Search Results (139)

We develop a residual-adaptive preconditioned quantum pseudo-spectral method for generalised $\psi$-Caputo initial-value problems containing a discrete delay, weakly singular hereditary memory, and nonlinear reaction terms. A $\psi$-fractional Chebyshev basis yields closed-form operational matrices that are exact on the chosen finite spectral space. To make the hereditary term compatible with block encoding, the power-law kernel is approximated by a sum of exponentials and supplemented by an explicit local near-field correction, converting global memory into finitely many local auxiliary modes. A structure-preserving preconditioner controls the condition number, while a residual-adaptive multidomain strategy and damped Newton iteration treat layers and nonlinearities. We prove well-posedness in Mittag–Leffler weighted graph spaces, derive a combined spectral–kernel–residual error estimate, and state the quantum linear-system complexity with explicit block-encoding normalisations and right-hand-side preparation assumptions. Numerical tests show high accuracy for solutions smooth in the $\psi$-coordinate, improved robustness for singular and layered solutions, substantial condition-number reduction, and lower history cost under sum-of-exponentials compression. To evaluate performance, we compare against L1 product integration and Jacobi collocation, systematically quantifying their respective accuracy, computational cost, and conditioning characteristics. Full article

This paper considers a class of fractional stochastic differential systems driven jointly by the Rosenblatt process and a compensated Poisson random measure. The Rosenblatt process captures non-Gaussian fluctuations with memory effects; the Poisson jumps model sudden random shocks. By employing the mild solution formulation associated with fractional resolvent operators, we establish the existence of solutions via Krasnoselskii’s fixed point theorem (KFPT) and prove uniqueness through the Banach contraction under suitable Lipschitz conditions on the drift, diffusion, and jump coefficients together with contraction assumptions. Furthermore, Ulam–Hyers stability is established, ensuring that approximate solutions remain close to exact solutions in the mean-square sense. We also establish Mittag–Leffler-type continuous dependence on initial data, demonstrating that solutions depend continuously on their initial histories in the mean-square sense. In addition, open-loop approximate trajectory realization is established through the construction of an explicit open-loop control law that steers the stochastic system along any prescribed admissible trajectory under suitable invertibility and regularity assumptions. An example validates the theoretical results, demonstrating fractional memory, Rosenblatt noise, and Poisson jumps. Full article

The current paper deals with the containment control issue of fractional-order complex networks (FCNs) under communication uncertainties occurring with both multiplicative and additive noises and denial-of-service attacks. The dynamics of the followers are incorporated through the use of Caputo fractional derivatives, which are able to capture the nature of memory and hereditary dynamics of the complex systems. In order to reduce stochastic noise caused by the noisy communication medium, a new distributed containment protocol is proposed that takes both the multiplicative and additive noise effects in the interactions between the leader and the followers. Using the Mittag–Leffler stability theory, stochastic Lyapunov analysis, Itô calculus, the derivation of necessary conditions to ensure that the followers converge to the convex hull of the leaders was done. The explicit stability conditions are stipulated based on system parameters and control gains as well as intensities of noise. In addition, the robustness of the protocol suggested for use against intermittent DoS attacks is critically examined. The theoretical findings are substantiated by simulation experiments that demonstrate that the suggested methodology guarantees containment and resilience to the fluctuations in the fractional order and communication breakdowns. The findings offer an inclusive framework in the development of robust distributed controllers of fractional-order MASs operating under adversarial and uncertain networked frameworks. Full article

In this paper, we investigate the stability analysis for a class of novel variable-order Caputo fractional-order dual switching systems. First, the short memory principle is adopted to construct the studied system model, where the Caputo fractional order is randomly time-varying, and the outer deterministic switching signal governs the overall dwell-time scheduling of subsystems. Under the designed event-triggered deterministic switching strategy, each fractional-order subsystem is characterized by an internal Markov random jumping processing. Secondly, combining the multiple Lyapunov functions method, fractional-order comparison lemma and average dwell time (ADT) technique, the corresponding sufficient stability criteria are established to guarantee the globally asymptotic stability almost surely (GAS a.s.) and the global Mittag–Leffler stability almost surely (GMLS a.s.). Finally, a numerical simulation example is presented to verify the feasibility and effectiveness of the derived theoretical results. Full article

