Search Results (595)

This study presents a qualitative analysis of the fractional coupled nonlinear Schrödinger equation (FCNSE) to obtain its complete set of solutions. An appropriate wave transformation is applied to reduce the FCNSE to a fourth-order dynamical system. Due to its non-Hamiltonian nature, this system poses significant analytical challenges. To overcome this complexity, the dynamical behavior is examined within a specific phase–space subspace, where the system simplifies to a two-dimensional, single-degree-of-freedom Hamiltonian system. The qualitative theory of planar dynamical systems is then employed to characterize the corresponding phase portraits. Bifurcation analysis identifies the physical parameter conditions that give rise to super-periodic, periodic, and solitary wave solutions. These solutions are derived analytically and illustrated graphically to highlight the influence of the fractional derivative order on their spatial and temporal evolution. Furthermore, when an external generalized periodic force is introduced, the model exhibits quasi-periodic behavior followed by chaotic dynamics. Both configurations are depicted through 3D and 2D phase portraits in addition to the time-series graphs. The presence of chaos is quantitatively verified by calculating the Lyapunov exponents. Numerical simulations demonstrate that the system’s behavior is highly sensitive to variations in the frequency and amplitude of the external force. Full article

This paper investigates soliton solutions of the time-fractional Biswas–Arshed (BA) equation using the Extended Simplest Equation Method (ESEM). The model is analyzed under two distinct fractional derivative operators: the $\beta$-derivative and the M-truncated derivative. These approaches yield diverse solution types, including kink, singular, and periodic-singular forms. Also, in this work, a nonlinear second-order differential equation is reconstructed as a planar dynamical system in order to study its bifurcation structure. The stability and nature of equilibrium points are established using a conserved Hamiltonian and phase space analysis. A bifurcation parameter that determines the change from center to saddle-type behaviors is identified in the study. The findings provide insight into the fundamental dynamics of nonlinear wave propagation by showing how changes in model parameters induce qualitative changes in the phase portrait. The derived solutions are depicted via contour plots, along with two-dimensional (2D) and three-dimensional (3D) representations, utilizing Mathematica for computational validation and graphical illustration. This study is motivated by the growing role of fractional calculus in modeling nonlinear wave phenomena where memory and hereditary effects cannot be captured by classical integer-order approaches. The time-fractional Biswas–Arshed (BA) equation is investigated to obtain diverse soliton solutions using the Extended Simplest Equation Method (ESEM) under the $\beta$-derivative and M-truncated derivative operators. Beyond solution construction, a nonlinear second-order equation is reformulated as a planar dynamical system to analyze its bifurcation and stability properties. This dual approach highlights how parameter variations affect equilibrium structures and soliton behaviors, offering both theoretical insights and potential applications in physics and engineering. Full article

This paper provides the modeling, implementation, and simulation of fractional-order processes associated with the production of the enriched ¹³C isotope due to chemical exchange processes between carbamate and CO₂. To demonstrate and simulate the process most effectively, an execution of a new approximating solution of fractional-order systems is required, which has become possible due to the utilization of advanced AI methods. As the separation process exhibits extremely strong nonlinearity and fractional-order-based performance, it was similarly necessary to utilize the fractional-order system theory to mathematically model the operation, which consists of the comparison of its output with an integrator function. The learning of the dynamic structure’s parameters of the derived fractional-order model is performed by neural networks, which are AI-based domain solutions. Thanks to the approximations executed, the concentration dynamics of the enriched ¹³C isotope can be simulated and predicted with a high level of precision. The solutions’ effectiveness is corroborated by the model’s response comparison with the reaction of the actual process. The current implementation uses neural networks trained specifically for this purpose. Furthermore, since the isotopic separation processes are long-settling-time processes, this paper proposes some control strategies that are developed for the ¹³C isotopic separation process, in order to improve the system performances and to avoid the loss of enriched product. The adaptive controllers were tuned by imposing them to follow the output of a first-order-type transfer function, using a PI or a PID controller. Finally, the paper confirms that AI solutions can successfully support the system throughout a range of responses, which paves the way for an efficient design of the automatic control for the ¹³C isotope concentration. Such systems can similarly be implemented in other industrial processes. Full article

We consider first-order impulses for impulsive stochastic differential equations driven by fractional Brownian motion (fBm) with Hurst parameter $H\in ({\displaystyle \frac{1}{2}},1)$ involving a nonlinear $\varphi$-Laplacian operator. The system incorporates both state and derivative impulses at fixed time instants. First, we establish the existence of at least one mild solution under appropriate conditions in terms of nonlinearities, impulses, and diffusion coefficients. We achieve this by applying a nonlinear alternative of the Leray–Schauder fixed-point theorem in a generalized Banach space setting. The topological structure of the solution set is established, showing that the set of all solutions is compact, closed, and convex in the function space considered. Our results extend existing impulsive differential equation frameworks to include fractional stochastic perturbations (via fBm) and general $\varphi$-Laplacian dynamics, which have not been addressed previously in tandem. These contributions provide a new existence framework for impulsive systems with memory and hereditary properties, modeled in stochastic environments with long-range dependence. Full article

