Search Results (1,401)

This paper advances a comprehensive controllability framework for Prabhakar fractional differential systems ($\mathcal{PFDS}$s) of order $\eta \in (1,2)$ with nonlocal initial conditions, where the second-order setting requires the joint specification of both an initial state and an initial velocity. Explicit solution representations for four structurally distinct classes of second-order Prabhakar systems are derived via the Laplace transform method and Neumann series expansions, revealing that the placement of the forcing term directly in the system or under the Prabhakar fractional integral operator produces fundamentally different convolution kernels. For linear integro-differential systems, necessary and sufficient controllability conditions are established through a Gramian rank criterion with an explicit norm-bounded control law, while for nonlinear systems, sufficient conditions are obtained via the Schauder fixed-point theorem under an asymptotic growth condition. Three numerical examples validate the theory: a three-dimensional linear system and a two-dimensional nonlinear integro-differential system achieve terminal errors of order ${10}^{-12}$ and ${10}^{-7}$, respectively, and a Prabhakar fractional mass–spring–damper system with viscoelastic hereditary damping demonstrates direct physical relevance, with all theoretical conditions verified and a terminal error of $7.42\times {10}^{-5}$ confirming precise rest-position steering by the Gramian-based control law. Full article

In this article, we investigate the existence of solutions for a class of fractional integro-differential equations (FIDE’s) involving the $\psi$-Caputo fractional derivative ($\psi$-CFD) subject to $\psi$-Caputo boundary conditions. The analysis is carried out in an appropriate Banach space setting using the Mönch fixed-point theorem. Furthermore, sufficient conditions ensuring the existence and uniqueness of solutions are derived by employing tools from nonlinear functional analysis. In addition, the obtained results contribute to the current literature by extending existing works on fractional differential equations (FDE’s) involving generalized Caputo-type operators. The novelty of this study lies in the incorporation of $\psi$-CFD’s together with $\psi$-Caputo boundary conditions under the framework of Mönch fixed-point theory. An illustrative example is provided to verify the applicability and effectiveness of the theoretical findings. Full article

In this study, a fractional-order epidemic compartmental model is formulated using the Caputo derivative to account for the memory effects of the chikungunya virus. Based on Banach contractions, fixed-point theorems are used to prove existence and uniqueness, and fundamental properties such as positivity and boundedness are established. Normalized forward sensitivity indices are employed to evaluate the relative impact of model parameters on the transmission dynamics and control of the disease. To reduce the spreading of infection, an optimal control problem is formulated by introducing time-dependent control measures with four control strategies that include public health prevention, treatment enhancement, and vector-control measures. Necessary conditions for optimality are derived using Pontryagin’s Maximum Principle. The predictor–corrector Adams–Bashforth–Moulton scheme is applied across different fractional orders and effectively reduces infection levels. The influence of the fractional order $\xi$ on the epidemic dynamics is investigated, showing that lower values of $\xi$ slow disease progression through a memory effect inherent in the Caputo operator. Moreover, an artificial neural network (ANN) trained via the Levenberg–Marquardt algorithm independently validates the numerical solutions. Full article

In this paper, we investigate the practical exponential stability of a class of nonlinear systems governed by the tempered $\varpi$-Caputo fractional derivative. A new Lyapunov-based criterion is established to derive sufficient conditions ensuring $\varpi$-practical exponential stability. The obtained result is formulated in a general framework involving suitable growth bounds on the Lyapunov function together with a tempered fractional derivative inequality and a boundedness condition on a weighted integral term. The proposed theorem provides an explicit practical exponential estimate for the system trajectories and extends existing stability results that are available for standard fractional and tempered fractional systems. To demonstrate the applicability of the developed theory, two applications are presented. First, the general criterion is applied to a class of perturbed tempered $\varpi$-fractional systems, for which verifiable sufficient conditions are derived in terms of quadratic Lyapunov functions and perturbation bounds. Second, a state-feedback stabilization result is established for a class of nonlinear tempered fractional control systems, showing that the proposed theorem can be used as an effective tool for closed-loop practical exponential stabilization. Finally, numerical examples are provided to validate the theoretical developments and to illustrate the effectiveness of the proposed approach. An additional test case with ${\eta }_3>0$ is included to demonstrate the nontrivial range of Theorem 1. Furthermore, a socio-economic tempered fractional cobweb model is incorporated to show how the proposed criterion applies to price-adjustment dynamics with memory and persistent market perturbations. Full article

