Search Results (122)

Ramanujan’s master theorem is considered to express particular values of the Liouville fractional derivatives. This fact is used to immediately obtain generalizations. Firstly, it is shown that the original formulation is suitable only for anti-causal functions. Consequently, an expression for causal functions is introduced. Secondly, the two formulations are extended to be valid for any real order and related to the Laplace transform, corresponding to transforms with left- or right-sided regions of convergence. This leads to introducing the concept of the unilateral MacLaurin series and the substitution of the Liouville derivative by the (Liouville–)Grünwald–Letnikov derivative, which essentially expresses a discrete formulation that is suitable for numerical implementations. The way in which this procedure arose and its relationship with the Liouville derivatives suggested other approaches, which are briefly discussed, such as replacing the MacLaurin series with the Mittag–Leffler series or the Liouville derivative with the Riesz, Feller, or Hadamard derivatives. Full article

Quadrotor unmanned aerial vehicles (QUAVs) exhibit strongly coupled nonlinear dynamics and are highly sensitive to disturbances and measurement noise, which can significantly degrade trajectory tracking performance and induce chattering in sliding mode-based controllers. In this work, a fractional-order backstepping sliding mode control (FO-BSMC) strategy is proposed for QUAV trajectory tracking. In contrast to existing fractional-order sliding mode approaches, where the fractional operator is typically introduced into the sliding surface or control law, the proposed methodology incorporates fractional-order behavior directly into the QUAV dynamic model through the Caputo definition, while the Grünwald–Letnikov approximation is adopted for numerical implementation. A conventional integer-order BSMC scheme is also developed, and Lyapunov-based stability analyses are presented for both the conventional BSMC and the proposed FO-BSMC formulations. The fractional order is selected using the PSO algorithm. The performance of both controllers is evaluated under external disturbances, perturbed initial conditions, and measurement noise. Monte Carlo simulations are further conducted to assess the sensitivity of the closed-loop system to initialization uncertainties. The simulation results demonstrate that the proposed FO-BSMC achieves lower tracking errors, faster convergence, improved robustness against external disturbances and measurement noise, and smoother control actions with reduced chattering than the conventional BSMC. Full article

Maximizing the dominant Lyapunov exponent ${\lambda }_1$ of an incommensurate fractional-order chaotic system, while respecting the dynamical conditions for a strange attractor, is a non-convex, gradient-free problem on a history-dependent landscape. Existing metaheuristic studies typically use hard-cutoff penalties that distort the fitness landscape and integer-order Lyapunov estimators that can be biased for strongly fractional regimes. This paper presents a constraint-faithful optimization framework combining (i) subtractive-hinge penalties that vanish on the feasible set, (ii) a memory-consistent Grünwald–Letnikov variational Lyapunov estimator with adaptive tail-sum truncation, (iii) joint search over parameters and incommensurate orders by the Marine Predators Algorithm, and (iv) a fractional conditional Lyapunov exponent (FCLE) that recovers the integer-order limit. Applied with a fixed configuration to the fractional-order Lorenz, Ma–Chen financial, Iqbal–Wang, and Hyper–Chen systems, the framework converges to feasible attractors with enlarged Lyapunov spectra. Dissipativity is rigorously verified; all selected optima have strictly negative Lyapunov trace at the reported precision. FCLE analysis on the optimized Lorenz attractor recovers the integer-order identity $c_{min}={\lambda }_1$ under full-state coupling, and shows that single-state x-coupling raises the threshold to ≈9${\lambda }_1^{*}$. The optimized fractional-order Lorenz attractor is employed as the random-number generator of a recent chaos-based image-encryption scheme, where it yields strong statistical results across standard benchmarks. Full article