The conventional sliding mode control (SMC) strategy for direct torque control of switched reluctance motors suffers from severe chattering and prolonged dynamic response. Accordingly, an enhanced SMC strategy is proposed to mitigate motor chattering and suppress torque ripple. On the basis of the conventional exponential approximation rate, a compensation factor and a fractional order are incorporated. Meanwhile, the sigmoid function, characterized by superior smoothness, is employed to replace the sign function that induces severe chattering, thereby attenuating the motor torque ripple. At the same time, in response to the challenge of parameter tuning arising from motor nonlinearity and the abundance of parameters, the sparrow search algorithm (SSA) is employed to optimize the controller parameters. The motor control models before and after the improvement are constructed in MATLAB/Simulink, and the sparrow search algorithm (SSA) is employed to optimize the controller parameters for both cases. Comparative results indicate that the improved control strategy and parameter optimization method can effectively suppress motor torque ripple and enhance the dynamic response characteristics of the system under various operating conditions and rotational speeds. Full article

Brushless DC (BLDC) motors are commonly used in electric vehicles (EVs) because of their efficiency, small size and great torque-speed performance. These motors have a few benefits such as low maintenance, increased reliability and power density. Nevertheless, BLDC motors are highly nonlinear and their dynamics are very complicated, in particular, under changing load and supply conditions. The above features require the design of strong and adaptable control methods that can ensure performance over a broad spectrum of disturbances and uncertainties. In order to overcome these issues, this paper uses a Fractional-Order Proportional-Integral-Derivative (FOPID) controller that offers better control precision, better frequency response, and an extra degree of freedom in tuning by using non-integer order terms. Although it has the benefits, there are three primary drawbacks: (i) it is not real-time adaptable, (ii) it is hard to choose appropriate initial gain values, and (iii) it is sensitive to big disturbances and parameter changes. A new control framework is suggested to address these problems. First, a Reinforcement Learning (RL) approach based on Deep Deterministic Policy Gradient (DDPG) is presented to optimize the FOPID gains online so that the controller can adjust itself continuously to the variations in the system. Second, Snake Optimization (SO) algorithm is used in fine-tuning of the FOPID parameters at the initial stages to guarantee stable convergence. Lastly, cascade control structure is adopted, where FOPID controllers are used in the inner (current) and outer (speed) loops. This construction adds robustness to the system as a whole and minimizes the effect of disturbances on the performance. In addition, the cascade design also allows more coordinated and smooth control actions thus reducing stress on the power electronic switches, which reduces switching losses and the overall efficiency of the drive system. The suggested RL-enhanced cascade FOPID controller is verified by Hardware-in-the-Loop (HIL) testing, which shows better performance in the aspects of speed regulation, robustness, and adaptability to realistic conditions of operation in EV applications. Full article

A class of boundary value problems for fractional differential systems involving variable-order derivatives is considered. Such problems can be transformed into some boundary value problems for nonlinear Caputo fractional differential systems. Here, the relations between linear Caputo fractional differential equations and their corresponding linear integral equations are investigated, and the results demonstrate that a proper Lipschitz-type condition is needed for studying nonlinear Caputo fractional differential equations. Then, an existence and uniqueness result is established in some vector subspaces by Banach’s fixed-point theorem and ${\parallel ·\parallel }_e$ norm. In addition, two examples are presented to illustrate the theoretical conclusions. Full article

This study introduces a novel numerical approach to analyze the axisymmetric bending behavior of functionally graded (FG) graphene platelet (GPL)-reinforced annular sandwich nanoplates featuring a porous core. The nanostructures are exposed to coupled magnetic and hygrothermal environments. The porosity distribution and GPL weight fraction are modeled as nonlinear functions through the thickness, capturing realistic gradation effects. The governing equations are derived using the virtual displacement principle, taking into account the Lorentz force and the interaction with an elastic foundation. To address the size-dependent behavior and thickness-stretching effects, the model employs the nonlocal strain gradient theory (NSGT) integrated with a modified version of Shimpi’s quasi-3D higher-order shear deformation theory (Q3HSDT). The differential quadrature method (DQM) is applied to obtain numerical solutions for the displacement and stress fields. A detailed parametric study is conducted to investigate the influence of various physical and geometric parameters, including the nonlocal parameter, strain gradient length scale, magnetic field strength, thermal effects, foundation stiffness, core thickness, and radius-to-thickness ratio. The findings support the development of smart, lightweight, and thermally adaptive nano-electromechanical systems (NEMS) and provide valuable insights into the mechanical performance of FG-GPL sandwich nanoplates. These findings have potential applications in transducers, nanosensors, and stealth technologies designed for ultrasound and radar detection. Full article