In this paper, we consider a reaction–diffusion system governed by incommensurate fractional time derivatives based on the Degn–Harrison model. Its formulation incorporates various memory effects on axial position through Caputo derivatives of variable orders, producing a more realistic modeling of the temporal dynamics. This paper starts with a study of the spatially homogeneous system and establishes conditions for local stability by using the Matignon criterion. The spectral decomposition method under Neumann boundary condition is then applied to study the complete reaction–diffusion system and describe diffusion-induced instabilities. Our results indicate that the noninteger fractional orders lead to significant changes in stability regions, as well as the initiation of pattern formation. Specifically, the orders of fractions induced as a control variable are regarded to be effective in controlling the stability of the system, thus they are global (or positive) control variables when their values achieved at some levels apply to the entire saturation, etc. Our numerical simulations are in excellent agreement with the theoretical predictions and show that memory asymmetry induces complex spatiotemporal dynamics not seen for classical integer-order systems. Full article

Nonlinear equations arise extensively in engineering and applied sciences, where efficient and reliable iterative solvers are required. This study introduces two fractional-order iterative schemes based on a common predictor–corrector structure: a Caputo-based method, NCFS${}_1$, and an Atangana–Baleanu–Caputo (ABC)-based variant, NFS${}_1^{abc}$. The proposed schemes incorporate a fractional order and two tunable parameters to improve flexibility in the iterative process. The local convergence behavior of the Caputo-based method is analyzed by means of fractional Taylor expansions, yielding an explicit error equation and convergence order, while analogous asymptotic considerations are discussed for the ABC-based variant. A dynamical-systems analysis is also performed through basins of attraction, the Convergence Area Index, and the Wada measure. Numerical experiments on application-motivated nonlinear models indicate that the proposed methods can provide faster error reduction, smaller residuals, and lower computational cost than selected existing fractional iterative schemes. These results suggest that the proposed framework is a flexible and effective approach for nonlinear root-finding problems, combining local convergence analysis with global dynamical assessment. 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

To address the key problems of low-frequency oscillation and insufficient regulation accuracy at the Point of Common Coupling (PCC) in hydro–wind–photovoltaic hybrid systems, which are caused by the randomness of wind and photovoltaic output, the water-hammer effect of hydropower units, and multi-source power coupling, a joint control strategy based on Fractional-Order Proportional Integral Derivative (FOPID) and Co-evolutionary Multi-objective Particle Swarm Optimization (CMOPSO) is proposed. First, a small-signal transfer function model of the system covering photovoltaic inverters, doubly fed induction generators (DFIGs), hydropower units and voltage-source converter-based high-voltage direct current (VSC-HVDC) converter stations is established to accurately characterize the water-hammer effect and multi-source dynamic coupling characteristics. Second, a Caputo-type FOPID controller is designed. Compared with traditional integer-order controllers with limited tuning flexibility, the FOPID controller utilizes its five degrees of freedom to address specific multi-source coupling challenges. This precisely compensates for the non-minimum phase lag caused by the water-hammer effect in hydropower units via the fractional derivative link, and effectively smooths the impact of stochastic wind–solar fluctuations on PCC voltage through the memory characteristics of the fractional integral link. This multi-parameter regulation mechanism prevents a trade-off between response speed and overshoot suppression, achieving effective decoupling of complex multi-source dynamic interactions. Third, a dual-objective optimization framework with the Integral of Time-weighted Absolute Error (ITAE) and Oscillatory Disturbance Risk Index (ODRI) as the objectives is constructed. The multi-population co-evolution mechanism of the CMOPSO algorithm is adopted to solve the Pareto-optimal solution set, realizing the coordinated optimization of dynamic response accuracy and oscillation instability risk. Finally, comparative simulations are carried out on the Simulink platform with traditional PI/FOPI controllers and optimization algorithms such as Multi-objective Particle Swarm Optimization based on the Decomposition/Simple Indicator-Based Evolutionary Algorithm (MPSOD/SIBEA). The results show that the proposed strategy can effectively suppress low-frequency oscillations in the range of 0~30 Hz. Compared with the traditional PI controller, the PCC voltage overshoot is reduced by more than 40%, the oscillation decay time is shortened by 33%, the ITAE and ODRI indices are decreased by 12.58% and 2.47%, respectively, and the stability of DC bus voltage is significantly improved. Its robustness and comprehensive control performance are superior to existing methods, providing an efficient and stable control scheme for power electronics-dominated complex new energy grid-connected systems. Full article