In this paper, the issue of robust sensor fault estimation for Takagi–Sugeno (T–S) fuzzy systems with Caputo–Hadamard fractional-order dynamics subject to monotone nonlinearities is addressed. An adaptive observer is designed for the joint estimation of the system state and a globally constant sensor bias fault. The Caputo–Hadamard operator is used to handle logarithmic memory effects, and the T–S fuzzy representation is used for multi-regime nonlinear dynamics through a convex interpolation structure. Sufficient linear matrix inequality (LMI) conditions are obtained to ensure generalized Mittag–Leffler stability of the augmented estimation error system under a constant-fault assumption, by combining a sector inequality for strongly monotone nonlinearities with a fractional Lyapunov approach. The stability conditions are directly posed in the decision variables and the observer gains are recovered through a standard change of variables. To broaden the engineering applicability of the result, a finite-horizon practical Mittag–Leffler stability theorem is also derived for absolutely-continuous time-varying sensor faults whose Caputo–Hadamard derivative is bounded on the operating horizon $[t_0,T]$, in which the augmented estimation error remains in a residual ball whose radius is proportional to that bound. An alternative design, called a polynomial gain-scheduled observer, is also developed to reduce the conservatism of the constant-gain design, with observer gains given as polynomials of a measurable, fault-free scheduling vector. Quantitative root-mean-square performance metrics, LMI feasibility margins and an adaptation-gain sensitivity study are reported, and the polynomial matrix inequality is certified both by a dense grid check and by a sum-of-squares (SOS) feasibility argument so that the polynomial design is supported by a constructive certificate over the admissible scheduling set. Three numerical scenarios with fractional order $0.8$ are provided: a strict constant-bias scenario that exactly validates the LMI theorem, a bounded-derivative ramp scenario that validates the practical Mittag–Leffler theorem, and a polynomial gain-scheduled scenario that validates the polynomial observer. Full article

This paper develops a direct analytical framework for synchronizing and controlling fractional-order octonion-valued fuzzy bidirectional associative memory neural networks (FOOVFBAMNNs). Octonion algebra is neither commutative nor associative, which limits the application of standard analytical tools. To address this challenge, we first propose a generalized Cauchy–Schwarz inequality tailored to the octonionic domain, which operates directly without relying on system decomposition. This inequality lays the groundwork for a Lyapunov-based stability analysis that retains the system’s inherent geometric structure to avoid decomposition into real-valued components. Based on this framework, we derive concise 2-norm inequality criteria, which are sufficient to guarantee Mittag-Leffler synchronization of the proposed model. We also employ a Particle Swarm Optimization (PSO) algorithm to systematically optimize the flexible parameters in the generalized inequality, enhancing the practical performance of the synchronization scheme. To validate the effectiveness of the proposed method, we apply it to a multi-domain image restoration task. Numerical experiments verify the performance of our method. In terms of Peak Signal-to-Noise Ratio (PSNR), the octonion-valued network with PSO-tuned parameters achieves better results than its non-optimized counterpart as well as models constructed in complex or quaternion domains. Full article

This study investigates the Mittag–Leffler-type stability properties of fractional perturbed systems with respect to their unperturbed counterparts by incorporating initial time differences into the analysis. In contrast to many existing studies in which initial time effects are neglected, the proposed framework explicitly considers time shifts together with the memory-dependent nature of fractional-order systems. Using Caputo fractional derivatives and Lyapunov-type functionals, new sufficient conditions are established for the stability behavior of perturbed systems relative to the corresponding unperturbed systems under shifted initial times. The obtained results extend existing stability criteria by simultaneously addressing fractional memory effects, perturbation terms, and variations in the initial time. To illustrate the applicability and effectiveness of the theoretical findings, representative examples, numerical simulations, graphical comparisons, and global error analyses are presented. The numerical part is based on the Caputo framework and is further supported by benchmark comparisons involving Riemann–Liouville and shifted Grünwald–Letnikov approaches. The proposed results provide a useful framework for the stability analysis of memory-dependent dynamical systems arising in engineering and applied sciences. Full article

In this paper, we present an applied numerical study inspired by recent theoretical advances on boundary feedback control for fractional coupled PDE–ODE systems. While earlier works have mainly focused on proving the existence, uniqueness, and stability of solutions within the fractional Lyapunov framework, our contribution lies in translating these theoretical results into a practical setting for graph neural networks (GNNs). In this model, the partial differential equation describes the diffusion of information signals across the network topology, while the fractional-order ordinary differential equation captures the nonlinear and memory-dependent update rules of the node states. By employing the backstepping algorithm, we design a boundary controller that guarantees Mittag–Leffler stability of the coupled system. Numerical simulations demonstrate that, in the absence of control, the network dynamics exhibit instability and divergence, whereas the proposed boundary control gradually stabilizes the information propagation process. These results underline the effectiveness of the method and its potential relevance for the fractional modeling and regulation of graph-based neural architectures. An example is given with a consensus problem that shows that the boundary controller stabilizes the network dynamics. Full article

