Search Results (74)

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

Accurate and trustworthy prediction of lithium-ion battery aging remains challenging due to multi-mechanistic degradation, cell-to-cell variability, and distribution shift between laboratory calibration and deployment. Fractional-order models have been proposed to capture long-memory effects in electrochemical systems; however, it remains unclear when such memory is empirically identifiable and beneficial within the common prognostics abstraction of state-of-health (SOH) versus cycle index. This work develops a fully reproducible computational pipeline for mechanistic battery aging based on a Caputo fractional differential equation (FDE) and evaluates its cross-cell generalization on open NASA cycling data. Parameters are identified using bounded robust nonlinear least squares and validated under a strict transfer protocol: calibration on cells B0005/B0006 and evaluation on held-out cells B0007/B0018 without refitting. The fractional model is benchmarked against a classical ODE surrogate, an ECM-inspired resistance-proxy baseline, and one-step-ahead machine-learning predictors. Uncertainty quantification is performed via parameter bootstrap and subsequently calibrated using conformal correction to target nominal coverage under transfer. Results show that the fractional order tends to collapse toward the integer-order limit (α → 1) in this dataset, indicating limited evidence of additional long-memory at the SOH-versus-cycle level under the considered protocol, while robust identification remains essential for stability. Calibrated prediction intervals achieve near-nominal coverage on held-out cells, highlighting the importance of UQ calibration under cell-to-cell shift. The proposed scripts and environment specifications enable direct replication and facilitate future extensions to stress-aware fractional models and hybrid physics–ML approaches. Full article

This work investigates a sophisticated class of neutral fuzzy fractional functional differential equations (N3FDEs), where the fractional order $\alpha$ satisfies $0<\alpha \le 1$. We present a comprehensive analysis of the existence, uniqueness, and well-posedness of solutions under the generalized Hukuhara framework. First, we examine the existence and uniqueness of solutions under the generalized Hukuhara framework, providing an refined iterative formula for linear systems. We further verify the system’s well-posedness, proving that solutions remain stable and respond continuously to changes in initial data and parameters. Second, we introduce a novel spectral Vieta–Lucas projection method to approximate the solution. By leveraging the unique properties of Vieta–Lucas polynomials, we transform complex memory-dependent fuzzy equations into a streamlined algebraic system. Finally, numerical examples and error analysis show the method is accurate and efficient. Full article

Accurate State of Charge (SOC) estimation is critical for the safety and efficiency of Battery Management Systems (BMS). While data-driven methods have shown promise, they often exhibit limited generalization capability due to the lack of physical constraints. Incorporating symmetry in the battery, such as through the use of Physics-Informed Neural Networks (PINNs), can mitigate this issue. However, PINNs typically rely on integer-order equivalent circuit model differential equations, which fail to accurately describe the complex electrochemical relaxation processes. To bridge this gap, we propose a novel Fractional Differential Physics-Informed Neural Network (FDE-PINN) framework. Unlike traditional approaches, this method embeds a Fractional-Order Equivalent Circuit Model (FO-ECM) into the Gated Recurrent Unit (GRU) architecture to explicitly capture the anomalous diffusion and long-memory effects inherent in battery polarization. Specifically, the network is trained by minimizing a composite loss function that integrates the data fitting error with residuals from fractional-order governing equations, including Coulomb counting and fractional voltage dynamics. Extensive experiments on the Panasonic 18650PF dataset and CALCE A123 dataset verify the method’s superiority. Results demonstrate that the proposed FDE-GRU model achieves an average MSE of $14.29\times {10}^{-4}$ (with an MAE of 2.43% and RMSE of 3.23%) on the NCA chemistry and $26.24\times {10}^{-4}$ (with an MAE of 3.75% and RMSE of 5.09%) on the LiFePO₄ chemistry, significantly outperforming traditional methods by reducing the estimation error by 35.6% and 26.2% compared to the standard GRU, respectively. Full article

This work develops and analyzes a novel fractional-order chemostat system (FOCS) with a Caputo fractional derivative (CFD) featuring a sliding memory window and periodic boundary conditions (PBCs), designed to model microbial pollutant degradation in sustainable water treatment. By incorporating the Caputo fractional derivative with sliding memory (CFDS), the model captures time-dependent behaviors and memory effects in biological systems more realistically than classical integer-order formulations. We reduce the two-dimensional fractional differential equations (FDEs) governing substrate and biomass concentrations to a one-dimensional FDE by utilizing the PBCs. The existence and uniqueness of non-trivial, periodic solutions are established using the Carathéodory framework and fixed-point theorems, ensuring the system’s well-posedness. We prove the positivity and boundedness of solutions, demonstrating that substrate concentrations remain within physically meaningful bounds and biomass concentrations stay strictly positive, with solution trajectories confined to a biologically feasible invariant set. Additionally, we analyze non-trivial equilibria under constant dilution rates and derive their stability properties. The rigorous mathematical results confirm the viability of FOCS models for representing memory-driven, periodic bioprocesses, offering a foundation for advanced water treatment strategies that align with Sustainable Development Goal 6 (Clean Water and Sanitation). This work establishes a comprehensive mathematical framework that bridges fractional calculus with sustainable water treatment applications, providing both theoretical foundations and practical implications for optimizing bioreactor performance in environmental biotechnology. Full article

