Search Results (83)

This research paper demonstrates how to manage Caputo fractional neutral integro-differential equations which include both integral and nonlinear elements through a unified framework that models dynamic systems with memory-based dynamics. The research establishes sufficient conditions for controllability through fixed point theory in a Banach space framework which requires particular assumptions while the study focuses on the ${\mathfrak{K}}_1<1$ condition which leads to the existence of a controllable solution. The proposed criteria are demonstrated through a numerical example which tests the theoretical results. The real-world case study uses artificial neural network (ANN) technology to predict Litecoin prices through the application of the fractional controllability model which analyzes historical financial data. The hybrid framework enables precise forecasting of nonlinear time series because it combines fractional calculus mathematical principles with ANN learning abilities. The proposed method demonstrates its predictive efficiency. The method shows robust performance through experimental results using cross-validation and performance metrics. The proposed model demonstrates competitive performance while providing additional advantages such as incorporation of memory effects and theoretical controllability. The research establishes a novel connection between fractional dynamical systems and machine learning which serves as an essential tool for studying complicated systems in theoretical research and practical applications. Full article

Video-based action recognition for neural rehabilitation—spanning stroke recovery, Parkinsonian gait assessment, and cerebral palsy monitoring—faces critical challenges, including temporal ambiguity, non-causal motion correlations, and the absence of causally grounded dynamics modeling. While transformer-based architectures achieve strong performance, they often exploit spurious temporal and environmental cues, limiting reliability in safety-critical clinical settings. We propose NeuroPrisma, a neuro-prismatic video framework that integrates frequency-domain spectral decomposition with causal intervention under Structural Causal Models (SCMs) via the backdoor criterion. NeuroPrisma introduces (i) a Prismatic Spectral Attention (PSA) module, which applies discrete Fourier transforms to decompose temporal features into multi-scale frequency bands, disentangling slow postural dynamics from rapid corrective movements, and (ii) a Causal Intervention Layer (CIL), which performs do-calculus-based backdoor adjustment to remove confounding influences and produce causally invariant representations. PSA preconditions representations prior to intervention, improving confounder estimation and causal robustness. Extensive evaluation against seven state-of-the-art models (I3D, SlowFast, TimeSformer, ViViT, Video Swin Transformer, UniFormerV2, and VideoMAE) demonstrates that NeuroPrisma achieves 98.7% Top-1 accuracy on UCF101, 82.4% on HMDB51, 71.2% on Something-Something V2, and 91.5%/95.8% on NTU RGB+D (Cross-Subject/Cross-View), consistently outperforming prior methods. It further reduces the Causal Confusion Score (CCS) by 42.3%, indicating substantially lower reliance on spurious correlations, while maintaining real-time performance with 23.4 ms latency per 16-frame clip on an NVIDIA A100 GPU. All improvements are statistically significant (p < 0.001, Cohen’s d = 0.72–1.24). Evaluation was conducted exclusively on benchmark datasets (UCF101, HMDB51, Something-Something V2, and NTU RGB+D) under controlled conditions, without direct clinical validation on neurological patient cohorts. Overfitting was mitigated using three random seeds (42, 123, 456), RandAugment, Mixup (α = 0.8), weight decay (0.05), and early stopping. Cross-dataset generalization from UCF101 to HMDB51 without fine-tuning achieved 76.2% Top-1 accuracy. Future work will focus on prospective clinical validation across stroke, Parkinson’s disease, and cerebral palsy populations, including correlation with standardized clinical assessment scales such as Fugl–Meyer, UPDRS, and GMFCS. These results establish NeuroPrisma as a causally grounded and computationally efficient framework for reliable, real-time movement assessment in clinical rehabilitation systems. Full article

