Search Results (675)

This study explores the spread of malware within a digital framework by introducing a unique fractional-order model that employs the Atangana–Baleanu–Caputo (ABC) derivative. As cyber threats grow increasingly sophisticated and widespread, traditional models using classical differential equations often prove inadequate, particularly in capturing long-term memory effects and historical dependencies inherent in real-world systems. To address these challenges, the proposed approach utilizes the non-local characteristics of fractional calculus, offering a more comprehensive framework for understanding malware behavior. The model includes the derivation of the basic reproduction number, ${\Re }_0$, to evaluate conditions for malware persistence or elimination, sensitivity analysis and examines equilibrium states to assess overall system stability. Theoretical analysis ensures the existence and uniqueness of solutions through fixed-point techniques. Through numerical simulations, the theoretical results are validated, emphasizing the significant impact of antidotal and recovery measures in controlling malware spread. These findings provide essential guidance for enhancing the protection and robustness of sophisticated cyber-physical and humanoid infrastructures. Full article

In this work, we present a fractional-order reaction–diffusion model for the spread of infectious diseases, incorporating incommensurate Caputo derivatives to capture memory effects and heterogeneous temporal behavior across compartments. Focusing on a generalized SI model with nonlinear incidence, we explore the local asymptotic stability of both disease-free and endemic equilibria. The model accommodates spatial diffusion, saturation effects, and varying fractional orders, yielding a more realistic depiction of epidemic propagation. Analytical techniques—ranging from linearization to spectral analysis—are employed to rigorously establish stability conditions. Numerical simulations support the theoretical findings, highlighting the impact of memory and spatial structure on long-term dynamics. This study offers a refined mathematical lens to understand the persistence or eradication of infectious diseases under memory-dependent and spatially heterogeneous environments. 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

Difference schemes for the numerical solution of fractional differential equations rely on discretizations of the fractional derivative. In this paper, we obtain the second-order expansion formula for the L1 approximation of the Caputo fractional derivative. Second-order approximations of the fractional derivative are constructed based on the expansion formula and parameter-dependent discretizations of the second derivative. Examples illustrating the application of these approximations to the numerical solution of ordinary and partial fractional differential equations are presented, and the convergence and order of the difference schemes are proved. Numerical experiments are also provided, confirming the theoretical predictions for the accuracy of the numerical methods. Full article

This study presents an efficient numerical scheme for solving the generalized nonlinear time-fractional Klein–Gordon equation. The Caputo time-fractional derivative is discretized using a conventional finite-difference approach, while the spatial domain is approximated with uniform hyperbolic polynomial B-splines. These discretizations are coupled through the $\theta$-weighted scheme. The uniform hyperbolic polynomial B-spline framework extends classical spline theory by incorporating hyperbolic functions, thereby enhancing flexibility and smoothness in curve and surface representations—features particularly useful for problems exhibiting hyperbolic characteristics. A rigorous stability and convergence analysis of the proposed method is provided. The effectiveness of the scheme is further validated through numerical experiments on benchmark problems. The results demonstrate up to two orders of magnitude improvement in $L_{\infty }$ error norms compared to prior spline methods. This substantial accuracy enhancement highlights the robustness and efficiency of the proposed approach for fractional partial differential equations. Full article

This paper proposes a dual-fractional framework for stochastic differential equations (SDEs) that integrates conformable fractional calculus into both the system dynamics and the stochastic driving noise. For the first time, a conformable formulation of fractional noise is introduced, replacing the traditional Caputo-based representation. This modification eliminates singular kernel functions while preserving the fundamental properties of classical calculus, thereby simplifying both the analysis and numerical implementation. A complete analytical study is presented, rigorously addressing the convergence properties, deriving explicit error estimates, and establishing the numerical stability of the proposed scheme. The framework is realized through an enhanced conformable fractional discrete Temimi–Ansari method (CFDTAM), which accommodates distinct fractional orders for the system dynamics and the stochastic component. The stability and accuracy of the proposed scheme are validated through comparisons with the stochastic Runge–Kutta method (SRK) as implemented in Mathematica 12. Applications to benchmark models—including the fractional Langevin, Ginzburg–Landau, and Davis–Skodje systems—further demonstrate the robustness of the framework, especially in regimes where the Hurst exponent $\overline{\mathrm{Ӊ}}$ greater than 0.5. Overall, the results establish the method as a rigorous and efficient tool for modelling and analyzing stochastic fractional systems in finance, biophysics, and engineering. Full article

