Search Results (1,588)

Textile dye effluents containing toxic organic compounds pose serious environmental challenges. In this study, novel Poly(1-vinyl imidazole)-Bis[2-(methacryloyloxy)ethyl] phosphate (PVIB) polymers were synthesized with crosslinker molar fractions ranging from 5% to 80% and were subsequently investigated as advanced adsorbents for textile dye removal. Procion Red (PR), a widely used reactive dye, was selected as the model pollutant. The materials were characterized using FTIR, TGA, DTG, SEM-EDX, WD-XRF, TEM, and BET analyses. Adsorption mechanisms were examined through kinetic, isotherm, and thermodynamic models. Among the synthesized formulations, PVIB20% achieved the best dye removal, reaching an experimental adsorption capacity of 330 mg g⁻¹ within 60 min under acidic to neutral conditions. The kinetic modeling studies identified the pseudo-first-order model as the best fit, indicating a surface-controlled process involving both physical and chemical interactions. Isotherm studies showed that the Langmuir and Redlich–Peterson models provided the best fit, yielding a maximum monolayer adsorption capacity of 765 mg g⁻¹. Thermodynamic analysis revealed that the adsorption was spontaneous, endothermic, and entropy-driven. Overall, PVIB20% demonstrated superior adsorption capacity, rapid kinetics, and strong dye–polymer interactions compared with many conventional and modified adsorbents, which highlights its potential as an efficient and durable material for anionic dye removal from wastewater. Full article

Fractional calculus in discrete-time is a recent field that has drawn much interest for dealing with multidisciplinary systems. A result of this tremendous potential, researchers have been using constant and variable-order fractional discrete calculus in the modelling of financial and economic systems. This paper explores the emergence of chaotic and regular patterns of the fractional four-dimensional (4D) discrete economic system with constant and variable orders. The primary aim is to compare and investigate the impact of two types of fractional order through numerical solutions and simulation, demonstrating how modifications to the order affect the behavior of a system. Phase space orbits, the 0-1 test, time series, bifurcation charts, and Lyapunov exponent analysis for different orders all illustrate the constant and variable-order systems’ behavior. Moreover, the spectral entropy (SE) and $C_0$ complexity exhibit fractional-order effects with variations in the degree of complexity. The results provide new insights into the influence of fractional-order types on dynamical systems and highlight their role in promoting chaotic behavior. Additionally, two types of control strategies are devised to guide the states of a 4D fractional discrete economic system with constant and variable orders to the origin within a specified amount of time. MATLAB simulations are presented to demonstrate the efficacy of the findings. Full article

Macroeconomic mathematical models are practical instruments structured to carry out theoretical analyses of macroeconomic developments. In this manuscript, the Caputo-like fractional operator of variable order is used to introduce and investigate the mechanism of the discrete macroeconomic model. The nature of the dynamics was established, and the emergence of chaos using a distinct variable fractional order, especially the stability of the trivial solution, is examined. The findings reveal that the variable-order discrete macroeconomic model displays chaotic dynamics employing bifurcation, the Largest Lyapunov exponent ($L{E}_{max}$), the phase portraits, and the 0–1 test. Furthermore, a complexity analysis is performed to demonstrate the existence of chaos and quantify its complexity using $C_0$ complexity and spectral entropy ($SE$). These studies show that the suggested variable-order fractional discrete macroeconomic model has more complex features than integer and constant fractional orders. Finally, MATLAB R2024b simulations are run to exemplify the outcomes of this study. Full article

The study of economic maps has consistently attracted researchers due to their rich dynamics and practical relevance. A deeper understanding of these systems enables the development of more effective control strategies. In this work, we examine the influence of the fractional order $\upsilon$ with the Caputo fractional difference on an economic market map. The primary contribution is the comprehensive analysis of how both commensurate and incommensurate fractional orders affect the stability and complexity of the map. Numerical investigations, including phase portraits, largest Lyapunov exponents, and bifurcation analysis, reveal that the system undergoes a cascade of period-doubling bifurcations before transitioning into chaos. To further characterize the dynamics, complexity is evaluated using the 0–1 test and $C_0$ complexity, both confirming chaotic behavior. Furthermore, two-dimensional control schemes are introduced and theoretically validated to both stabilize the chaotic economic market map and achieve synchronization with a combined response map. The theoretical and numerical results are validated through MATLAB 2016 simulations. Full article