The Mittag-Leffler function holds significant importance in fractional calculus due to its extensive applications in addressing challenges across science, engineering, biology, hydrology, and earth sciences. Notably, the closed-form solution of a time-fractional model naturally emerges as the Mittag-Leffler function (MLF), necessitating precise and efficient computations. Consequently, numerical approximations are essential for accurately calculating the Mittag-Leffler function. In this study, we develop a straightforward yet precise real pole rational approximation for the Mittag-Leffler function. We demonstrate first-order convergence and L-acceptability, which aid in mitigating unwanted oscillations. Additionally, we create an effective and precise first-order generalized exponential time differencing scheme to solve the time-fractional reaction–diffusion equations. We obtain and prove the convergence result using Grönwall-type inequality. Several numerical experiments are conducted to confirm the efficiency and accuracy of the proposed numerical scheme compared with exact solutions. The computational efficiency of the proposed method is compared with another existing first-order numerical technique. Furthermore, our proposed scheme is crucial for developing higher-order predictor–corrector schemes for solving time-fractional models. Full article

This paper investigates an optimal cooperative guidance strategy for the active defense of an early-warning aircraft (EWA) escorted by two fighters against an incoming missile. The proposed framework extends classical three-body defense models (Target–Missile–Interceptor) into a more realistic four-body engagement (Target–Missile–Interceptor 1–Interceptor 2), allowing explicit coordination among multiple defenders. By projecting the 3D engagement kinematics onto two orthogonal 2D planes—a validated simplification for typical aerial combat geometries—a tractable dynamic model is obtained. Within this model, an analytical cooperative guidance law is derived using optimal control theory and the calculus of variations, minimizing a multi-objective cost function that combines miss distance, control effort, intercept geometry, and coordination terms. Extensive Monte Carlo simulations across 23 attack directions and multiple initial ranges demonstrate that the proposed method achieves an interception success rate of 99%, with an average miss distance of below 5 m. Robustness tests further confirm stable performance under target maneuver uncertainty, sensor noise, and modeling deviations. The algorithm features closed-form control commands with low computational complexity, enabling real-time onboard implementation. Full article

This paper develops a structured analytical framework and a robust numerical methodology for the spectral stability of time-varying bilateral quaternion differential equations of the form $\dot{q}=A\left(t\right)q+qB\left(t\right)$. By systematically extending classical real matrix theory to non-commutative dynamical systems via exact isometric real representations, this study utilizes the Kronecker product of real adjoint matrices to rigorously elucidate the underlying tensor structure of the bilateral evolution operator. This tensor-based reformulation proves that the Floquet multipliers of the bilaterally coupled system can be strictly decoupled into the product of the spectra corresponding to the left and right unilateral subsystems. Second, a “Scalar-Vector Stability Separation Principle” based on logarithmic norms is proposed, demonstrating that the transient energy evolution of the system is governed exclusively by the Hermitian real parts of the coefficient matrices, remaining entirely independent of the anti-Hermitian imaginary parts (rotation terms). Furthermore, for constant-coefficient and slowly varying systems, the Riesz projection from holomorphic functional calculus is introduced to establish algebraic criteria for exponential dichotomies, thereby revealing a cubic scaling law that relates the robustness threshold to the spectral gap (${\epsilon }_0\propto {\beta }^3$). Numerically, a Quaternion Chebyshev Spectral Collocation Method (Q-CSCM) is embedded within this exact vectorization framework to ensure that the algebraic symmetries of the bilateral system are strictly preserved through the isomorphic mapping. By explicitly constructing the fully discrete Kronecker product matrix via the exact real vectorization isomorphism, discrete energy estimates are utilized to rigorously prove that the numerical scheme successfully inherits the intrinsic spectral accuracy of the Chebyshev approximation. Comprehensive numerical experiments demonstrate that, within the low-dimensional regime, this methodology exhibits substantial temporal approximation efficiency advantages and superior numerical robustness compared to an alternative Legendre spectral baseline, as well as traditional explicit and state-of-the-art implicit symplectic Runge–Kutta methods, particularly when solving stiff and critically stable problems such as nonlinear Riccati oscillators. Full article