This research employs a Caputo fractional-derivative model to investigate the effects of magnetic field inclination and thermal radiation on the unsteady flow of a Casson fluid over an inclined plate in a porous medium. The model incorporates memory effects to generalize the classical formulation, while also accounting for internal heat generation and a chemical reaction. The governing equations are solved analytically using the Laplace transform, yielding power-series solutions in the time domain. Convergence analysis and benchmarking confirm the reliability and accuracy of the derived solutions. Key physical parameters are analyzed, and their impacts on the system are presented both graphically and in tabular form. The results indicate that increasing the inclination of the plate and magnetic field significantly suppresses the velocity distribution and reduces the associated boundary-layer thickness. Conversely, a higher fractional-order parameter enhances the velocity, temperature, and species concentration profiles by reducing memory effects. This study makes a significant contribution to the fractional modeling of unsteady heat and mass transfer in complex non-Newtonian fluids and provides valuable insights for the precise control of transport processes in industrial, chemical, and biomedical applications. Full article

Researchers have devised numerous methods to model intricate behaviors in phenomena that unfold in multiple stages. This work focuses on a specific category of piecewise hybrid terminal systems characterized by delay. To account for hereditary memory effects, which are absent in standard integer-order systems, our framework partitions the time interval into two distinct phases. The initial phase employs the classical derivative, while the subsequent phase utilizes the Atangana–Baleanu–Caputo (ABC) fractional derivative. We establish conditions that guarantee both the existence and uniqueness of solutions through the application of suitable fixed-point arguments. Furthermore, Hyers–Ulam (H-U) stability is investigated to ascertain the robustness and reliability of the derived solutions. To exemplify these theoretical findings, we present a fractional-order tuberculosis treatment model that incorporates the development of drug resistance, alongside a general numerical example. Numerical simulations reveal that changes in the fractional order influence the dynamic behavior of the disease. Full article

In this paper, we study the local stability of an incommensurate fractional reaction–diffusion glycolysis model. The glycolysis process, fundamental to cellular metabolism, exhibits complex dynamical behaviors when formulated as a nonlinear reaction–diffusion system. To capture the heterogeneous memory effects often present in biochemical and chemical processes, we extend the classical model by introducing incommensurate fractional derivatives, where each species evolves with a distinct fractional order. We linearize the system around the positive steady state and derive sufficient conditions for local asymptotic stability by analyzing the eigenvalues of the associated Jacobian matrix under fractional-order dynamics. The results demonstrate how diffusion and non-uniform fractional orders jointly shape the stability domain of the system, highlighting scenarios where diffusion destabilizes homogeneous equilibria and others where incommensurate memory effects enhance stability. Numerical simulations are presented to illustrate and validate the theoretical findings. Full article

Time-fractional interface problems arise in systems where interacting materials exhibit memory effects or anomalous diffusion. These models provide a more realistic description of physical processes than classical formulations and appear in heat conduction, fluid flow, porous media diffusion, and electromagnetic wave propagation. However, the presence of complex interfaces and the nonlocal nature of fractional derivatives makes their numerical treatment challenging. This article presents a numerical scheme that combines radial basis functions (RBFs) with the finite difference method (FDM) to solve time-fractional partial differential equations involving interfaces. The proposed approach applies to both linear and nonlinear models with constant or variable coefficients. Spatial derivatives are approximated using RBFs, while the Caputo definition is employed for the time-fractional term. First-order time derivatives are discretized using the FDM. Linear systems are solved via Gaussian elimination, and for nonlinear problems, two linearization strategies, a quasi-Newton method and a splitting technique, are implemented to improve efficiency and accuracy. The method’s performance is assessed using maximum absolute and root mean square errors across various grid resolutions. Numerical experiments demonstrate that the scheme effectively resolves sharp gradients and discontinuities while maintaining stability. Overall, the results confirm the robustness, accuracy, and broad applicability of the proposed technique. Full article