This paper introduces convex contractions of order two in complete suprametric spaces and establishes a conditional fixed point theorem for such mappings. The suprametric setting produces a nonlinear Picard recurrence with a quadratic term, requiring explicit orbit-smallness and diameter conditions to ensure convergence. Under these hypotheses, we prove the existence and uniqueness of a fixed point and the geometric convergence of the Picard sequence, recovering Istrăţescu’s classical theorem when the suprametric parameter is zero. Examples are provided to illustrate both the role and applicability of the conditions. The result is further applied to fractional Volterra–Fredholm integro-differential equations and fractional discrete-time neural networks, yielding existence, uniqueness, iterative convergence, and Mittag-Leffler stability of solutions. Full article

Fractional differential equations are highly beneficial in economics because they can be used to analyze nonlinear systems with memory effects. This research investigates a group of nonlinear fractional Riccati equations that show up in models of inventory and economic growth. The present work is a combined semi-analytical method for finding deterministic series solutions in the Caputo sense: the Adomian Decomposition–Sumudu Transform Method. Limited studies have examined its usage in memory-affected economic models. This method is effective with nonlinearities due to its ability to operate without the necessity of linearizing or discretizing them. The Mittag-Leffler function is employed to demonstrate that the series converges in a strict manner if it converges in a manner that is both absolute and uniform when the conditions are met. Finally, a Lyapunov stability study is conducted to ensure that the solution can accommodate modifications to the original data. Numerical models with different fractional orders show that the behavior of the system is controlled by the fractional parameter. When the fractional order is small, memory effects are increasing. As the order approaches closer to one, the solutions start to act like classical ones. These results indicate that the current methodology can be used for practical applications such as short-term currency exchange rates and volatility in financial markets. Full article

This paper develops an accountability-aware fractional control framework for embodied intelligent systems in shared human environments. The approach combines a Caputo fractional-order stabilizing law, an intent-evidence realization with softmax belief reconstruction, and a conditional proxemic safety layer. Sufficient conditions are established for local Mittag-Leffler stability of the augmented error dynamics and forward invariance of the safe set. Numerical results are presented as a theorem-validation benchmark. For the base case with $\alpha =0.9$, the augmented error norm decays from $1.2359$ to $9.90\times {10}^{-3}$ while the safety margin remains strictly positive, and the robustness condition is satisfied with a margin of $1.8641$. An $\alpha$-sweep and a step-size convergence study further show that the fractional order induces a systematic safety–performance trade-off and that the reported behaviors are numerically stable. Additional simulations with four intent classes, bounded observation noise, and Monte Carlo uncertainty stress tests are included to strengthen the numerical evidence beyond the two-intent theorem-validation case. The manuscript also clarifies the quantitative interpretation of the accountability index, the conditional nature of the safety theorem, and an implementable sampled safety-filter realization for concrete robotic platforms. The results support the proposed framework as a mathematically consistent tool for shaping the balance between regulation and proxemic safety. Full article

This article employs a fractional differential equation model to probe the dynamic mechanism of animal avoidance learning and memory retention. This model encompasses both linear and nonlinear scenarios. We first obtain the series-type analytical solution for the linear scenario and its absolute uniform convergence by Laplace transform and Mittag–Leffler function. Secondly, we establish the existence, uniqueness and Ulam–Hyers stability for the nonlinear scenario via the fixed point theorem and analytical techniques. Eventually, some examples and numerical simulations are provided to examine the effectiveness and availability of the main findings. Full article

This paper introduces a novel memristor-based hyperchaotic system in which the integer-order derivatives are replaced by a variable-order fractal-fractional operator. The dynamical properties of the system, including equilibrium points, Lyapunov exponents, bifurcation diagrams with respect to the variable orders, and the Kaplan–Yorke dimension, are analyzed. A synchronization scheme based on active control is designed for the master–slave configuration, and global Mittag–Leffler stability of the error dynamics is established using a suitable variable-order Lyapunov function. The synchronized states are then applied to an image encryption algorithm. Numerical simulations, security analyses, and NIST randomness tests demonstrate the effectiveness and enhanced performance of the proposed framework compared to existing fixed-order and classical fractional-order methods. Full article