Personalized instruction remains a major bottleneck in higher education, especially in large classes where timely, individualized feedback is difficult to achieve. Existing automation typically relies on rigid rule-based pipelines or computationally heavy deep learning models, making it difficult to simultaneously achieve interpretability, instructional usability, and scalable deployment. In this study, we present CalcTutor, a generative-AI-based assessment and feedback system designed to support open-ended handwritten calculus problem solving. The system organizes instructional support through three coordinated components: (1) a multi-agent large language model (LLM) mechanism that evaluates solution processes and produces diagnostic feedback, (2) a retrieval-augmented generation (RAG) pipeline that links diagnosed difficulties to aligned instructional materials, and (3) real-time learner analytics for both students and instructors, forming an integrated instructional support workflow rather than an automated answer-checking tool. In offline evaluation and a pilot classroom deployment, the multi-agent grader achieved a weighted agreement accuracy of 0.931 and an F1-score of 0.934 on 1055 handwritten solutions. Participant feedback and workflow testing indicated that CalcTutor can be stably integrated into routine classroom use and enables students to interpret and act upon the provided feedback. These results indicate that automated assessment, diagnostic feedback, and targeted review can operate coherently within a single instructional process that supports instructor-led assessment practices. Using undergraduate calculus as an application domain for open-ended handwritten mathematical assessment, the study demonstrates the operational feasibility of a closed-loop assessment–feedback–revision workflow and provides a deployable instructional infrastructure for formative instructional support in real classroom contexts. Full article

Modern enterprise information systems must simultaneously support complex organizational structures, ensure robust security, and remain scalable and maintainable over time. Traditional Role-Based Access Control (RBAC) models, while effective for permission management, operate primarily as post-design security layers and do not provide a unified methodology for structuring system architecture. This paper introduces the Zoned Role-Based (ZRB) model, a mathematically formalized and comprehensive framework that integrates organizational modeling, system design, implementation, access control, and long-term maintenance. ZRB models an organization as a hierarchy of zones, each containing its own roles, applications, operations, and users, forming a recursive Zone Tree that directly mirrors real organizational semantics. Through formally defined role hierarchies, zone-scoped permission sets, and inter-zone inheritance mappings, ZRB provides a context-aware permission calculus that unifies authentication and authorization across all zones. The paper presents the theoretical foundations of ZRB, a multi-phase engineering methodology for constructing integrated enterprise systems, and a complete implementation architecture with permission inference, navigation design, administrative subsystems, and deployment models. Primary validation and evaluations across several developed systems demonstrate significant improvements in permission accuracy, administrative efficiency, scalability, and maintainability. ZRB thus offers a rigorously defined and practically validated framework for building secure, scalable, and organizationally aligned enterprise information systems. Full article

The Ujlayan–Dixit (UD) fractional calculus provides a powerful fractional extension of the Lomax distribution, offering a suitable framework for representing complex behaviors beyond classical approaches. In this paper, we adopt the UD fractional Lomax distribution and establish its statistical theory. Based on the adopted density, we derive closed-form expressions for the cumulative distribution, survival, and hazard functions, as well as the mode. Several UD fractional statistical measures of the Lomax random variable are derived, including the fractional moments, fractional information theoretic measures, including UD fractional Shannon and Tsallis entropy measures, and the probability density function of the $k^{th}$ order statistic under the UD fractional framework. Finally, a real data application concerning the time to break down an insulating fluid is used to illustrate the usefulness of the proposed distribution in modeling real data applications. The fitting performance of the suggested model is compared with several extensions of the Lomax distribution. The comparative results show that the UD fractional Lomax distribution outperforms several well-known extensions of Lomax distribution. This framework provides researchers with many robust tools for advanced reliability assessment, uncertainty quantification, and risk modeling, providing insights into phenomena not captured by the classical Lomax distribution. Moreover, when the fractional parameter $q\to 1^{-}$, the proposed approach converges to the classical Lomax results, bridging fractional and classical perspectives. Full article