This paper investigates a new class of mixed impulsive fractional boundary value problems (BVPs) in which the mixing occurs both in the governing fractional differential equations—through the combined presence of $\psi$-Caputo and quantum (q-difference) fractional derivatives—and in the boundary conditions formulated via fractional integral constraints. By incorporating two distinct operators within the same dynamical framework, the proposed model is capable of capturing both memory effects and discrete-scale behaviors inherent in complex hybrid systems. Using the Banach contraction mapping principle and the Leray–Schauder nonlinear alternative, sufficient conditions ensuring the existence and uniqueness of solutions are established. The theoretical results unify and extend several known fractional models. Owing to its flexible structure, the proposed framework may serve as a useful mathematical tool for modeling impulsive phenomena in systems where non-local memory and scale-transition mechanisms coexist, such as in engineering, physics, and applied sciences. Finally, numerical examples are provided to illustrate the applicability and qualitative behavior of the solutions. Full article

Diabetes mellitus is a chronic disease with complex progression dynamics. This study introduces a fractional order compartmental model based on the Caputo derivative, a singular-kernel derivative, to describe disease progression across four compartments: susceptible, insulin-resistant, diabetic without complications, and diabetic with complications. The model is novel for integrating memory effects into disease-stage transitions while maintaining dimensional consistency. Key mathematical properties, including existence, uniqueness, positivity, boundedness, equilibrium analysis, and both local and global stability, are established. Ulam–Hyers stability is also examined to evaluate the robustness of the model solutions. Numerical approximations are obtained using the Adomian Decomposition Method and its Laplace variant. Simulations indicate that lower fractional orders enhance memory effects, slow disease progression, and influence long-term dynamics. These results demonstrate that the proposed approach provides a flexible and robust framework for studying chronic disease progression and makes a meaningful contribution to the literature on fractional diabetes models. Full article

This paper investigates a class of boundary value problems involving Caputo fractional derivatives of order $\nu \in (2,3]$. We begin by establishing two novel comparison principles. Subsequently, by employing the monotone iterative technique coupled with upper and lower solutions, we demonstrate the existence of extremal solutions for the corresponding fractional differential equations. Finally, an illustrative example is provided to validate our main findings. 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