We propose a dimensionally consistent fractional spatio-temporal PDE framework for modelling immune-mediated demyelination in multiple sclerosis (MS). The system couples effector and regulatory T cells, M1/M2 macrophage polarisation, pro- and anti-inflammatory cytokines, oligodendrocyte dynamics, and time-dependent therapeutic controls within a unified distributed-parameter structure. In contrast to ad hoc replacements of integerorder derivatives by Caputo fractional derivatives, the fractional extension proposed here is derived from an underlying continuous-time random walk (CTRW) process with Mittag–Leffler-distributed residence times. This stochastic derivation yields a governing system in which a single commensurate fractional order $\alpha \in (0,1]$, together with a characteristic memory timescale ${\tau }_0$, ensures dimensional consistency and mass balance across all coupled components. The model is formulated as a system of nonlinear reaction–diffusion equations with cross-regulatory and multiplicative interaction terms governing immune amplification, cytokine feedback, and the demyelination–remyelination balance. Analytical interpretation shows how non-Markovian residence times induce Mittag–Leffler-type relaxation and thereby modify effective growth, decay, and stability properties. Numerical simulations compare classical and fractional dynamics, revealing that memory-driven kinetics prolong effector T-cell and M1-macrophage activity, attenuate reparative M2 and oligodendrocyte responses, and extend the effective action of bang–bang therapy inputs representing IFN-$\beta$ and glatiramer acetate beyond their dosing windows. The results indicate that integer-order models may underestimate chronic inflammatory persistence and demyelination severity, while providing a mathematically and physically well-posed platform for memory-aware immune modelling and therapy evaluation in MS. Full article

This paper presents a unified analytical and computational framework for the study of typhoid fever transmission dynamics governed by a Caputo fractional-order compartmental model of order $\kappa \in (0,1]$. The population is stratified into five epidemiological classes, namely susceptible $\left(\mathcal{S}\right)$, asymptomatic $\left(\mathcal{A}\right)$, symptomatic $\left(\mathcal{I}\right)$, hospitalised $\left(\mathcal{H}\right)$, and recovered $\left(\mathcal{R}\right)$, and the governing system explicitly incorporates asymptomatic transmission, treatment dynamics, and temporary immunity with waning. The use of the Caputo fractional derivative is motivated by the well-documented existence of chronic asymptomatic Salmonella Typhi carriers, whose heavy-tailed sojourn times in the carrier state are naturally encoded by the Mittag–Leffler waiting-time distribution arising from the fractional operator. A complete qualitative analysis of the fractional system is carried out: the basic reproduction number ${\mathcal{R}}_0$ is derived via the next-generation matrix method; local and global asymptotic stability of both the disease-free equilibrium ${\mathcal{E}}_0$ (when ${\mathcal{R}}_0\le 1$) and the endemic equilibrium ${\mathcal{E}}^{*}$ (when ${\mathcal{R}}_0>1$) are established using fractional Lyapunov theory and the LaSalle invariance principle; and the normalised sensitivity indices of ${\mathcal{R}}_0$ are computed to identify transmission-amplifying and transmission-suppressing parameters. Existence, uniqueness, and Ulam–Hyers stability of solutions are established via Banach and Leray–Schauder fixed-point arguments. To complement the analytical results, a fractional physics-informed neural network ($\mathcal{PINN}$) framework is developed to simultaneously reconstruct compartmental trajectories and identify unknown biological parameters from sparse synthetic observations. $\mathcal{PINN}$ embeds the L1-Caputo discretisation directly into the training residuals and employs a four-stage Adam–L-BFGS optimisation strategy to recover five trainable parameters $\mathbf{\Theta }$ = $\{\varphi ,\mu ,\sigma ,\psi ,\beta \}$ across three fractional orders $\kappa \in \{1.0,0.95,0.9\}$. The estimated parameters show strong agreement with the true values at the classical limit $\kappa =1.0$ ($\mathrm{MAPE}=2.27\%$), with the natural mortality rate $\mu$ recovered with $\mathrm{APE}\le 0.51\%$ and the transmission rate $\beta$ with $\mathrm{APE}\le 3.63\%$ across all fractional orders, confirming the structural identifiability of the model. Pairwise correlation analysis of the learned parameters establishes the absence of equifinality, validating that $\beta$ can be reliably included in the trainable set. Noise robustness experiments under Gaussian perturbations of $1\%$, $3\%$, and $5\%$ demonstrate graceful degradation ($\mathrm{MAPE}$: $0.82\%\to 3.10\%\to 7.31\%$), confirming the reliability of the proposed framework under realistic observational conditions. Full article