Because they are useful for both enabling numerical simulations and containing well-defined physical phenomena, discrete fractional reaction–diffusion models have attracted a great deal of interest from academics. Within the family of fractional reaction–diffusion models, a discrete form is examined in detail in this study. Furthermore, we investigate the complex synchronization dynamics of a suggested discrete master–slave reaction–diffusion system using the accuracy of linear control techniques combined with a fractional discrete Lyapunov approach. This study’s deviation from the behavior of equivalents with integer orders makes it very fascinating. Like the non-local nature inherent in Caputo fractional derivatives, it creates a memory Lyapunov function that is closely linked to the historical background of the system. The investigation provides a strong basis to the theoretical results.

This study investigates the stability of nonlinear systems, particularly those characterized by eigenvalues. We introduce dynamic Lyapunov functions as a mechanism for stability analysis, especially when explicit solutions are not available. The authors provide stability criteria at the equilibrium point, demonstrating exponential stability and ensuring a return to equilibriumfollowing disturbances. The results have a big effect on the design and analysis of control systems because they provide a new way to achieve stability without using complicated calculations or assumptions. The abstract delineates the Riemann–Liouville fractional integral, Caputo fractional integrals and derivatives, and the Mittag–Leffler function. The research employs the root–Horwitz criteria and introduces a novel formulation of the superchaotic Chen system. Fractional superchain systems (FHCS) represent a sophisticated framework for investigation.

The aim of this manuscript is to introduce a novel concept called Jaggi-type hybrid $(\varphi$ -$F)$-contraction and establish some fixed point results for this class of contractions in the framework of G-metric space. The validity of the main result is shown by a suitable example and the realized improvements with respect to the corresponding literature are highlighted. By using the constructed example, it is observed that the results established herein cannot be deduced from their analogs in previously announced results in the literature. As an application, the existence and uniqueness of solutions to certain nonlinear Volterra integral equations are investigated to illustrate the utility of our obtained results.

We investigate chaos control and hybrid synchronization in a variable-order fractional Arneodo system by treating the differentiation order $\alpha \left(t\right)$ as a closed-loop control variable. A hybrid chaos indicator, combining a tracking error with a windowed estimate of the largest Lyapunov exponent, drives both static and dynamic order modulation laws. The presence and uniqueness of solutions are demonstrated through two distinct methodologies: a piecewise constant-order decomposition with an explicit convergence rate and a direct contraction-mapping argument on the variable-order Volterra operator. Local stability is analyzed via Matignon’s spectral criterion under a quasi-static (frozen-time) approximation. The modulation laws are designed to steer $\alpha \left(t\right)$ below the critical order ${\alpha }_c\approx 0.8632$, at which the nontrivial equilibria $E_{1,2}=(\pm \sqrt{5.5},0,0)$ become locally asymptotically stable. A second-order predictor–corrector scheme attains its expected convergence rate. A controlled ablation study over 200 Monte Carlo runs demonstrates that the proposed laws reduce the terminal tracking error by 81% relative to the best fixed-order baseline, while requiring approximately eight orders of magnitude less control effort than classical active control. Hybrid synchronization (complete in $(u,v)$ and anti-synchronization in w) is successfully achieved in the variable-order setting.

The research paper primarily focuses on the theoretical framework and mathematical methodology for solving the Sawada Kotera Ito with time fraction (SKI) equation using the Elzaki transform decomposition technique, including the Caputo-Fabrizio derivative and the Atangana-Baleanu derivative, which are crucial for comprehending the fractional SKI (1). Various numerical examples are given to illustrate the application of the Elzaki transform in solving the fractional SKI equation. We utilize efficient basis functions, namely fractional Lagrange functions, for the interpolation of temporal variables. By integrating the primary equation with the initial-boundary conditions andemploying the relevant operational matrices for spatial and fractional temporal variables, the proposed model is converted into a system of nonlinear algebraic equations, which can be resolved using effective iterative solvers. Furthermore, we conduct a comprehensive analysis of the method’s convergence. Furthermore, we evaluate multiple test issues to assess the proposed scheme, which demonstrates its superior accuracy and less computational expense compared to contemporary numerical methods in the literature.

The paper discusses a numerical approach to solving complicated partial differential equations, with a particular emphasis on fractional advection-dispersion equations of space-time variable order. With the use of fractional derivative matrices, Legendre polynomials, and numerical examples and comparisons, it surpasses current methods by utilizing spectral collocation techniques. It resolves equations involving spatial and time variables that are variable-order fractional advection–dispersion (VOFADE). Legendre polynomials serve as basis functions in this method, whereas Legendre operational matrices are employed for fractional derivatives. The technique is more computationally efficient since it reduces fractional advection–dispersion equations to systems of algebraic equations. Numerical examples and a comparison with current approaches illustrate the method’s superior performance in solving complicated partial differential equations, especially in the context of transport processes.