Sandflies spread the neglected vector-borne disease anthroponotic cutaneous leishmaniasis (ACL), which only affects humans. Despite decades of control, asymptomatic carriers, vector pesticide resistance, and low public awareness prevent eradication. This study proposes a fractional-order optimal control model that integrates biological and behavioral aspects of ACL transmission to better understand its complex dynamics and intervention responses. We model asymptomatic human illnesses, insecticide-resistant sandflies, and a dynamic awareness function under public health campaigns and collective behavioral memory. Four time-dependent control variables—symptomatic treatment, pesticide spraying, bed net use, and awareness promotion—are introduced under a shared budget constraint to reflect public health resource constraints. In addition, Caputo fractional derivatives incorporate memory-dependent processes and hereditary effects, allowing for epidemic and behavioral states to depend on prior infections and interventions; on the other hand, standard integer-order frameworks miss temporal smoothness, delayed responses, and persistence effects from this memory feature, which affect optimal control trajectories. Next, we determine the optimality conditions for fractional-order systems using a generalized Pontryagin’s maximum principle, then solve the state–adjoint equations numerically with an efficient forward–backward sweep approach. Simulations show that fractional (memory-based) dynamics capture behavioral inertia and cumulative public response, improving awareness and treatment efforts. Furthermore, sensitivity tests indicate that integer-order models do not predict the optimal allocation of limited resources, highlighting memory effects in epidemiological decision-making. Consequently, the proposed method provides a realistic and flexible mathematical basis for cost-effective and sustainable ACL control plans in endemic settings, revealing how memory-dependent dynamics may affect disease development and intervention efficiency. Full article

High-performance tracking control for the hydraulic manipulator should address the challenges of the uncertainties and unknowns associated with the electro-hydraulic servo system (EHSS). This paper presents an extended state observer-based chattering-free terminal sliding-mode (ESO-CFTSM) control scheme for hydraulic manipulators. A third-order integral chain model is developed to characterize the system dynamics, where uncertainties and unknowns are considered as disturbances and estimated by the ESO. Meanwhile, a full-order TSM manifold is designed to stabilize the closed-loop system in finite-time. For this proposed scheme, the feedforward compensation of disturbances is introduced in the equivalent control law. Furthermore, the composite reaching law and a low-pass filter are used to realize the chattering-free control. The singularity is avoided because there are no derivatives of terms with fractional powers in the control law. The stability of the overall system is proved by Lyapunov technique. The simulations using the physical model of a hydraulic manipulator with coupled dynamics show the effectiveness of the proposed scheme for trajectory tracking problems. Simulation results indicate that the proposed ESO-CFTSM can achieve superior performance without being affected by lumped disturbances. Full article

This study presents a comprehensive Lyapunov-based framework for analyzing partial practical stability in nonlinear tempered fractional-order systems (TFOS). We develop novel stability concepts including ${\beta }^{*}$-practical uniform generalized Mittag–Leffler stability (${\beta }^{*}$-PUGMLS) and ${\beta }^{*}$-practical uniform exponential stability (${\beta }^{*}$-PUES) with respect to system substates. Through carefully constructed Lyapunov functions, we establish sufficient conditions under which the system’s states converge to a predefined neighborhood of the origin. The theoretical framework provides Mittag–Leffler and exponential stability criteria for tempered fractional-order systems, extending classical stability theory to this important class of systems. Furthermore, we apply these stability results to design stabilizing feedback controllers for a specific class of triangular TFOS, demonstrating the practical utility of our theoretical developments. The efficacy of the proposed stability criteria and control strategy is validated through several illustrative examples, showing that system states converge appropriately under the derived conditions. This work contributes significantly to the stability theory of fractional-order systems and provides practical tools for controlling complex nonlinear systems in the tempered fractional calculus framework. Full article

This study investigates stochastic bifurcation in a generalized tristable Rayleigh–Duffing oscillator with fractional inertial force under both additive and multiplicative recycling noises. The system’s dynamic behavior is influenced by its inherent spatial symmetry, represented by the potential function, as well as by temporal symmetry breaking caused by fractional memory effects and recycling noise. First, an approximate integer-order equivalent system is derived from the original fractional-order model using a harmonic balance method, with minimal mean square error (MSE). The steady-state probability density function (sPDF) of the amplitude is then obtained via stochastic averaging. Using singularity theory, the conditions for stochastic P bifurcation (SPB) are identified. For different fractional derivative’s orders, transition set curves are constructed, and the sPDF is qualitatively analyzed within the regions bounded by these curves—especially under tristable conditions. Theoretical results are validated through Monte Carlo simulations and the Radial Basis Function Neural Network (RBFNN) approach. The findings offer insights for designing fractional-order controllers to improve system response control. Full article

