Economic dispatch in wind-integrated power systems is a critical challenge, yet many recent metaheuristics suffer from premature convergence, heavy parameter tuning, and limited ability to escape local optima in non-smooth valve-point landscapes. This study proposes a new hybrid optimization framework, the Fractional Grasshopper Optimization algorithm (FGOA), which integrates fractional-order calculus into the standard Grasshopper Optimization algorithm (GOA) to enhance its search efficiency. The FGOA method is applied to the economic load dispatch (ELD) problem, a nonlinear and nonconvex task that aims to minimize fuel and wind-generation costs while satisfying practical constraints such as valve-point loading effects (VPLEs), generator operating limits, and the stochastic behavior of renewable energy sources. Owing to the increasing role of wind energy, stochastic wind power is modeled through the incomplete gamma function (IGF). To further improve computational accuracy, FGOA is hybridized with Sequential Quadratic Programming (SQP), where FGOA provides global exploration and SQP performs local refinement. The proposed FGOA-SQP approach is validated on systems with 3, 13, and 40 generating units, including mixed thermal and wind sources. Comparative evaluations against recent metaheuristic algorithms demonstrate that FGOA-SQP achieves more accurate and reliable dispatch outcomes. Specifically, the proposed approach achieves fuel cost reductions ranging from 0.047% to 0.71% for the 3-unit system, 0.31% to 27.25% for the 13-unit system, and 0.69% to 12.55% for the 40-unit system when compared with state-of-the-art methods. Statistical results, particularly minimum fitness values, further confirm the superior performance of the FGOA-SQP framework in addressing the ELD problem under wind power uncertainty.

Fractional calculus, the study of derivatives and integrals of arbitrary order [...]

This study has developed a unified framework for modeling economic growth through Caputo fractional differential equations. The framework has established the existence and uniqueness of solutions by employing a generalized fixed-point approach. In particular, the analysis has introduced and utilized new classes of symmetric operators, including symmetric Lipschitz-type mappings, symmetric Kannan-type contractions, and symmetric Chatterjea-type contractions. These mappings are based on a refined symmetric Lipschitz condition that enables the examination of the behavior of their iterative sequences. The study has focused on several forms of symmetric contractions defined on metric spaces endowed with a binary relation, providing a setting that generalizes and unifies various existing fixed-point theorems. This framework has extended classical results by Goebel and Sims, Goebel and Japon-Pineda, and others. Finally, to illustrate the practical significance of the theoretical findings, the developed results have been applied to demonstrate the existence of solutions for fractional models of economic growth and a related Fredholm integral equation.

This work develops and analyzes a novel fractional-order chemostat system (FOCS) with a Caputo fractional derivative (CFD) featuring a sliding memory window and periodic boundary conditions (PBCs), designed to model microbial pollutant degradation in sustainable water treatment. By incorporating the Caputo fractional derivative with sliding memory (CFDS), the model captures time-dependent behaviors and memory effects in biological systems more realistically than classical integer-order formulations. We reduce the two-dimensional fractional differential equations (FDEs) governing substrate and biomass concentrations to a one-dimensional FDE by utilizing the PBCs. The existence and uniqueness of non-trivial, periodic solutions are established using the Carathéodory framework and fixed-point theorems, ensuring the system’s well-posedness. We prove the positivity and boundedness of solutions, demonstrating that substrate concentrations remain within physically meaningful bounds and biomass concentrations stay strictly positive, with solution trajectories confined to a biologically feasible invariant set. Additionally, we analyze non-trivial equilibria under constant dilution rates and derive their stability properties. The rigorous mathematical results confirm the viability of FOCS models for representing memory-driven, periodic bioprocesses, offering a foundation for advanced water treatment strategies that align with Sustainable Development Goal 6 (Clean Water and Sanitation). This work establishes a comprehensive mathematical framework that bridges fractional calculus with sustainable water treatment applications, providing both theoretical foundations and practical implications for optimizing bioreactor performance in environmental biotechnology.

Theaim of this research is to establish existence and uniqueness results for the Riemann–Liouville fractional integral equation of order αϰ(t)=f(t)+λΓα∫0tt−sα−1gs,ϰ(s)ds,t∈[0,1], by developing common fixed point theorems for generalized contractions involving control functions of two variables in the framework of complex valued suprametric spaces. The proposed results extend and generalize several existing findings in the literature, and some illustrative examples are provided to demonstrate the novelty and applicability of the main theorem.

This paper proposes a novel fractional-order African vulture optimization algorithm (FO-AVOA) for solving the optimal reactive power dispatch (ORPD) problem. By integrating fractional calculus into the conventional AVOA framework, the proposed method enhances the exploration–exploitation balance, accelerates convergence, and improves solution robustness. The ORPD problem is formulated as a constrained optimization task with the objective of minimizing real power losses while satisfying generator voltage limits, transformer tap ratios, and reactive power compensator constraints. The general optimization capability of the FO-AVOA is verified using the CEC 2017, 2020, and 2022 benchmark functions. In addition, the method is applied to the IEEE 30-bus and IEEE 57-bus test systems. The results demonstrate significant power loss reductions of up to 15.888% and 24.39% for the IEEE 30-bus and IEEE 57-bus systems, respectively, compared with the conventional AVOA and other state-of-the-art optimization algorithms, along with strong robustness and stability across independent runs. These findings confirm the effectiveness of the FO-AVOA as a reliable optimization tool for modern power system applications.