Agricultural pest infestations cause substantial global crop losses. Morphological similarities across species and structural variations across developmental stages render accurate identification a persistently expert-dependent and time-consuming process. Recent deep learning approaches have advanced automated pest classification; however, most efforts have concentrated on architectural design, while optimization strategies have received comparatively little attention. This study proposes a novel optimization framework, hereafter referred to as Adaptive Fractional Gradient Descent (AFGD), that integrates the Grünwald–Letnikov (GL) fractional derivative into the backpropagation process of deep convolutional neural networks. Unlike standard gradient descent, the proposed method maintains a weighted history of past gradients. It dynamically adjusts the fractional order α via Bayesian optimization at regular training intervals, enabling the model to adaptively balance exploiting gradient memory against exploring new gradients throughout training. Experiments conducted on the IP102 benchmark dataset using DenseNet121, ResNet101, and EfficientNetB0 backbones demonstrated consistent accuracy improvements over standard gradient descent across all configurations. In the untrained setting, absolute test accuracy improved by 20.73, 11.51, and 11.01 percentage points for DenseNet121, ResNet101, and EfficientNetB0, although the absolute accuracy levels in this configuration remain modest. Under ImageNet pre-training, the proposed method yielded absolute gains of 6.69, 7.39, and 3.76 percentage points over the corresponding standard gradient baselines, with the highest absolute test accuracy of 70.81% recorded for DenseNet121. These findings indicate that fractional-order gradient control is a promising, architecturally complementary optimization strategy for robust pest classification, with broader implications for deep learning applications in precision agriculture. Full article

Optimal coordination of directional overcurrent relays (DOCRs) in medium-to-large power systems constitutes a computationally demanding, mixed-integer, nonlinear optimisation problem whose complexity escalates rapidly with system size, making the simultaneous minimisation of relay operating time and computational cost a critical open challenge. This study presents a two-level hierarchical framework in which Type-2 Fuzzy C-Means (T2FCM) clustering partitions 226 fault scenarios into subproblems at the upper level, while the Hybrid Fractional Entropy Evolution (HFEE) algorithm independently optimises relay settings for each cluster at the lower level. HFEE integrates fractional-order velocity updates—derived from the Grünwald–Letnikov formulation—with a Shannon entropy diversity-control mechanism to prevent premature convergence. T2FCM captures inherent fault-current uncertainty through interval-valued type-2 fuzzy memberships, yielding more robust cluster assignments near protection-zone boundaries than crisp partitioning methods. The framework is validated on the extended IEEE 30-bus system. An ablation study demonstrates that standalone HFEE achieves a 29.19% improvement in $T_{\mathrm{op}}$ over the prior best-reported result; however, a comprehensive parameter sweep over cluster counts $K\in \{2,\dots ,8\}$ and fractional orders $\alpha \in \{0.1,\dots ,0.9\}$ across 50 independent runs per configuration shows that the proposed clustering-decomposed method achieves 3.68–66.67% lower wall-clock computation time while maintaining zero CTI violations across all active relay pairs. The communicationless, entirely offline framework demonstrates scalability for simultaneous sub-transmission and distribution protection coordination and offers a practically deployable strategy for modern power networks. Full article

This paper proposes a novel hybrid numerical scheme that augments the classical L1 finite-difference approximation of the Caputo fractional derivative of order $\alpha \in (0,1]$ with a selective shifted Grünwald–Letnikov correction (controlled by a shift parameter $\beta \in [0,1)$) applied only to the most recent time increment. When $\beta =0$, the scheme reduces exactly to the classical L1 scheme and retains its optimal convergence rate $O\left(h^{2-\alpha }\right)$, where h denotes the uniform time-step size. For $\beta >0$ (optimally chosen as $\beta =1-\alpha /2$), extra numerical damping is introduced at the cost of a mildly reduced convergence order $O\left(h^{1-\alpha }\right)$, while long-term stability is significantly improved. The scheme is applied to the fractional Kelvin-Voigt and Standard Linear Solid models to analyze creep and relaxation responses. Numerical simulations demonstrate that the proposed hybrid scheme achieves improved accuracy, long-term stability, and computational efficiency compared to classical integer-order models and several existing fractional schemes reported in the recent literature. Results show that fractional orders capture anomalous creep behavior more accurately, aligning with experimental data from recent studies. The proposed method offers improved computational performance for real-time structural health monitoring applications. Full article