In this study, a time-fractional extension of the classical Theis problem with an exponential source term is investigated in a confined aquifer. The governing equation is modeled using two different fractional derivatives—the Caputo and Atangana–Baleanu–Caputo (ABC) operators—to account for memory effects in groundwater flow. The Fractional Reduced Differential Transform Method (FRDTM) is applied to obtain approximate series solutions up to the fifth order. The impact of the fractional order $\alpha$, the nature of the fractional kernel and the localized source term on the hydraulic head are explored at different radial positions. The comparative analysis between the Caputo and Atangana–Baleanu–Caputo (ABC) models reveals how memory effects and operator choice significantly influence the hydraulic head response, offering insights into selecting suitable models for aquifers with varying recharge characteristics. Full article

In this article, mathematical modeling and stability analysis of Lyme disease and its transmission dynamics using Caputo fractional-order derivatives is presented. The compartmental model has been formulated to analyze the spread of Borrelia burgdorferi virus through tick vectors and mammalian hosts. The feasible region is established, and the boundedness of the model is verified. Analytically, the disease-free equilibrium and the basic reproduction number $\left({\Re }_0\right)$ has been determined to assess outbreak potential. By virtue of the fixed-point theory, the existence and uniqueness of solutions has been established. The numerical simulations are obtained via the Runge–Kutta 4 method, demonstrating the model’s ability to capture realistic disease progression. Finally, sensitivity analysis and control strategies (tick population reduction, host vaccination, public awareness, and early treatment) are evaluated, revealing that integrated control measures significantly reduce infection rates and enhance recovery. Full article

This paper presents a Crank–Nicolson mixed finite element method along with its reduced-order extrapolation model for a fourth-order nonlinear diffusion equation with Caputo temporal fractional derivative. By introducing the auxiliary variable $v=-{\epsilon }^2\Delta u+f(u)$, the equation is reformulated as a second-order coupled system. A Crank–Nicolson mixed finite element scheme is established, and its stability is proven using a discrete fractional Gronwall inequality. Error estimates for the variables u and v are derived. Furthermore, a reduced-order extrapolation model is constructed by applying proper orthogonal decomposition to the coefficient vectors of the first several finite element solutions. This scheme is also proven to be stable, and its error estimates are provided. Theoretical analysis shows that the reduced-order extrapolation Crank–Nicolson mixed finite approach reduces the degrees of freedom from tens of thousands to just a few, significantly cutting computational time and storage requirements. Numerical experiments demonstrate that both schemes achieve spatial second-order convergence accuracy. Under identical conditions, the CPU time required by the reduced-order extrapolation Crank–Nicolson mixed finite model is only $1/60$ of that required by the Crank–Nicolson mixed finite scheme. These results validate the theoretical analysis and highlight the effectiveness of the methods. Full article

This paper aims to explore the numerical solution of non-linear fractional-order Burgers’-Huxley equation based on Caputo’s formulation of fractional derivatives. The equation serves as a versatile tool for analyzing a wide range of physical, biological, and engineering systems, facilitating valuable insights into nonlinear dynamic phenomena. The fractional operator provides a comprehensive mathematical framework that effectively captures the non-locality, hereditary characteristics, and memory effects of various complex systems. The approximation of temporal differential operator is carried out through finite difference based L1 scheme, while spatial discretization is performed using modified cubic B-spline basis functions. The stability as well as convergence analysis of the approach are also presented. Additionally, some numerical test experiments are conducted to evaluate the computational efficiency of a modified fourth-order cubic B-spline (M43BS) approach. Finally, the results presented in the form of tables and graphs highlight the applicability and robustness of M43BS technique in solving fractional-order differential equations. The proposed methodology is preferred for its flexible nature, high accuracy, ease of implementation and the fact that it does not require unnecessary integration of weight functions, unlike other numerical methods such as Galerkin and spectral methods. Full article

The paper is devoted to the study of a class of singular high-order fractional integro-differential equations with p-Laplacian operator, involving both the Riemann–Liouville fractional derivative and the Caputo fractional derivative. First, we investigate the problem by the method of reducing the order of fractional derivative. Then, by using the Schauder fixed point theorem, the existence of solutions is proved. The upper and lower bounds for the unique solution of the problem are established under various conditions by employing the Banach contraction mapping principle. Furthermore, four numerical examples are presented to illustrate the applications of our main results. Full article