The outbreaks of large-scale epidemics, such as COVID-19 in 2019–2022, challenge modelers. Beside the effect of the incubation period of the virus, the delay property of detection should be also stressed. This kind of memory effect affects the entire change rate, which cannot be reflected by the conventional instantaneous derivative. The fractional derivative (FD) meets this request to some extent. Yet the shortcoming of it limits its usage. Through a strict modeling approach, a new susceptible–infective–removed (SIR) model with the memory-dependent derivative (MDD) has been constructed. The numerical simulations indicate that (1) the neglecting of the incubation period may underestimate the number of susceptible individuals and overestimate the infected ones; (2) the neglecting of the treatment period may badly overestimate the removed individuals; (3) the consequence of tardy detection intervention may be very serious, and the infectious rate may increase rapidly with a postponed peak time; and (4) the SIR model with the FD yields bad estimations, not only in the primary stage but also in the subsequent evolution. Due to the reasonability of the new SIR model with the MDD, it is suggested to epidemic researchers.

The fractal extension of the time-dependent Ginzburg–Landau (TDGL) equation, formulated within the framework of Scale Relativity, generalizes superconducting dynamics to non-differentiable space–time. Although analytically well established, its numerical solution remains difficult because of the strong coupling between amplitude and phase curvature. Here we develop two complementary deep learning solvers for the fractal TDGL (FTDGL) system. The Fractal Physics-Informed Neural Network (F-PINN) embeds the Scale-Relativity covariant derivative through automatic differentiation on continuous fields, whereas the Fractal Graph Neural Network (F-GNN) represents the same dynamics on a sparse spatial graph and learns local gauge-covariant interactions via message passing. Both models are trained against finite-difference reference data, and a parametric study over the dimensionless fractality parameter D quantifies its influence on the coherence length, penetration depth, and peak magnetic field. Across multivortex benchmarks, the F-GNN reduces the relative L2 error on ψ2 from 0.190 to 0.046 and on Bz from approximately 0.62 to 0.36 (averaged over three seeds). This ≈4× improvement in condensate-density accuracy corresponds to a substantial enhancement in vortex-core localization—from tens of pixels of uncertainty to sub-pixel precision—and yields a cleaner reconstruction of the 2π phase winding around each vortex, improving the extraction of experimentally relevant observables such as ξeff, λeff, and local Bz peaks. The model also preserves flux quantization and remains robust under 2–5% Gaussian noise, demonstrating stable learning under experimentally realistic perturbations. The D—scan reveals broader vortex cores, a non-monotonic variation in the penetration depth, and moderate modulation of the peak magnetic field, while preserving topological structure. These results show that graph-based learning provides a superior inductive bias for modeling non-differentiable, gauge-coupled systems. The proposed F-PINN and F-GNN architectures therefore offer accurate, data-efficient solvers for fractal superconductivity and open pathways toward data-driven inference of fractal parameters from magneto-optical or Hall-probe imaging experiments.

This work focuses on the finite-time synchronization control (FTSC) for fractional-order chaotic systems (FOCSs) subject to uncertain dynamics, unknown parameters and input nonlinearities. In the control law design, the uncertain dynamics of the FOCSs are addressed by using fuzzy logic systems (FLSs), while the unknown control direction caused by unknown input nonlinearity is handled through applying the Nussbaum gain function (NGF) method. Parameter adaptive laws are derived to estimate the unknown parameters of the given FOCSs, the parameter vectors of the FLSs, and unknown bounded constants, respectively. By integrating these parameter-adaptive laws with the FT backstepping control framework and FO Lyapunov direct method, an adaptive fuzzy FTSC strategy is developed. This strategy ensures that the synchronization error (SE) can converge to a small neighborhood of zero (SNoZ) within a FT and all signals of the closed-loop system (CLS) remain ultimately bounded. In the end, three simulation cases are utilized to demonstrate the efficiency of the proposed control method.

In this study, a time-fractional extension of the classical Theis problem with an exponential source term is investigated in a confined aquifer. The governing equation is modeled using two different fractional derivatives—the Caputo and Atangana–Baleanu–Caputo (ABC) operators—to account for memory effects in groundwater flow. The Fractional Reduced Differential Transform Method (FRDTM) is applied to obtain approximate series solutions up to the fifth order. The impact of the fractional order α, the nature of the fractional kernel and the localized source term on the hydraulic head are explored at different radial positions. The comparative analysis between the Caputo and Atangana–Baleanu–Caputo (ABC) models reveals how memory effects and operator choice significantly influence the hydraulic head response, offering insights into selecting suitable models for aquifers with varying recharge characteristics.