Search Results (14)

Fractional optimal control problems provide an effective mathematical framework for modeling dynamical systems with memory, hereditary behavior, and anomalous diffusion effects. However, the nonlocal nature of Caputo fractional operators and the reduced regularity of fractional solutions pose significant challenges for the development of accurate and efficient computational methods. In this paper, we develop a spectral-fractional Physics-Informed Neural Network (Spectral-fPINN) framework for solving fractional optimal control problems governed by Caputo fractional differential equations. The proposed methodology combines normalized shifted Legendre spectral approximations, fractional operational matrix formulations, and physics-informed optimization within a unified computational framework. Unlike conventional PINN and fPINN approaches, which directly approximate the unknown solution variables, the proposed framework predicts the spectral coefficient vectors associated with the shifted Legendre basis functions, yielding a low-dimensional global representation with improved approximation efficiency. Caputo fractional derivatives are evaluated through spectral operational matrices, while the resulting optimization problem is discretized using Gauss–Legendre quadrature and solved through gradient-based optimization. In addition, a theoretical analysis of the proposed Spectral-fPINN framework is presented, including approximation, consistency, stability, and convergence results, together with error estimates and residual control properties. Several benchmark linear and nonlinear fractional optimal control problems are investigated to validate the proposed methodology. The numerical results demonstrate excellent agreement with exact solutions, very small residual errors, and rapid spectral coefficient decay, confirming the high-order accuracy and robustness of the proposed approach. Overall, the proposed Spectral-fPINN framework provides an accurate, stable, and computationally efficient methodology for solving a broad class of fractional optimal control problems. Full article

This research paper demonstrates how to manage Caputo fractional neutral integro-differential equations which include both integral and nonlinear elements through a unified framework that models dynamic systems with memory-based dynamics. The research establishes sufficient conditions for controllability through fixed point theory in a Banach space framework which requires particular assumptions while the study focuses on the ${\mathfrak{K}}_1<1$ condition which leads to the existence of a controllable solution. The proposed criteria are demonstrated through a numerical example which tests the theoretical results. The real-world case study uses artificial neural network (ANN) technology to predict Litecoin prices through the application of the fractional controllability model which analyzes historical financial data. The hybrid framework enables precise forecasting of nonlinear time series because it combines fractional calculus mathematical principles with ANN learning abilities. The proposed method demonstrates its predictive efficiency. The method shows robust performance through experimental results using cross-validation and performance metrics. The proposed model demonstrates competitive performance while providing additional advantages such as incorporation of memory effects and theoretical controllability. The research establishes a novel connection between fractional dynamical systems and machine learning which serves as an essential tool for studying complicated systems in theoretical research and practical applications. Full article

This work analyses the existence, uniqueness, and Ulam-type stability of neutral fractional functional differential equations with recursively defined state-dependent delays. Employing the Caputo fractional derivative of order $\alpha \in (0,1)$ with respect to a strictly increasing function $\phi$, the analysis extends classical results to nonuniform memory. The neutral term and delay chain are defined recursively by the solution, with arbitrary continuous initial data. Existence and uniqueness of solutions are established using Krasnosel’skii’s fixed point theorem. Sufficient conditions for Ulam–Hyers stability are obtained via the Volterra-type integral form and a $\phi$-fractional Grönwall inequality. Examples illustrate both standard and nonlinear time scales, including a Hopfield neural network with iterated delays, which has not been previously studied even for integer-order equations. Fractional neural networks with iterated state-dependent delays provide a new and effective model for the description of AI processes—particularly machine learning and pattern recognition—as well as for modelling the functioning of the human brain. Full article

This article considers a class of Caputo fractional-order fuzzy cellular neural networks (CFOFCNNs) with transmission delays and uncertain perturbations. In particular, nonlinear activations and fuzzy operators AND and OR are investigated in the drive-response neural networks (NNs). To achieve practical finite-time (PFT) synchronization and finite-time (FT) synchronization of the studied systems, we design new nonlinear controllers including four feedback terms in this paper, and each carries a different role in the control process. Integrating different comparison principles and nonlinear feedback schemes, straightforward synchronization criteria of the CFOFCNNs are derived. Unlike existing works, a significant finding is that adjusting the feedback coefficients and parameters can enable synchronization switching. Namely, changing one of the feedback terms from positive to negative can cause PFT synchronization to switch to FT synchronization via adjusted control parameters, making our control methods applicable to different scenarios. The settling time depends explicitly on feedback coefficients, initial conditions, and fractional order. Full article

