FMH). A multiscale network centrality measure is built based on high-frequency return dependencies to measure the self-similar, scale-invariant nature of inter-stock dependencies. The network factor and portfolio returns are then broken down with the wavelet packet decomposition (WPD) to obtain frequency-domain profiles, which characterize the variability of risk transmission in relation to investment horizons. The profiles are consistent with scaling properties of fractal, but the decomposition does not identify causal pathways on its own. Estimation of fractal dimension by use of the box-counting technique aided by the Hurst exponent analysis reveals that the A-share of China market exhibited long-range dependence and multifractal scaling. Network factor has the largest explanatory power in mid-frequency between the D5 and D6 bands of 32 to 128 days. This intermediary frequency concentration is consistent with the hypothesis of heterogeneous markets, in which the groups of investors with varying time horizons generate scale-related price dynamics. The addition of the network factor to a 6-factor specification lowers the GRS under the 5-factor specification by 31.45 to 17.82 on the same test-asset universe, indicating better cross-sectional coverage in the sample. The estimates of the Lyapunov exponents (0.039) as well as the correlation dimension ($D_2=4.7$) confirm the presence of low-dimensional chaotic processes of the network factor series, but these values are specific to the Chinese A-share market over the 2005–2023 sample period. These results provide a frequency-disaggregated use of network-based factor modeling and suggest that it can be applicable in multiscale portfolio risk management where the investor horizon is not uniform. Full article

Rolling bearing faults are the primary cause of rotating machinery failure. Under complex operating conditions, the weak fault impact signals are easily overwhelmed by strong noise and exhibit significant non-stationary characteristics, posing severe challenges to accurate diagnosis. To address this, this paper proposes an improved Transformer-based fault diagnosis method driven by the improved black-winged kite algorithm-variational mode decomposition (IBKA-VMD) and hierarchical fractional-order attention entropy (HFrAttE). The method employs the integrated multi-strategy IBKA to adaptively determine the optimal parameters of VMD, utilizes HFrAttE to construct highly discriminative feature sets, and further builds an improved Transformer model integrating bidirectional attention mechanisms and feature decoupling structures for deep feature mining. The classification decision is finalized by the twin extreme learning machine (TELM). Experimental results on the case western reserve university (CWRU) bearing dataset under different noise environments (−2 dB, −5 dB) demonstrate that the proposed method maintains 100% accuracy, recall, and F1-score under −5 dB noise interference, significantly outperforming comparative models. It exhibits excellent anti-noise performance and feature extraction capability, providing an efficient solution for intelligent operation and maintenance of rotating machinery under complex operating conditions. Full article

The full-bridge LLC resonant converter is one of the most suitable converters for high-power, high-efficiency applications. Although the design methodologies for full-bridge LLC resonant converters are already well-established, the development of the fractional-order domain has brought new flexibility to converter design. Based on the fact that inductors and capacitors have fractional-order characteristics, this paper presents a de-normalized fractional-order FHA gain model, which reveals the impact of fractional-order characteristics of practical inductors and capacitors on the converter gain. By maintaining the convenience of the FHA design method, this work identifies the fractional orders of a resonant tank inductor and capacitor and incorporates them into the parameter design as part of the design requirements, making the design results more accurate than the conventional FHA design method. Specifically, compared with the conventional FHA-based design, the proposed approach improves the DC voltage gain margin of the full-bridge LLC converter by 26% and expands the ZVS operating range margin by 23.3%. Full article

Precise and smooth actuation is a central requirement in surgical robotics, where small tracking errors or oscillations can directly affect task quality and safety. This paper studies the control of an induction-motor-driven surgical joint using a sliding-mode strategy enhanced by fractional-order operators and implemented within a DTC–SVM structure. The motivation is to improve motion smoothness and disturbance rejection without sacrificing the fast dynamic response offered by direct torque control. A dynamic model of the actuator is developed by combining the electrical equations of the induction motor with the mechanical dynamics of a robotic joint, including inertia, viscous friction, gravity-induced torque, and Coulomb friction. Fractional-order sliding surfaces are introduced for both position and flux regulation, and the closed-loop stability is examined through Lyapunov-based arguments. Simulation results show accurate trajectory tracking with limited overshoot and smooth transient responses. The motor speed remains well regulated, while stator flux and currents stay within admissible bounds. The electromagnetic torque adapts to load variations with reduced ripple, and the rotor pulsation remains bounded. Within the limits of numerical evaluation, these results indicate that the proposed fractional-order sliding-mode DTC–SVM scheme is suitable for precision-oriented surgical robotic actuation. Full article