This paper proposes a new behavioral model for radio-frequency power amplifiers (RF PAs) by extending the two-dimensional Polar Volterra series to fractional derivative order, using a numerical Mittag–Leffler-based formulation of fractional orthonormal generating functions. The motivation stems from the increasing need for accurate and computationally efficient models to represent nonlinearities and memory effects in wideband RF PAs, especially in energy-efficient 5G systems. The proposed method significantly reduces model complexity by lowering the number of estimated parameters while maintaining or improving modeling fidelity. To evaluate its performance, three different RF PA devices were used as test cases. The results demonstrated that the proposed approach achieved an over 81.5% reduction in the number of model parameters and improved modeling accuracy. Besides that, in a scenario with the same number of parameters, normalized mean square error (NMSE) gains of up to 8.72 dB were obtained. These findings support the method’s potential for practical use in RF PA behavioral modeling and digital predistortion applications. Full article

Continuous decrease in inertia and sensitivity to load/generation fluctuation are significant challenges for present-day power networks. The primary reason for these issues is the increased penetration capabilities of renewable energy sources. An imbalanced load with significant power output has a substantial impact on the frequency and voltage characteristics of electrical networks. Various load frequency control (LFC) technologies are widely used to address these issues. Existing LFC approaches in the literature are inadequate in addressing system uncertainty, parameter fluctuation, structural changes, and disturbance rejection. As a result, the purpose of this work is to suggest a better LFC approach that makes use of a combination of a one plus tilt fractional filtered derivative (1+TDF^λ) cascaded controller and a fractional order proportional–integral–derivative (PI^λD^μ) controller, which is referred to as the recommended 1+TDF^λ/PI^λD^μ controller. Drawing inspiration from the dynamics of religious societies, including the roles of followers, missionaries, and leaders, and the organization into religious and political schools, this paper proposes a new application of the efficient divine religions algorithm (DRA) to improve the design of the 1+TDF^λ/PI^λD^μ controller. A triple-area test system is constructed to analyze a realistic power system, taking into account certain physical restrictions such as nonlinearities as well as the impact of PV and wind energy integration. The effectiveness of the presented 1+TDF^λ/PI^λD^μ controller is evaluated by comparing their frequency responses to those of other current controllers like PID, FOPID, 2DOF-PID, and 2DOF-TID^μ. The integral time absolute error (ITAE) criterion was employed as the objective function in the optimization process. Comparative simulation studies were conducted using the proposed controller, which was fine-tuned by three recent metaheuristic algorithms: the divine religions algorithm (DRA), the artificial rabbits optimizer (ARO), and the wild horse optimizer (WHO). Among these, the DRA demonstrated superior performance, yielding an ITAE value nearly twice as optimal as those obtained by the ARO and WHO. Notably, the implementation of the advanced 1+TDF^λ/PI^λD^μ controller, optimized via the DRA, significantly minimized the objective function to $0.4704\times {10}^{-4}$. This reflects an approximate enhancement of $99.5\%$ over conventional PID, FOPID, and 2DOF-TID^μ controllers, and a $99\%$ improvement relative to the 2DOF-PID controller. The suggested case study takes into account performance comparisons, system modifications, parameter uncertainties, and variations in load/generation profiles. Through the combination of the suggested 1+TDF^λ/PI^λD^μ controller and DRA optimization capabilities, outcomes demonstrated that frequency stability has been significantly improved. Full article

Automatic control in biomedicine has attracted the attention of clinicians to mitigate the side effects resulting from drug overdoses administered to patients. To provide the most optimal and accurate results, the computer-controlled systems in biomedical engineering require more advanced tuning procedures that tackle patient variability and ensure the robustness of the control system. This has been enhanced over the past two decades through the replacement of standard PID controllers with fractional-order controllers. However, most of the developed fractional-order control methods address only the robustness with respect to gain variations. In this study, a novel fractional-order control algorithm that is robust to time constant variations is developed. The control algorithm is designed for second-order plus dead time systems. A graphical solution is chosen to solve the nonlinear system of equations for the proposed approach. Three biomedical applications are employed as case studies. The first one consists in the control of the bispectral index in general anesthesia, the second one refers to the blood glucose level control for diabetic patients, and finally, the third one tackles computerized control in chemotherapy. The closed-loop simulation results validate the efficiency of the tuning method according to the accepted values of the performance specifications in the scientific literature. Full article