Underwater object detection using forward-looking sonar presents fundamental challenges absent from terrestrial imagery: low-contrast single-channel inputs, multi-scale acoustic shadows, and object classes spanning a wide range of acoustic scattering characteristics. Three coordinated modifications to the YOLOv8 framework are proposed to address structural limitations of standard bottleneck chains for this domain. A fractional-order feature propagation mechanism based on Grunwald–Letnikov discretization enables each bottleneck to access a decaying-weighted history of all prior intra-chain feature states via a single learnable scalar per block. A boundary-aware gating module with joint spatial-channel attention selectively suppresses fractional correction at geometric boundary locations. A parameter-free energy-based attention module applied in the detection neck exploits the local statistical distinctiveness of genuine acoustic features during multi-scale fusion. Evaluated on the Underwater Acoustic Target Detection dataset, the proposed system achieves mAP50 of 0.8635 and mAP50-95 of 0.3964, improvements of 0.0188 and 0.0136 respectively over the YOLOv8n baseline at less than 2.0% parameter overhead, surpassing larger generic YOLOv8 variants on mAP50. Full article

This study investigates the Mittag–Leffler-type stability properties of fractional perturbed systems with respect to their unperturbed counterparts by incorporating initial time differences into the analysis. In contrast to many existing studies in which initial time effects are neglected, the proposed framework explicitly considers time shifts together with the memory-dependent nature of fractional-order systems. Using Caputo fractional derivatives and Lyapunov-type functionals, new sufficient conditions are established for the stability behavior of perturbed systems relative to the corresponding unperturbed systems under shifted initial times. The obtained results extend existing stability criteria by simultaneously addressing fractional memory effects, perturbation terms, and variations in the initial time. To illustrate the applicability and effectiveness of the theoretical findings, representative examples, numerical simulations, graphical comparisons, and global error analyses are presented. The numerical part is based on the Caputo framework and is further supported by benchmark comparisons involving Riemann–Liouville and shifted Grünwald–Letnikov approaches. The proposed results provide a useful framework for the stability analysis of memory-dependent dynamical systems arising in engineering and applied sciences. Full article

Fractional differential operators provide an effective approach for modeling complex technological processes, particularly physical phenomena in continuum mechanics characterized by memory and non-local effects. Different types of fractional derivatives require different numerical approximation schemes; in this study, the Caputo and Grünwald–Letnikov derivatives are considered. The aim of this work was to develop and validate a fractional differential model of longitudinal oscillations in a suspended monorail system that accounts for nonlinear and memory-dependent effects. In contrast to classical integer-order approaches, the proposed framework incorporates multiscale surface irregularity effects, including rail roughness, friction, and other disturbances influencing system dynamics, through a fractional-order formulation. A fractional differential mathematical model describing the motion of longitudinal oscillations of a large-sized cargo transported along a suspended monorail is proposed. A numerical algorithm based on finite-difference approximation of fractional operators was developed for its implementation. The scientific contribution lies in integrating multiscale surface irregularity effects into a fractional-order modeling framework to improve the accuracy of dynamic response prediction. Numerical experiments demonstrated the effectiveness of the approach, and the results were validated through comparison with existing models of monorail dynamics. Additionally, statistical validation based on correlation analysis confirmed good agreement with the experimental data. The proposed model can be applied to the design and optimization of suspended transport systems, improving vibration control, reliability, and operational safety under real dynamic loading conditions. Full article

Image enhancement is of pivotal importance in the detection of defects in wood veneers. However, acquired images frequently exhibit signs of blurring, uneven illumination, and insufficient contrast, which can lead to a reduction in the accuracy of defect recognition. In this study, an algorithm based on Grünwald–Letnikov (G–L) fractional-order differentiation is proposed for the enhancement of wood veneer defect images. Initially, the gain characteristics of differential amplitude-frequency responses on high- and low-frequency image components are analyzed, and the feasibility of the method is demonstrated by linking these characteristics with the frequency-domain distributions of live knot, dead knot, and crack defects. Secondly, an eight-direction mask operator is constructed based on the G–L definition, and a DC component preservation factor is introduced to eliminate the luminance drift caused by mask truncation. The application of the mask is performed independently on the R, G, and B channels, and a dynamic blending mechanism is designed to achieve a balance between texture enhancement and structural fidelity. Finally, a set of six evaluation metrics (AG, E, PSNR, RMSE, SSIM, and VIF) is employed to assess the quality of enhanced images. The proposed algorithm is then compared with five existing algorithms (SSR, MSR, MSRCR, CLAHE, and AGC) under both noise-free and additive white Gaussian noise conditions. The findings indicate that the G–L fractional-order differentiation algorithm facilitates a more balanced representation of image features, thereby enhancing contrast, brightness, and textural contours. This approach results in more authentic color reproduction and superior visual quality. Full article