This study examines a comprehensive class of coupled nonlinear $\varrho$-Hilfer fractional neutral impulsive integro-differential systems with mixed delays and non-local initial conditions. The primary contribution of this study is the creation of a unified analytical framework that encompasses coupled interactions, neutral-type dependencies, and impulsive disturbances, which have been studied separately by researchers. We utilize the Banach contraction principle and Krasnoselskii’s fixed-point theorem to provide suitable conditions for the existence and uniqueness of solutions within the product space of piecewise continuous weighted functions. In addition to existence, we examine Ulam–Hyers–Rassias (UHR) stability using a generalized Gronwall inequality, which guarantees the system’s robustness against functional perturbations. We also develop a controllability framework and a feedback control law that steer the system towards the desired terminal states. The theoretical results are supported by a numerical simulation using a complex kernel, implemented via a modified predictor-corrector algorithm, which validates the practical effectiveness of the proposed control and stability outcomes. Full article

This paper studies an inverse problem associated with the time-fractional Schrödinger equation in a field-free potential. To address the severe ill-posedness of the problem, a mollification regularization method with truncated kernels is employed to obtain stable approximate solutions. Both a priori and a posteriori strategies for selecting the regularization parameter are developed, and corresponding error estimates for the regularized solutions are derived. The effectiveness of the proposed approach is demonstrated through numerical simulations. Full article

This study presents a comprehensive investigation of exact fractional wave solutions and bifurcation analysis for the (3 + 1)-dimensional extended shallow water wave (3D-eSWW) equation with $\beta$-derivative, which models nonlinear wave phenomena in fluid dynamics and coastal engineering. Leveraging the flexibility of the fractional derivative, the model provides a more generalized and adaptable framework for describing shallow water wave propagation. The Modified Extended Direct Algebraic Method (MEDAM) is systematically employed to derive a broad spectrum of novel exact analytical solutions. These include the following: dark solitary waves, singular solitons, singular periodic waves, periodic solutions expressed via trigonometric and Jacobi elliptic functions, polynomial solutions, hyperbolic wave patterns, combined dark–singular structures, combined hyperbolic–linear waves, and exponential-type wave profiles. Each solution family is presented with explicit parameter constraints that ensure both mathematical consistency and physical relevance, thereby offering a robust classification of wave regimes under diverse conditions. A thorough bifurcation analysis is conducted on the reduced dynamical system to examine parametric dependence and stability transitions. Critical bifurcation thresholds are identified, and distinct solution branches are mapped in the parameter space spanned by wave numbers, nonlinear coefficients, external forcing, and the fractional order $\beta$. The analysis reveals how solution dynamics undergo qualitative transitions—such as the emergence of solitary waves from periodic patterns or the appearance of singular structures—driven by the interplay of nonlinearity, dispersion, and fractional-order effects. These insights are crucial for understanding wave stability, predictability, and the onset of extreme events in shallow water contexts. Graphical representations of selected solutions validate the analytical results and illustrate the influence of $\beta$ on wave morphology, propagation, and stability. The simulations demonstrate that varying the fractional order can significantly alter wave profiles, highlighting the role of fractional calculus in capturing complex real-world behaviors. This work demonstrates the efficacy of the MEDAM technique in handling high-dimensional fractional nonlinear PDEs and provides a systematic framework for predicting and classifying wave regimes in real-world shallow water environments. The findings not only enrich the solution inventory of the 3D-eSWW equation but also advance the analytical toolkit for studying complex spatio-temporal dynamics in fractional mathematical physics and fluid mechanics. Ultimately, this research contributes to the development of more accurate models for coastal protection, tsunami forecasting, and marine engineering applications. Full article