The Wind Farm Layout Optimization Problem (WFLOP) aims to improve wind energy utilization and reduce wake-induced power losses through optimal placement of wind turbines. Genetic Algorithms (GAs) and Particle Swarm Optimization (PSO) have been widely adopted due to their suitability for discrete optimization tasks, yet they suffer from limited global exploration and insufficient convergence depth. Differential evolution (DE), while effective in continuous optimization, lacks adaptability in discrete and nonlinear scenarios such as WFLOP. To address this, the fractional-order differential evolution (FODE) algorithm introduces a memory-based difference mechanism that significantly enhances search diversity and robustness. Building upon FODE, this paper proposes FQFODE, which incorporates reinforcement learning to enable adaptive adjustment of the evolutionary process. Specifically, a Q-learning mechanism is employed to dynamically guide key search behaviors, allowing the algorithm to flexibly balance exploration and exploitation based on problem complexity. Experiments conducted across WFLOP benchmarks involving three turbine quantities and five wind condition settings show that FQFODE outperforms current mainstream GA-, PSO-, and DE-based optimizers in both solution quality and stability. These results demonstrate that embedding reinforcement learning strategies into differential frameworks is an effective approach for solving complex combinatorial optimization problems in renewable energy systems. Full article

Fractional-order calculus (FOC) has gained significant attention in electric vehicle (EV) energy storage and management systems, as it provides enhanced modeling and analysis capabilities compared to traditional integer-order approaches. This review presents a comprehensive survey of recent advancements in the application of FOC to EV energy storage systems, including lithium-ion batteries (LIBs), supercapacitors (SCs), and fuel cells (FCs), as well as their integration within energy management systems (EMS). The review focuses on developments in electrochemical, equivalent circuit, and data-driven models formulated in the fractional-order domain, which improve the representation of nonlinear, memory-dependent, and multi-scale dynamics of energy storage devices. It also discusses the benefits and limitations of current FOC-based models, identifies open challenges such as computational feasibility and parameter identification, and outlines future research directions. Overall, the findings indicate that FOC offers a robust framework with significant potential to advance next-generation EV energy storage and management systems. Full article

Fractional-order models provide a powerful framework for capturing memory-dependent and viscoelastic dynamics in mechanical systems, which are often inadequately represented by classical integer-order characterizations. This study addresses the identification of dynamic parameters in both single-degree-of-freedom (1-DOF) and three-degree-of-freedom (3-DOF) Duffing oscillators with fractional damping, modeled using the Grünwald–Letnikov characterization. The 1-DOF system includes a cubic nonlinear restoring force and is excited by a harmonic input to induce steady-state oscillations. For both systems, time domain simulations are conducted to capture long-term responses, followed by Fourier decomposition to extract steady-state displacement, velocity, and acceleration signals. These components are combined with a GL-based fractional derivative approximation to construct structured regressor matrices. System parameters—including mass, stiffness, damping, and fractional-order effects—are then estimated using pseudoinverse techniques. The identified models are validated through a comparison of reconstructed and original trajectories in the phase space, demonstrating high accuracy in capturing the underlying dynamics. The proposed framework provides a consistent and interpretable approach for frequency domain system identification in fractional-order nonlinear systems, with relevance to applications such as mechanical vibration analysis, structural health monitoring, and smart material modeling. Full article

In the integrated hydro–wind–solar–storage system, the strong output fluctuations of wind and solar power, along with prominent system nonlinearity and time-varying characteristics, make it difficult for traditional PID controllers to achieve high-precision and robust dynamic control. This paper proposes a fuzzy fractional-order PID control strategy based on a multi-objective optimization algorithm, aiming to enhance the system’s frequency regulation, power balance, and disturbance rejection capabilities. The strategy combines the adaptive decision-making ability of fuzzy control with the high-degree-of-freedom tuning features of fractional-order PID. The multi-objective optimization algorithm AGE-MOEA-II is employed to jointly optimize five core parameters of the fuzzy fractional-order PID controller (K_p, K_i, K_d, λ, and μ), balancing multiple objectives such as system dynamic response speed, steady-state accuracy, suppression of wind–solar fluctuations, and hydropower regulation cost. Simulation results show that compared to traditional PID, single fractional-order PID, or fuzzy PID controllers, the proposed method significantly reduces system frequency deviation by 35.6%, decreases power overshoot by 42.1%, and improves renewable energy utilization by 17.3%. This provides an effective and adaptive solution for the stable operation of hydro–wind–solar–storage systems under uncertain and variable conditions. Full article