Cryptocurrency markets exhibit persistent temporal dependence and multifractal scaling behavior, yet these properties remain only partially incorporated into existing deep learning architectures. This study proposes the Fractional Attention-Driven LSTM–N-BEATS (FA-LSTM-NBEATS) framework, integrating Grünwald–Letnikov fractional memory operators, adaptive attention, interpretable N-BEATS decomposition, and an asymmetric loss function within a unified forecasting and risk estimation model. The framework is evaluated using daily BTC, ETH, and BNB data from 2018 to 2025 through hierarchical ablation analysis, walk-forward validation, Diebold–Mariano testing, residual diagnostics, and Fractional Value-at-Risk (VaR) evaluation. Results indicate persistent scaling behavior with Hurst exponents above 0.76 and multifractal spectrum widths of $\mathrm{\Delta }\alpha \approx 0.51–0.54$. FA-LSTM-NBEATS achieves the strongest relative forecasting performance for BNB, with the lowest RMSE (0.02789) and MAE (0.01950) among all evaluated models. The learned fractional parameter $\left(\alpha ≈0.6106\right)$ remains stable across assets, suggesting convergence toward a persistent memory regime. In addition, Fractional VaR produces coverage ratios closer to unity than Historical and LSTM-based benchmarks under high-volatility conditions. Full article

Long-memory degradation processes in rotating machinery often exhibit nonlinear evolution, nonlocal temporal dependence, and hereditary characteristics, which are difficult to fully capture using conventional integer-order models or standard Markovian decision frameworks. To address this issue, this study proposes a hybrid fractional-dynamics and deep reinforcement learning framework for predictive maintenance of memory-dependent degradation systems. First, the Grünwald–Letnikov fractional difference operator is introduced to construct a fractional-memory representation of degradation trajectories, enabling the model to explicitly encode long-range dependence and accumulated historical degradation effects. Then, a bidirectional gated recurrent unit network is employed to learn sequential degradation representations from the fractional-memory state space, while a deep Q-network is designed to optimize maintenance decisions under uncertain degradation evolution. Experimental results on the IEEE PHM 2012 bearing dataset show that the proposed FM-BiGRU-DQN with safety-guided execution achieved a mean maintenance lead time of 33.9 ± 12.6 steps, an in-band rate of 0.85 ± 0.06, a failure rate of 0.00 ± 0.00, and a deployment reliability of 1.00 ± 0.00 over 10 independent random seeds. Compared with NM-BiGRU-DQN, the in-band rate increased from 0.55 ± 0.10 to 0.85 ± 0.06, with a paired-test p-value of 0.013. Cross-dataset validation on the XJTU-SY bearing dataset further achieved an in-band rate of 0.80 and a failure rate of 0.00. These results indicate that embedding fractional-memory dynamics into deep reinforcement learning improves maintenance timing accuracy, policy robustness, and deployment reliability for complex memory-dependent degradation systems. Full article