Unmanned aerial vehicles (UAVs) are a promising technology for future sixth-generation (6G) wireless networks. They are airborne vehicles that act either as as flying relays or base stations (BS) to provide the line-of-sight (LOS) transmission, enable wide-area coverage, and increase the spectral efficiency. In this work, a UAV is employed to forward information from the BS to distant users using a decode-and-forward (DF) protocol. The BS serves ground users through UAV by employing non-orthogonal multiple access (NOMA). The UAV relay will be wirelessly powered and harvests energy from the BS by applying a simultaneous wireless information and power transfer (SWIPT) technique. To further improve overall performance, the near user will act as a full-duplex (FD) relay to forward the far user’s information by applying cooperative non-orthogonal multiple access (C-NOMA). The proposed scheme considers a practical detection order using a feasible successive interference cancellation (SIC) operation. Additionally, a relay power control method is introduced for the near user to guarantee a reliable cooperative link. In the proposed scheme, a low-complexity closed-form power allocation is derived to maximize the minimum achievable rate. Numerical results demonstrate that the power allocation scheme significantly improves the far user’s rate performance, and the proposed scheme guarantees a higher target rate and outperforms the conventional NOMA, half-duplex (HD) C-NOMA, and FD C-NOMA with fixed power allocation (FPA) and fractional transmit power allocation (FTPA) schemes. Full article

Thermoelectric materials enable the direct conversion of heat into electricity, but progress is often limited by challenges in reproducibility and stability. Bi₂O₂Se has recently attracted attention as a promising candidate; however, reported transport properties of undoped polycrystalline samples vary by several orders of magnitude, complicating its use as a baseline for doping studies. In this work, we investigate the sources of variability and identify key factors including precursor contamination, reactions with quartz ampoules and graphite dies, grain size effects, and surface oxidation. To mitigate these issues, we employed calcination of Bi₂O₃ precursors, synthesis with controlled temperature gradients, coarse-fraction powders, and hot pressing in Si₃N₄ dies. The resulting polycrystalline Bi₂O₂Se exhibits improved reproducibility, reduced sensitivity to thermal cycling, and characteristic transport values around σ_RT ≈ 500 S·m⁻¹ and S ≈ −300 μV·K⁻¹ at room temperature. This is a good starting point for further doping studies and a prerequisite of thermoelectric efficiency studies in the future, which can reveal the true thermoelectric potential of this material. Full article

This article presents several key findings for fractional-order delay differential equations. First, we establish the existence and uniqueness of solutions using two distinct approaches, the Chebyshev norm and the Bielecki norm, thereby providing a comprehensive understanding of the solution space. Notably, the uniqueness of the solution is rigorously demonstrated using the Lipschitz condition, ensuring a single solution under specific constraints. Additionally, we examine a specific form of constant delay and apply Burton’s method to further confirm the uniqueness of the solution. Furthermore, we conduct an in-depth investigation into the Hyers–Ulam stability of the problem, providing valuable insights into the behavior of solutions under perturbations. Notably, our results eliminate the need for contraction constant conditions that are commonly imposed in the existing literature. Finally, numerical simulations are performed to illustrate and validate the theoretical results obtained in this study. Fractional-order delay differential equations play a crucial role in real-life applications in systems where memory and delayed effects are essential. In biology and epidemiology, they model disease spread with incubation delays and immune memory. In control systems and robotics, they help design stable controllers by accounting for time-lagged responses and past behavior. Full article