This article deals with the global existence and uniqueness of solutions to a fractional-order SIQR epidemic model, alongside its intricate chaotic and complex dynamics as functions of the fractional order. The well-posedness of the model solutions, including global existence, uniqueness, and positivity, is established by constructing appropriate Lyapunov functions. The local and global stability analyses are conducted for both the disease-free and endemic equilibria of the model. An asymptotic solution of the system in the form of series is derived by the Laplace–Adomian decomposition method (L–ADM), and its convergence is rigorously proved. Subsequently, numerical analysis determines and interprets the optimal truncation order of this asymptotic solution. Numerical simulations are performed based on the asymptotic solution, and the dynamics and chaos of the dynamic system with respect to the fractional order are analyzed and illustrated in terms of the maximum Lyapunov exponent and structural complexity. Finally, a local sensitivity analysis is conducted for each state variable with respect to the model parameters. Full article

Tight sandstone reservoirs are characterized by highly heterogeneous pore structures, in which multiscale pore–throat systems jointly control the shapes of capillary pressure curves and relative permeability, thereby exerting a fundamental influence on water production behavior and the overall development performance of gas reservoirs. The Ordos Basin is generally characterized by the development of tight sandstone. The tight sandstones exhibit porosities of 2–13% and permeabilities of 0.01–10 × 10⁻³ μm². To quantitatively elucidate the controlling mechanisms of multiscale pore structure on capillary pressure curve morphology and relative permeability, this study systematically investigates the fractal and multifractal characteristics of pore structures in tight sandstones based on high-pressure mercury intrusion (MICP) and nuclear magnetic resonance (NMR) experimental data, and establishes a quantitative relationship between fractal parameters and the capillary pressure curve shape parameter λ. First, capillary pressure curves were fitted using the Brooks–Corey model within the effective saturation interval to extract the shape parameter λ, which characterizes the concentration degree of pore-size distribution and the drainage behavior. Subsequently, based on NMR T₂ spectra, the small-pore fractal dimension D₁, large-pore fractal dimension D₂, and the multifractal singularity spectrum width Δα were extracted to quantitatively describe the geometric complexity of pore structures at different scales. On this basis, the correlations between λ and D₁, D₂, and Δα were systematically analyzed, and the predictive performance of λ under different parameter combinations was compared. The results indicate that: (1) the pore structures of tight sandstones exhibit pronounced fractal and multifractal characteristics at the NMR T₂ scale, with significant differences among samples; (2) λ shows an overall negative correlation with fractal parameters, among which the correlations with the large-pore fractal dimension D₂ and the multifractal spectrum width Δα are the most significant; (3) compared with models using a single fractal dimension, the multiparameter model incorporating Δα provides a more comprehensive characterization of multiscale pore heterogeneity, leading to a substantial improvement in the accuracy and stability of λ prediction; and (4) λ exerts a clear control on the shape of relative permeability curves, where a larger λ corresponds to earlier initiation and forward-shifted rising segments of water-phase flow, while a smaller λ results in overall flatter relative permeability curves. From the perspectives of fractal and multifractal theory, this study establishes an intrinsic linkage among pore structure, capillary pressure curve shape parameters, and relative permeability, providing a novel quantitative framework for constraining relative permeability curve morphology in tight sandstones under conditions where systematic relative permeability experiments are unavailable. Full article

Robust controller tuning is essential for the accurate regulation of nonlinear dynamic plants operating under variable conditions. This study proposes an enhanced gradient-based optimizer, termed the quadratic wavelet–enhanced gradient-based optimizer (QWS–GBO), which integrates quadratic interpolation mutation (QIM) and a wavelet mutation strategy (WMS). QIM reinforces population diversity, while WMS mitigates stagnation and strengthens local refinement through adaptive perturbations, yielding a more effective balance between global exploration and local exploitation. QWS–GBO is employed in a reference–follower control framework based on Bode’s ideal response, where the follower is realized by a fractional-order proportional–integral–derivative (FOPID) controller. The FOPID parameters are optimized using QWS–GBO and evaluated in two stages. First, performance is assessed on the CEC2020 benchmark suite under a uniform protocol. Second, the approach is applied to DC motor speed regulation. On the CEC2020 functions, QWS–GBO consistently achieves lower mean objective values and faster convergence than GBO, dwarf mongoose optimization (DMO), the arithmetic optimization algorithm (AOA), and the salp swarm algorithm (SSA) with only minor computational overhead (35.90 s per trial versus 34.00 s for GBO). In the DC motor case, the QWS–GBO–tuned FOPID controller attains a rise time of 0.0216 s, settling time of 0.0350 s, zero overshoot, and peak time of 0.0509 s. Robustness tests under four operating conditions showed limited deviations (maximum 0.0058 s in rise time, 0.0113 s in settling time, 0.465% in overshoot, and 0.0131 s in peak time). Additional analyses confirmed that both QIM and WMS individually contribute measurable gains, validating their joint integration. Implementation details and parameter settings are provided to ensure reproducibility. Full article