Dynamic edge-server allocation in mobile edge computing (MEC) networks is a challenging multi-objective optimization problem due to highly dynamic user demands, spatiotemporal traffic variations, and the need to simultaneously minimize service latency and workload imbalance. Existing heuristic and metaheuristic-based approaches for this problem often suffer from premature convergence, limited exploration–exploitation balance, and inadequate adaptability to dynamic network conditions, leading to suboptimal edge-server placement and inefficient resource utilization. Moreover, most existing methods lack memory-aware search mechanisms, which restrict their ability to capture long-term system dynamics. To address these limitations, this paper proposes a Fractional-Order Multi-Objective African Vulture Optimization Algorithm (FO-MO-AVOA) for dynamic edge-server allocation. By integrating fractional-order calculus into the standard multi-objective AVOA framework, the proposed method introduces long-memory effects that enhance convergence stability, search diversity, and adaptability to time-varying workloads. The performance of FO-MO-AVOA is evaluated using realistic MEC network scenarios and benchmarked against several well-established metaheuristic algorithms. Simulation outcomes reveal that FO-MO-AVOA achieves 40–46% lower latency, 38–45% reduction in workload imbalance, and up to 28–35% reduction in maximum workload compared to competing methods. Extensive experiments conducted on real-world telecom network data demonstrate that FO-MO-AVOA consistently outperforms state-of-the-art multi-objective optimization algorithms in terms of convergence behaviour, Pareto-front quality, and overall system performance. Full article

This paper introduces a novel Bayesian Fractional Weibull (BFW) regression framework, which generalizes the classical Weibull accelerated failure time model using fractional calculus. The proposed methodology addresses key challenges in big data reliability engineering and predictive maintenance by incorporating a fractional order parameter that provides adaptive flexibility when classical Weibull assumptions are violated. We obtain fractional score equations using Caputo derivatives and provide theoretical consistency by proving that classical maximum likelihood estimation emerges as a special case when the fractional order approaches unity. A Bayesian implementation enables full uncertainty quantification and robust inference, particularly in reliability applications characterized by limited data and complicated failure mechanisms. Comprehensive numerical experiments demonstrate the efficacy of the framework: synthetic data validate theoretical properties under both well-specified and misspecified scenarios, while a real-world case study using the NASA C-MAPSS turbofan engine dataset—a standard benchmark in recent reliability literature—shows substantial improvements in predictive performance. The BFW model achieves a 21.7% improvement in predictive performance. This framework combines the theoretical rigor of fractional calculus with the practical advantages of Bayesian inference, directly addressing the need for interpretable and robust methods in big data reliability analytics. Full article

This paper uses the theory of self-affine fractal functions to model the dynamic flight graphs of starling flocks, integrating the fractional calculus of self-affine fractal functions to quantitatively characterize the intrinsic nonlinear dynamics and memory effects within the system, employing statistical inference methods to find the fractal fit for the images. The changes in box dimensions over time could characterize the phase transition process of the starling flight flocks. By analyzing the rate of change of fractal dimensions, we identify critical points corresponding to phase transitions during collective flight behavior. During the flight of the starling flocks, a real-time phase transition process for evading attacks and effective advancement has been identified. Experimental data confirms the effectiveness of controlling the phase transition. Full article