Hybrid PV-TEG systems can harvest both solar electrical and thermoelectric power, but their operating point drifts with irradiance, temperature gradients, partial shading, and load changes—often yielding multi-peak P-V characteristics. Conventional MPPT (e.g., P&O) and fixed-structure integer-order PID struggle to remain fast, stable, and globally optimal in these conditions. To address fast, robust tracking in these conditions, we propose an adaptive fractional-order PID (FOPID) MPPT whose parameters (K_p, K_i, K_d, λ, μ) are auto-tuned by the red-billed blue magpie optimizer (RBBMO). RBBMO is used offline to set the controller’s search ranges and weighting; the adaptive law then refines the gains online from the measured ΔV, ΔI slope error to maximize the hybrid PV-TEG output. The method is validated in MATLAB R2024b/Simulink 2024b, on a boost-converter–interfaced PV-TEG using five testbeds: (i) start-up/search, (ii) stepwise irradiance, (iii) partial shading with multiple local peaks, (iv) load steps, and (v) field-measured irradiance/temperature from Shanxi Province for spring/summer/autumn/winter. Compared with AOS, PSO, MFO, SSA, GHO, RSA, AOA, and P&O, the proposed tracker is consistently the fastest and most energy-efficient: 0.06 s to reach 95% MPP and 0.12 s settling at start-up with 1950 W·s harvested (vs. 1910 W·s AOS, 1880 W·s PSO, 200 W·s P&O). Under stepwise irradiance, it delivers 0.95–0.98 kJ at t = 1 s and under partial shading, 1.95–2.00 kJ, both with ±1% steady ripple. Daily field energies reach 0.88 × 10⁻³, 2.95 × 10⁻³, 2.90 × 10⁻³, 1.55 × 10⁻³ kWh in spring–winter, outperforming the best baselines by 3–10% and P&O by 20–30%. Robustness tests show only 2.74% power derating across 0–40 °C and low variability (Δv^max typically ≤ 1–1.5%), confirming rapid, low-ripple tracking with superior energy yield. Finally, the RBBMO-tuned adaptive FOPID offers a superior efficiency–stability trade-off and robust GMPP tracking across all five cases, with modest computational overhead. Full article

This study investigates the control of the phase-structural state in Ni–45Ti–xCu (x = 5, 7 at.%) shape memory alloys fabricated via a shortened powder metallurgy route: mechanical activation → spark plasma sintering (SPS) → heat treatment. Compact samples were produced from mechanically alloyed powders (650–750 rpm, up to 5 h) and sintered at 900 °C. The structure and microstructure were characterized using X-ray diffraction (to identify B2/B19′/Ni₄Ti₃ phases and assess ordering) and SEM–BSE/EDS (to analyze morphology, porosity, and Ni-rich precipitates). Two post-processing treatments were applied: single-stage annealing (500 °C, 2 h) and a three-stage treatment (900 °C/30 min → water quenching → 300 °C/20 min). Mechanical alloying transformed the initial elemental powder mixture (fcc-Ni, hcp-Ti, fcc-Cu) into a supersaturated fcc-(Ni, Cu, Ti) solid solution with emerging NiTi phases, with a minimum particle size achieved after ~300 min at 750 rpm. SPS compaction yielded a high-density matrix consisting predominantly of the B2 phase. Single-stage annealing preserved B19′ martensite and Ni₄Ti₃ precipitates, particularly in the 5 at.% Cu alloy. In contrast, the three-stage treatment dissolved the Ni₄Ti₃ precipitates, suppressed the formation of B19′ and R phases, and stabilized a highly ordered B2 matrix. Increasing the Cu content from 5 to 7 at.% significantly enhanced the B2 phase fraction, reduced secondary nickel-rich phases, and improved structural homogeneity, evidenced by a continuous neck network and closed porosity. The optimized condition—7 at.% Cu combined with the three-stage annealing—produced a microstructure with >95% B2 phase, <1% Ni₄Ti₃, and ~98% relative density. This forms the prerequisite microstructural state for a narrow transformation hysteresis and high functional cyclic stability. Full article

Controllers designed on the basis of an approximate discrete time model can fail when they are applied to the real processes. This is a problem of interest in adaptive symmetry control system theory. In this paper, we introduce the certain polynomial for discretizing the $n$-th order integrator in the case of a fractional-order hold (FROH) and delta operator. Moreover, the corresponding FROH discretization pulse transfer function and sampling data model in normal form are derived. On the basis of these results, we present FROH discrete system zero dynamics and sampling zero dynamics where their asymptotic symmetry properties are obtained. In addition, the implications incorporating sampling zero dynamics in the discrete-time models used for system identification become evident as the sampling period approaches zero. It is a further extension of the corresponding results. Full article

An optical vortex is a typical structured light field characterized by a helical wavefront and a central phase singularity. With its expanding applications in modern information technology, the demand for generating vortex beams with diverse topological charges continues to grow. Existing methods for modulating the topological charges of vortex beams involve complex operations and high costs. This study proposes a novel approach to modulate the topological charges of optical vortices through edge diffraction of a high-order Hermit–Gaussian (HG) mode laser. First, a high-order HG mode laser is built using off-axis pumping configuration. By selectively obscuring specific lobes of the high-order HG beam, various optical vortices are generated using a cylindrical lens mode converter. The topological charge can be continuously tuned by controlling the number of obscured lobes. This method substantially improves the efficiency of topological charge modulation, while also enabling the generation of fractional vortex states. These advancements show potential in mode-division-multiplexed optical communications and encryption. Full article