Neural networks, mimicking the structural and functional aspects of the human brain, have found widespread applications in diverse fields such as pattern recognition, control systems, and information processing. A critical phenomenon in these systems is synchronization, where multiple neurons or neural networks harmonize their dynamic behaviors to a common rhythm, contributing significantly to their efficient operation. However, the inherent complexity and nonlinearity of neural networks pose significant challenges in understanding and controlling this synchronization process. In this paper, we focus on the synchronization of a class of fractional-order, delayed, and non-autonomous neural networks. Fractional-order dynamics, characterized by their ability to capture memory effects and non-local interactions, introduce additional layers of complexity to the synchronization problem. Time delays, which are ubiquitous in real-world systems, further complicate the analysis by introducing temporal asynchrony among the neurons. To address these challenges, we propose a straightforward yet powerful global synchronization framework. Our approach leverages novel state feedback control to derive an analytical formula for the synchronization controller. This controller is designed to adjust the states of the neural networks in such a way that they converge to a common trajectory, achieving synchronization. To establish the asymptotic stability of the error system, which measures the deviation between the states of the neural networks, we construct a Lyapunov function. This function provides a scalar measure of the system’s energy, and by showing that this measure decreases over time, we demonstrate the stability of the synchronized state. Our analysis yields sufficient conditions that guarantee global synchronization in fractional-order neural networks with time delays and Caputo derivatives. These conditions provide a clear roadmap for designing neural networks that exhibit robust and stable synchronization properties. To validate our theoretical findings, we present numerical simulations that demonstrate the effectiveness of our proposed approach. The simulations show that, under the derived conditions, the neural networks successfully synchronize, confirming the practical applicability of our framework. Full article

This study delves into the synchronization issues of the impulsive fractional-order, mainly the Caputo derivative of the order between 0 and 1, bidirectional associative memory (BAM) neural networks incorporating the diffusion term at a fixed time (FXT) and a predefined time (PDT). Initially, this study presents certain characteristics of fractional-order calculus and several lemmas pertaining to the stability of general impulsive nonlinear systems, specifically focusing on FXT and PDT stability. Subsequently, we utilize a novel controller and Lyapunov functions to establish new sufficient criteria for achieving FXT and PDT synchronizations. Finally, a numerical simulation is presented to ascertain the theoretical dependency. Full article

Fractional calculus plays a pivotal role in modern scientific and engineering disciplines, providing more accurate solutions for complex fluid dynamics phenomena due to its non-locality and inherent memory characteristics. In this study, Caputo’s time fractional derivative operator approach is employed for heat and mass transfer modeling in unsteady Maxwell fluid within a cylinder. Governing equations within a cylinder involve a system of coupled, nonlinear fractional partial differential equations (PDEs). A machine learning technique based on the Levenberg–Marquardt scheme with a backpropagation neural network (LMS-BPNN) is employed to evaluate the predicted solution of governing flow equations up to the required level of accuracy. The numerical data sheet is obtained using series solution approach Homotopy perturbation methods. The data sheet is divided into three portions i.e., $80\%$ is used for training, $10\%$ for validation, and $10\%$ for testing. The mean-squared error (MSE), error histograms, correlation coefficient (R), and function fitting are computed to examine the effectiveness and consistency of the proposed machine learning technique i.e., LMS-BPNN. Moreover, additional error metrics, such as R-squared, residual plots, and confidence intervals, are incorporated to provide a more comprehensive evaluation of model accuracy. The comparison of predicted solutions with LMS-BPNN and an approximate series solution are compared and the goodness of fit is found. The momentum boundary layer became higher and higher as there was an enhancement in the value of Caputo, fractional order α = 0.5 to α = 0.9. Higher thermal boundary layer (TBL) profiles were observed with the rising value of the heat source. Full article

Unlike existing memristive neural networks or fuzzy neural networks, this article investigates a class of Caputo fractional-order uncertain memristive neural networks (CFUMNNs) with fuzzy operators and transmission delay to realistically model complex environments. Especially, the fuzzy symbol AND and the fuzzy symbol OR as well as nonlinear activation behaviors are all concerned in the generalized master-slave networks. Based on the characteristics of the neural networks being studied, we have designed distinctive information feedback control protocols including three different functional sub-modules. Combining comparative theorems, inequality techniques, and stability theory, novel delay-independent conditions can be derived to ensure the finite-time synchronization (FTS) of fuzzy CFUMNNs. Besides, the upper bound of the settling time can be effectively evaluated based on feedback coefficients and control parameters, which makes the achievements of this study more practical for engineering applications such as signal encryption and secure communications. Ultimately, simulation experiments show the feasibility of the derived results. Full article

To construct Caputo fractional-order time-delayed cellar neural networks (FOTDCNNs) that characterize real environments, this article introduces partial uncertainties, fuzzy operators, and nonlinear activation functions into the network models. Specifically, both the fuzzy AND operator and the fuzzy OR operator are contemplated in the master–slave systems. In response to the properties of the considered cellar neural networks (NNs), this article designs a new class of mixed control protocols that utilize both the error feedback information of systems and the sampling information of impulse moments to achieve network synchronization tasks. This approach overcomes the interference of time delays and uncertainties on network stability. By integrating the fractional-order comparison principle, fractional-order stability theory, and hybrid control schemes, readily verifiable asymptotic synchronization conditions for the studied fuzzy cellar NNs are established, and the range of system parameters is determined. Unlike previous results, the impulse gain spectrum considered in this study is no longer confined to a local interval (−2, 0) and can be extended to almost the entire real number domain. This spectrum extension relaxes the synchronization conditions, ensuring a broader applicability of the proposed control schemes. Full article