Gearbox fault diagnosis under controlled bench-test conditions with known speed variations and noise interference remains challenging because nonstationarity, background noise, and operating-condition fluctuations can easily submerge weak localized fault features. To address this issue, this study proposes a fault diagnosis method based on a fractional-order-enhanced dual-view representation and VibrMamba–VMamba collaborative modeling. First, this study introduces a Grünwald–Letnikov fractional-order differential enhancement module with a fractional order of $\alpha =0.6$ to strengthen fault-sensitive impulsive components and improve the representation of nonstationary vibration signals. The framework then uses the enhanced signal to construct dual-view inputs: a fractional-order-enhanced one-dimensional vibration sequence and a fractional-order-enhanced synchrosqueezing transform (SST) time–frequency image. Subsequently, the framework constructs a VibrMamba temporal branch and a VMamba visual branch to extract dynamic temporal features and global structural features, respectively. Instead of using simple feature concatenation, this study designs a sample-adaptive collaborative fusion mechanism with gated weighting and cross-branch residual enhancement to integrate complementary temporal–visual representations. Bench-level experiments show that the proposed method achieves 98.90% diagnostic accuracy under clean test conditions and maintains 91.52% accuracy at −5 dB signal-to-noise ratio (SNR). These results should be interpreted as bench-level validation under controlled laboratory conditions rather than as direct evidence of field-level generalization. This framework provides a methodological solution that integrates fractional-order signal enhancement, dual-view representation, and Mamba-style collaborative state-space modeling for gearbox fault classification under controlled laboratory conditions with known speed variations and noise disturbances. Full article

In this paper, we propose and numerically investigate a fractional T system. As a fractional generalization of the classical T model, the fractional order serves as a memory parameter governing the system dynamics. By employing the fractional stability criterion, the local stability of the equilibrium points is analyzed, and the existence of Hopf bifurcation is characterized. To efficiently simulate the long-time dynamics induced by fractional memory, a linear semi-implicit numerical scheme accelerated by a sum-of-exponentials approximation of the Caputo derivative is developed. The proposed scheme is shown to be stable and enables a significant reduction in computational cost compared with classical L1 and Grünwald–Letnikov methods. Numerical experiments, including time series, phase portraits, Lyapunov exponent computations, and bifurcation diagrams, demonstrate that varying the fractional order leads to transitions among stable, periodic, and chaotic regimes. In particular, pronounced transient dynamics are observed as the fractional order approaches its critical value, highlighting the memory-induced effects inherent in fractional-order systems. Full article

Population-based metaheuristic algorithms are widely used for multi-objective city evacuation planning, yet their opaque internal dynamics limit practitioner trust in safety-critical contexts. This study introduces, to the best of our knowledge, the first unified coupling of fractional calculus and fractal analysis with the EvoMapX process-level explainability framework in the context of evacuation optimization. In contrast with classical integer-order EvoMapX paired with exponential moving averages of operator credit, the proposed formulation embeds long-range memory directly into the explainability pipeline through Caputo and Grünwald–Letnikov derivatives. The Operator Attribution Matrix (OAM), Population Evolution Graph (PEG), and Convergence Driver Score (CDS) are extended with fractional-order formulations employing Caputo and Grünwald-Letnikov fractional derivatives with adaptive memory parameters, alongside Mittag–Leffler urgency escalation dynamics. A Fractional-Order PSO variant (FO-EPSO) with segment-specific fractional velocity updates and a fractal fitness landscape analysis module for adaptive parameter tuning are introduced. The framework incorporates nine evacuation-specific operators, a spatial OAM for zone-level attribution, and a multi-stakeholder explanation pipeline. Experiments across 520 disaster scenarios demonstrate that explainability and optimization performance are not mutually exclusive: the EvoMapX-integrated NSGA-II achieved a mean hypervolume of 0.731 versus 0.728 for the standard variant, with less than 5% computational overhead. The OAM revealed disaster-type-specific operator patterns invisible to conventional analysis. Real-world validations on Beijing Chaoyang District and Kigali, Rwanda, confirmed these findings. From an operational standpoint, the most consequential outcome of this work concerns its impact on human decision-makers: a controlled study with 45 emergency-management professionals showed that incorporating EvoMapX explanations cut the time required to commit to an evacuation plan by 24.9%, raised reported decision confidence by 20.3%, and lifted self-assessed algorithm understanding from 18.1% to 78.9% (all p < 0.001). Equally important for real-time disaster response, this entire layer of process-level transparency is delivered with a runtime penalty of under 5% relative to the non-explainable baselines, which we view as a key practical advantage for field deployment. This work establishes fractional-order process-level transparency as a feasible and beneficial paradigm for interpretable optimization in safety-critical domains. Full article