Fractional differential equations offer a natural framework for describing systems in which present states are influenced by the past. This work presents a nonlinear Caputo-type fractional differential equation (FDE) with a nonlocal initial condition and attempts to describe a model of memory-dependent behavioral adaptation. The proposed framework uses a fractional-order derivative $\eta \in (0,1)$ to discuss the long-term memory effects. The existence and uniqueness of solutions are demonstrated by Banach’s and Krasnoselskii’s fixed-point theorems. Stability is analyzed through Ulam–Hyers and Ulam–Hyers–Rassias benchmarks, supported by sensitivity results on the kernel structure and fractional order. The model is further employed for behavioral despair and learned helplessness, capturing the role of delayed stimulus feedback in shaping cognitive adaptation. Numerical simulations based on the convolution-based fractional linear multistep (FVI–CQ) and Adams–Bashforth–Moulton (ABM) schemes confirm convergence and accuracy. The proposed setup provides a compact computational and mathematical paradigm for analyzing systems characterized by nonlocal feedback and persistent memory. Full article

In this article, we present an efficient and highly accurate numerical scheme that achieves exponential convergence for solving nonlinear Riesz distributed-order fractional differential equations (RDFDEs) in one- and two-dimensional initial–boundary value problems. The proposed method is based on a two-stage collocation framework. In the first stage, spatial discretization is performed using the shifted Legendre–Gauss–Lobatto (SL-G-L) collocation method, where the approximate solutions and spatial derivatives are expressed in terms of shifted Legendre polynomial expansions. This reduces the original problem to a system of fractional differential equations (FDEs) for the expansion coefficients. Then, the temporal discretization is achieved in the second stage via Romanovski–Gauss–Radau collocation approach, which converts the system into a system of algebraic equations that can be solved efficiently. The method is applied to one- and two-dimensional nonlinear RDFDEs, and numerical experiments confirm its spectral accuracy, computational efficiency, and reliability. Existing numerical approaches to distributed-order fractional models often suffer from poor accuracy, instability in nonlinear settings, and high computational costs. By combining the efficiency of Legendre polynomials for bounded spatial domains with the stability of Romanovski polynomials for temporal discretization, the proposed two-stage framework effectively overcomes these limitations and achieves superior accuracy and stability. Full article

A novel numerical scheme is developed in this work to approximate solutions (APPSs) for nonlinear fractional differential equations (FDEs) governed by Robin boundary conditions (RBCs). The methodology is founded on a spectral collocation method (SCM) that uses a set of basis functions derived from generalized shifted Jacobi (GSJ) polynomials. These basis functions are uniquely formulated to satisfy the homogeneous form of RBCs (HRBCs). Key to this approach is the establishment of operational matrices (OMs) for ordinary derivatives (Ods) and fractional derivatives (Fds) of the constructed polynomials. The application of this framework effectively reduces the given FDE and its RBC to a system of nonlinear algebraic equations that are solvable by standard numerical routines. We provide theoretical assurances of the algorithm’s efficacy by establishing its convergence and conducting an error analysis. Finally, the efficacy of the proposed algorithm is demonstrated through three problems, with our APPSs compared against exact solutions (ExaSs) and existing results by other methods. The results confirm the high accuracy and efficiency of the scheme. Full article

This study introduces a novel hybrid framework that integrates machine learning (ML) with fractional-order differential equations (FDE) to enhance the prediction and clinical management of psoriasis, leveraging real-world data from the UCI Dermatology Dataset. By optimizing ML models, particularly the Voting Ensemble, to inform FDE parameters, and developing a user-friendly graphical user interface (GUI) for real-time diagnostics, the approach bridges computational efficiency with physiological realism, capturing memory-dependent disease progression beyond traditional integer-order models. Key findings reveal that the Voting Ensemble achieves a precision of 0.986 ± 0.007 and an AUC of 0.992 ± 0.005. At the same time, the fractional-order model, with an optimized order of 0.6781 and a mean square error (MSE) of 0.0031, accurately simulates disease trajectories, closely aligning with empirical trends for features such as Age and SawToothRete. The GUI effectively translates these insights into clinical tools, demonstrating probabilities ranging from 0% to 100% based on input features, supporting early detection and personalized planning. The framework’s robustness and potential for broader application to chronic conditions highlight its significance in advancing healthcare. Full article