When bridge towers are subjected to strong winds, they exhibit significant displacements. This displacement change can serve as an important indicator for structural performance warning. The displacement and acceleration collected in real time by the intelligent bridge monitoring system are disturbed by various noises, resulting in missed alarms in the monitoring system and causing huge economic losses. This study employs the fractional-order Butterworth lowpass filter method, eliminating the maximum value method, triple standard deviation method, etc. for preprocessing abnormal monitoring data characterized by missing values and outlier points. A fractional-order multi-rate Kalman fusion is proposed to process and model the correlation of structural displacement and acceleration data, and the simulated data and measured data are analyzed and verified respectively. Spectral analysis confirmed that by effectively fusing the low-frequency GPS signal with the high-frequency accelerometer signal, the fractional-order multi-rate Kalman fusion displacement measurement has a relatively high accuracy. Displacements obtained by the fractional-order multi-rate Kalman fusion method are adopted for correlation modeling, and residuals generated from this fractional-order fusion modeling are used for structural performance warning testing. The effectiveness of this structural performance warning is quantitatively validated through statistical assessment of warning accuracy. Full article

This work constructs an orthogonal function system on bounded intervals $[0,R]$ associated with the Atangana–Baleanu–Caputo (ABC) fractional derivative for $\alpha \in (1/2,1)$. Starting from a fractional Laguerre-type equation involving the ABC operator, solutions are obtained via a generalized Frobenius method, yielding series representations with characteristic exponent $\alpha -1$. Rather than postulating a weight function by analogy with classical or Caputo settings, the weight is derived directly from the requirement that the underlying fractional operator be formally self-adjoint on a suitable admissible domain. This operator-theoretic approach leads to the explicit Mittag–Leffler weight $w_{\alpha }\left(x\right)=x^{-(2\alpha -1)}{E}_{\alpha }(-x^{\alpha })$, which intrinsically reflects the nonlocal memory structure of the ABC kernel. A similarity transformation removes the universal singular factor and produces regularized eigenfunctions that are continuous on $[0,R]$ and orthogonal in the weighted $L^2$ space. The weight identity and formal self-adjointness are rigorously verified through a right-Volterra uniqueness argument. Numerical experiments confirm orthogonality up to machine precision, demonstrate spectral convergence for a model ABC differential equation, and illustrate consistency with classical Laguerre polynomials in the limit $\alpha \to 1^{-}$. The resulting framework provides a self-consistent orthogonal system suitable for spectral approximations of problems governed by the ABC operator on bounded domains. Full article

This study explores the interconnection between the variables of ChatGPT usage, energy consumption, water consumption, and carbon dioxide CO₂ emissions. A new integer and fractional order model using the Caputo derivative is proposed to comprehend the long-term dependencies of these variables. Boundedness, and global and local stability are examined for the fractional order model. The equilibrium points of these variables are shown to determine the stability of the model. The Runge–Kutta 7 numerical method is employed for the integer order model, whereas the semi-implicit linear interpolation (L1) method is used for the fractional order model. The parameter sensitivity is conducted on the system’s parameters to understand the variables’ impact by varying the relevant parameters for the system. To increase the efficacy of our analysis, we used machine learning approaches to model and predict the dynamics of CO₂ emissions, energy and water consumption, and ChatGPT usage. The Prophet ML model stood out among the other methods because it is adept at identifying long-term growth trends, seasonal changes, and the impact of outside variables in intricate time-series data. It is extremely beneficial for research centered on sustainability, where accurate projections are essential for wellinformed decision-making, because it can produce robust, interpretable forecasts against missing values and outliers. Using the Prophet ML model, our research guarantees precise and expandable predictions and provides valuable information that can direct tactics to balance environmental sustainability and technological progress. Full article