This paper investigates the long-time behavior of fractional-order complex memristive neural networks in order to analyze the synchronization of both anatomical and functional brain networks, for predicting therapy response, and ensuring safe diagnostic and treatments of neurological disorder (such as epilepsy, Alzheimer’s disease, or Parkinson’s disease). A new mathematical brain connectivity model, taking into account the memory characteristics of neurons and their past history, the heterogeneity of brain tissue, and the local anisotropy of cell diffusion, is proposed. This developed model, which depends on topology, interactions, and local dynamics, is a set of coupled nonlinear Caputo fractional reaction–diffusion equations, in the shape of a fractional-order ODE coupled with a set of time fractional-order PDEs, interacting via an asymmetric complex network. In order to introduce into the model the connection structure between neurons (or brain regions), the graph theory, in which the discrete Laplacian matrix of the communication graph plays a fundamental role, is considered. The existence of an absorbing set in state spaces for system is discussed, and then the dissipative dynamics result, with absorbing sets, is proved. Finally, some Mittag–Leffler synchronization results are established for this complex memristive neural network under certain threshold values of coupling forces, memristive weight coefficients, and diffusion coefficients. Full article

Finding all the roots of a nonlinear equation is an important and difficult task that arises naturally in numerous scientific and engineering applications. Sequential iterative algorithms frequently use a deflating strategy to compute all the roots of the nonlinear equation, as rounding errors have the potential to produce inaccurate results. On the other hand, simultaneous iterative parallel techniques require an accurate initial estimation of the roots to converge effectively. In this paper, we propose a new class of global neural network-based root-finding algorithms for locating real and complex polynomial roots, which exploits the ability of machine learning techniques to learn from data and make accurate predictions. The approximations computed by the neural network are used to initialize two efficient fractional Caputo-inverse simultaneous algorithms of convergence orders $\varsigma +2$ and $2\varsigma +4$, respectively. The results of our numerical experiments on selected engineering applications show that the new inverse parallel fractional schemes have the potential to outperform other state-of-the-art nonlinear root-finding methods in terms of both accuracy and elapsed solution time. Full article

This paper proposes an approximation algorithm based on the Legendre and Chebyshev artificial neural network to explore the approximate solution of fractional Lienard and Duffing equations with a Caputo fractional derivative. These equations show the oscillating circuit and generalize the spring–mass device equation. The proposed approach transforms the given nonlinear fractional differential equation (FDE) into an unconstrained minimization problem. The simulated annealing (SA) algorithm minimizes the mean square error. The proposed techniques examine various non-integer order problems to verify the theoretical results. The numerical results show that the proposed approach yields better results than existing methods. Full article

The current study presents the numerical solutions of a fractional order monkeypox virus model. The fractional order derivatives in the sense of Caputo are applied to achieve more realistic results for the nonlinear model. The dynamics of the monkeypox virus model are categorized into eight classes, namely susceptible human, exposed human, infectious human, clinically ill human, recovered human, susceptible rodent, exposed rodent and infected rodent. Three different fractional order cases have been presented for the numerical solutions of the mathematical monkeypox virus model by applying the stochastic computing performances through the artificial intelligence-based scaled conjugate gradient neural networks. The statics for the system were selected as 83%, 10% and 7% for training, testing and validation, respectively. The exactness of the stochastic procedure is presented through the performances of the obtained results and the reference Adams results. The rationality and constancy are presented through the stochastic solutions together with simulations based on the state transition measures, regression, error histogram performances and correlation. Full article

One of the main properties of solutions of nonlinear Caputo fractional neural networks is stability and often the direct Lyapunov method is used to study stability properties (usually these Lyapunov functions do not depend on the time variable). In connection with the Lyapunov fractional method we present a brief overview of the most popular fractional order derivatives of Lyapunov functions among Caputo fractional delay differential equations. These derivatives are applied to various types of neural networks with variable coefficients and time-varying delays. We show that quadratic Lyapunov functions and their Caputo fractional derivatives are not applicable in some cases when one studies stability properties. Some sufficient conditions for stability of equilibrium of nonlinear Caputo fractional neural networks with time dependent transmission delays, time varying self-regulating parameters of all units and time varying functions of the connection between two neurons in the network are obtained. The cases of time varying Lipschitz coefficients as well as nonLipschitz activation functions are studied. We illustrate our theory on particular nonlinear Caputo fractional neural networks. Full article