Human motivation is governed by a long-memory cognitive process in which the depth of temporal integration—how far into the past the system draws upon accumulated experience—is not fixed, but dynamically compressed under cognitive stress. Despite extensive empirical evidence that acute stress impairs working memory and narrows temporal integration in decision-making, no existing mathematical framework has formally coupled the memory depth of the governing operator to a physiologically grounded stress indicator. To address this gap, we propose a stress-adaptive variable-order fractional model for motivational intensity $M\left(t\right)$, in which the Caputo fractional order $\alpha \left(t\right)$ varies inversely with an aggregated stress indicator $\sigma \left(t\right)$ through the Hill-type coupling $\alpha \left(t\right)={\alpha }_{min}+({\alpha }_{max}-{\alpha }_{min})C/(C+\sigma \left(t\right))$, thereby encoding the empirically documented shift from deep integrative to shallow heuristic processing as cognitive load increases. Rather than deriving the model by algebraic manipulation of a differential equation, we formulate it directly as a causally consistent type-III Volterra integral equation, in which the memory kernel is evaluated at the history time s, ensuring that the weight assigned to each past state reflects the memory depth that was physiologically active when that state was experienced. Well-posedness is established rigorously via the Banach fixed-point theorem with explicit contraction constants, uniform boundedness and non-negativity of solutions are derived through the fractional Gronwall inequality, and numerical solutions are computed using an Adams–Bashforth–Moulton predictor–corrector scheme adapted to the variable-order kernel. Five numerical experiments demonstrate that stress-induced variation in $\alpha \left(t\right)$ produces qualitatively richer dynamics compared with the tested constant-order baselines: the proposed model achieves a steeper peak decline rate (0.48 versus 0.19–0.45), a larger burnout gap (3.15 versus 1.92–2.81), and faster recovery to ninety percent of peak motivation (4.2 versus 3.9–7.3 time units), while the empirically observed numerical convergence approaches $O\left(h^2\right)$ for sufficiently small step sizes. The framework offers a principled phenomenological substrate for memory-adaptive cognitive modelling, with direct implications for stress-aware intelligent tutoring systems that are capable of inferring $\alpha \left(t\right)$ in real time from biometric signals such as heart rate variability or galvanic skin response, and adjusting instructional complexity accordingly. Empirical calibration against learning-analytics and psychophysiological datasets, together with stochastic extensions for probabilistic burnout-risk prediction, are identified as immediate priorities for future research. Full article

This study presents an extended cubic B-spline Galerkin scheme for the numerical solution of multi-term time-fractional differential equations. The proposed formulation employs extended cubic B-splines together with the Caputo fractional derivative to model the time-fractional operators. Gauss quadrature is used to accurately evaluate the resulting integral. A stability analysis of the scheme is provided and its accuracy is assessed through $L_2$ and $L_{\infty }$ error norms over different spatial nodes and mesh refinements. The numerical results demonstrate excellent agreement with the exact solutions, as illustrated in the tables and figures. These findings confirm the robustness, efficiency and reliability of the proposed method for solving multi-term time-fractional differential equations. Full article

Thisarticle develops and analyzes a fractional-order Leslie–Gower prey–predator–parasite system incorporating two discrete delays and nonlocal spatial diffusion. The model’s central novelty lies in the simultaneous integration of three biologically realistic features that have not previously been combined: (i) fractional-order memory effects via a Caputo derivative of order $\alpha \in (0,1]$, (ii) two distinct biological delays—an infection transmission delay ${\tau }_1$ and a predator handling delay ${\tau }_2$—and (iii) nonlocal spatial dispersal modeled through fractional Laplacian operators ${(-\Delta )}^{\gamma /2}$. This triple integration enables the model to capture long-range temporal memory, delayed biological responses, and nonlocal spatial interactions simultaneously, offering insights into dynamics that are challenging to capture with classical integer-order or single-delay formulations. The fractional Laplacian generalizes classical diffusion by allowing long-range dispersal events (Lévy flights), where individuals can occasionally move over large distances with heavy-tailed step-size distributions—a phenomenon observed in many animal movement patterns but absent from standard diffusion models. We provide rigorous proofs of solution existence, uniqueness, non-negativity, and boundedness in both temporal and spatiotemporal settings. Local asymptotic stability conditions are derived for all feasible equilibrium states via characteristic equation analysis. The coexistence equilibrium undergoes a Hopf bifurcation when either delay crosses a critical threshold, with fractional order $\alpha$ modulating the bifurcation point and post-bifurcation oscillation frequency. A Lyapunov functional demonstrates global asymptotic stability of the infection-free equilibrium under biologically interpretable conditions. Turing instability analysis reveals conditions for spontaneous pattern formation, with the fractional exponent $\gamma$ controlling pattern wavelength and correlation length. Numerical simulations validate theoretical predictions, including spatial patterns, traveling waves, and chaos. To bridge theory with potential applications, we outline a statistical framework for parameter estimation and uncertainty quantification, suggesting that $\beta$, $\alpha$, and ${\tau }_1$ may be priority targets for parameter estimation. Full article