The present paper examines a novel exact solution to nonlinear fractional partial differential equations (FDEs) through the Sardar sub-equation method (SSEM) coupled with Jumarie’s Modified Riemann–Liouville derivative (JMRLD). We take the (3+1)-dimensional space–time fractional modified Korteweg-de Vries (KdV) -Zakharov-Kuznetsov (ZK) equation as a case study, which describes some intricate phenomena of wave behavior in plasma physics and fluid dynamics. With the implementation of SSEM, we yield new solitary wave solutions and explicitly examine the role of the fractional-order parameter in the dynamics of the solutions. In addition, the sensitivity analysis of the results is conducted in the Galilean transformation in order to ensure that the obtained results are valid and have physical significance. Besides expanding the toolbox of analytical methods to address high-dimensional nonlinear FDEs, the proposed method helps to better understand how fractional-order dynamics affect the nonlinear wave phenomenon. The results are compared to known methods and a discussion about their possible applications and limitations is given. The results show the effectiveness and flexibility of SSEM along with JMRLD in forming new categories of exact solutions to nonlinear fractional models. Full article

Most solutions of fractional differential equations (FDEs) that model real-world phenomena in various fields of science, industry, and engineering are complex and cannot be solved analytically. This paper mainly aims to present some useful results for studying the qualitative properties of solutions of FDEs involving the new generalized Hattaf mixed (GHM) fractional derivative, which encompasses many types of fractional operators with both singular and non-singular kernels. In addition, this study also aims to unify and generalize existing results under a broader operator. Furthermore, the obtained results are applied to some linear systems arising from medicine. Full article

Fractional differential equations (FDEs) have attracted more and more attention in the last years; among them, equations of Caputo type allow for “more natural” initial conditions when the order is greater than one. As a result, many numerical methods have been devised and investigated for approximating their solution: the Matlab© codes of some of them are also available. The aim of this paper is a systematic comparison of such codes on a selected set of test problems. The obtained results are available on the web. Full article

In this study, we suggest a straightforward analytical/semi-analytical method based on the Elzaki transform (ET) method to find the solution to a number of differential fractional boundary value problems with initial conditions (ICs). The suggested approach not only resolves the issue of some equation nonlinearity but also transforms the issue into a simpler algebraic recurrence problem. In science and engineering, fractional differential equations (FDEs) can be solved with the help of this basic but effective concept. Some illustrative cases are used to demonstrate the efficacy and value of the suggested technique. Full article

Although fractional calculus has evolved significantly since its origin in the 1695 correspondence between Leibniz and L’Hôpital, the numerical treatment of multi-order fractional differential equations remains a challenge. Existing methods are often either computationally expensive or reliant on complex operational frameworks that hinder their broader applicability. In the present study, a novel numerical algorithm is proposed based on orthogonal hybrid functions (HFs), which were constructed as linear combinations of piecewise constant sample-and-hold functions and piecewise linear triangular functions. These functions, belonging to the class of degree-1 orthogonal polynomials, were employed to obtain the numerical solution of multi-order fractional differential equations defined in the Caputo sense. A generalized one-shot operational matrix was derived to explicitly express the Riemann–Liouville fractional integral of HFs in terms of the HFs themselves. This allowed the original multi-order fractional differential equation to be transformed directly into a system of algebraic equations, thereby simplifying the solution process. The developed algorithm was then applied to a range of benchmark problems, including both linear and nonlinear multi-order FDEs with constant and variable coefficients. Numerical comparisons with well-established methods in the literature revealed that the proposed approach not only achieved higher accuracy but also significantly reduced computational effort, demonstrating its potential as a reliable and efficient numerical tool for fractional-order modeling. Full article

The current research study presents a comprehensive analysis of the local discontinuous Galerkin (LDG) method for solving multi-order fractional differential equations (FDEs), with an emphasis on circuit modeling applications. We investigated the existence, uniqueness, and numerical stability of LDG-based discretized formulation, leveraging the Liouville–Caputo fractional derivative and upwind numerical fluxes to discretize governing equations while preserving stability. The method was validated through benchmark test cases, including comparisons with analytical solutions and established numerical techniques (e.g., Gegenbauer wavelets and Dickson collocation). The results demonstrate that the LDG method achieves high-accuracy solutions (e.g., with a relatively large time step size) and reduced computational costs, which are attributed to its element-wise formulation. These findings position LDG as a promising tool for complex scientific and engineering applications, particularly in modeling fractional-order systems such as RL, RLC circuits, and other electrical circuit equations. Full article