This study presents a novel softsign-function-based fractional-order proportional–integral–derivative (softsign-FOPID) controller optimized using the fungal growth optimizer (FGO) for the stabilization and precise position control of an unstable magnetic ball suspension system. The proposed controller introduces a smooth nonlinear softsign function into the conventional FOPID structure to limit abrupt control actions and improve transient smoothness while preserving the flexibility of fractional dynamics. The FGO, a recently developed bio-inspired metaheuristic, is employed to tune the seven controller parameters by minimizing a composite objective function that simultaneously penalizes overshoot and tracking error. This optimization ensures balanced transient and steady-state performance with enhanced convergence reliability. The performance of the proposed approach was extensively benchmarked against four modern metaheuristic algorithms (greater cane rat algorithm, catch fish optimization algorithm, RIME algorithm and artificial hummingbird algorithm) under identical conditions. Statistical analyses, including boxplot comparisons and the nonparametric Wilcoxon rank-sum test, demonstrated that the FGO consistently achieved the lowest objective function value with superior convergence stability and significantly better (p < 0.05) performance across multiple independent runs. In time-domain evaluations, the FGO-tuned softsign-FOPID exhibited the fastest rise time (0.0089 s), shortest settling time (0.0163 s), lowest overshoot (4.13%), and negligible steady-state error (0.0015%), surpassing the best-reported controllers in the literature, including the sine cosine algorithm-tuned PID, logarithmic spiral opposition-based learning augmented hunger games search algorithm-tuned FOPID, and manta ray foraging optimization-tuned real PIDD². Robustness assessments under fluctuating reference trajectories, actuator saturation, sensor noise, external disturbances, and parametric uncertainties (±10% variation in resistance and inductance) further confirmed the controller’s adaptability and stability under practical non-idealities. The smooth nonlinearity of the softsign function effectively prevented control signal saturation, while the fractional-order dynamics enhanced disturbance rejection and memory-based adaptability. Overall, the proposed FGO-optimized softsign-FOPID controller establishes a new benchmark in nonlinear magnetic levitation control by integrating smooth nonlinear mapping, fractional calculus, and adaptive metaheuristic optimization. Full article

This work explores the issue of identifying sub-synchronous oscillations (SSOs). Regular detection techniques face issues with response timings to variations in viewpoint and adaptability to variations in conditions of the system but our proposed method overcomes them. We have actually come up with a new framework called Tempered Fractional Deterministic Learning (TF-DL) that successfully combines tempered fractional calculus with deterministic learning theory. This method makes a memory-based learner that works best for oscillatory dynamics. This lets SSO identification happen faster through a recursive structure that can run in real time. Theoretical analysis validates exponential convergence in the context of persistent excitation. Simulations show that detection time is 62.7% shorter than gradient descent, with better convergence and better parameters. Full article

This study introduces Itô-RMSProp, a novel extension of the RMSProp optimizer inspired by Itô stochastic calculus, which integrates adaptive Gaussian noise into the update rule to enhance exploration and mitigate overfitting during training. We embed this optimizer within Gated Recurrent Unit (GRU) networks for stock price forecasting, leveraging the GRU’s strength in modeling long-range temporal dependencies under nonstationary and noisy conditions. Extensive experiments on real-world financial datasets, including a detailed sensitivity analysis over a wide range of noise scaling parameters ($\epsilon$), reveal that Itô-RMSProp-GRU consistently achieves superior convergence stability and predictive accuracy compared to classical RMSProp. Notably, the optimizer demonstrates remarkable robustness across all tested configurations, maintaining stable performance even under volatile market dynamics. These findings suggest that the synergy between stochastic differential equation frameworks and gated architectures provides a powerful paradigm for financial time series modeling. The paper also presents theoretical justifications and implementation details to facilitate reproducibility and future extensions. Full article

The accurate estimation of parameters in dynamical systems stands for an open key research issue in modeling, control, and fault diagnosis. The presence of noise in input and output signals poses a serious challenge for accurate real-time dynamical system parameter estimation. This paper proposes a new robust algebraic parameter estimation methodology for integer-order dynamical systems that explicitly incorporates the signal filtering dynamics within the estimator structure and enhances noise attenuation through fractional differentiation in frequency domain. The introduced estimation methodology is valid for Liouville-type fractional derivatives and can be applied to estimate online the parameters of differentially flat, oscillating or vibrating systems of multiple degrees of freedom. The parametric estimation can be thus implemented for a wide class of oscillating or vibrating, nth-order dynamical systems under noise influence in measurement and control signals. Positive values are considered for the inertia, stiffness, and viscous damping parameters of vibrating systems. Parameter identification can be also used for development of actuators and control technology. In this sense, validation of the algebraic parameter estimation is performed to identify parameters of a differentially flat, permanent-magnet direct-current motor actuator. Parameter estimation for both open-loop and closed-loop control scenarios using experimental data is examined. Experimental results demonstrate that the new parameter estimation methodology combining signal filtering dynamics and fractional calculus outperforms other conventional methods under presence of significant noise in measurements. Full article