The large-scale rapid deployment of renewable generation and energy storage is transforming traditional power system dynamics through intermittency, reduced inertia, and pronounced long-range temporal dependence. Existing power system modeling frameworks are primarily based on short-memory assumptions and integer-order dynamics, which are unable to capture the persistence and oscillatory behavior of emerging renewable-dominant power systems. This structural mismatch leads to inaccurate system representation and degraded long-horizon prediction performance. Although fractional calculus has been applied to specific control and forecasting tasks in power systems, the joint system-level modeling of renewable generation and load demand using real-world data remains largely unexplored. In this paper, we develop a data-driven fractional-order dynamic modeling framework that explicitly incorporates long-memory effects into the governing equations through fractional differential equations based on the Caputo formulation. Using publicly available high-resolution datasets of load and renewable generation, empirical analysis reveals power-law decaying autocorrelations and dominant low-frequency spectral characteristics that motivate the use of fractional-order dynamics. Fractional orders and model parameters are jointly identified through prediction-error minimization to ensure consistency between modeled trajectories and observed persistence. The numerical results demonstrate that the proposed approach achieves a root–mean–square error of $3.12$, compared to $5.64$ and $4.98$ for integer-order and finite-memory models, respectively, and reduces the normalized root–mean–square error from $0.156$ and $0.132$ to $0.087$. Residual and spectral analyses further confirm that long-memory behavior is effectively captured by the proposed dynamics. The framework provides a scalable and physically interpretable foundation for the data-driven modeling of renewable-dominant power systems. Full article

As a core non-mechanical beam-steering device, cascaded liquid crystal polarization gratings (CLCPGs) suffer from insufficient pointing accuracy due to inherent fabrication and assembly errors, which must be compensated by liquid crystal optical phased array (LCOPA). However, in practical working conditions, LCOPA is vulnerable to coupled internal and external disturbances as well as inherent time delays, which prevent accurate compensation and limit the performance of the integrated system. To overcome these challenges, this paper proposes a novel composite control strategy. An improved observer and an improved Smith predictor are designed to estimate and compensate for the total disturbances and time delays, and parameter tuning is accomplished using the phase margin method. The effectiveness of the proposed strategy is validated on a LCOPA coarse–fine two-stage compensation system experimental platform. The results demonstrate that the strategy can effectively suppress disturbances and compensate for LCOPA errors, reducing the overall pointing error by more than 30% and increasing the dynamic response speed by 25%, while exhibiting excellent robustness and stability. This study provides theoretical and technical support for the engineering application of high-precision CLCPG scanning systems. Full article

This paper develops a functional operator-theoretic framework for nonlinear Erdelyi–Kober (EK) fractional dynamical systems formulated in Banach spaces endowed with the compact-open topology. Within this setting, sufficient conditions for existence, uniqueness, and Ulam–Hyers stability of solutions are established using the Banach and Schaefer fixed-point theorems. The continuity, boundedness, and Lipschitz properties of the associated nonlinear operators are analyzed to ensure well-posedness of the fractional system. As a constructive complement to the theoretical results, a power series iterative method (PSIM) is employed to obtain an explicit fractional series representation of the solution in the case $0<\alpha <1$. The applicability of the theoretical framework is illustrated through a nonlinear fractional dynamical Belousov–Zhabotinsky system (DBZS), where the assumptions of the main theorems are verified and the solution is constructed via the proposed series scheme. The results provide a coherent link between abstract fixed-point analysis and a constructive semi-analytical representation of solutions for EK fractional systems. Full article

This comprehensive review examines the application of fractal theory and technology in the study of cementitious materials, with a focus on cement and concrete. We begin by introducing the fundamental concepts of fractal geometry and the various fractal dimensions used to quantify material features. We then explore how fractal analysis has been applied to key aspects of cementitious materials, including pore structure, particle size distribution, fracture surfaces, and crack propagation. Each section highlights the methodologies employed, the insights gained, and the implications for material design and performance. Additionally, we discuss the use of fractal-based techniques in the non-destructive testing and monitoring of structures. Finally, we address the challenges and limitations of fractal approaches and propose future directions for research in this interdisciplinary field. Fractal theory can become a useful tool in the study of cementitious materials, aiding a deeper understanding of their physical properties and long-term durability, and guiding the design of more durable and efficient construction materials by giving engineers the required knowledge on the technology and its limitations. Full article