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Long-term memory is a feature observed in systems ranging from neural networks to epidemiological models. The memory in such systems is usually modeled by the time delay. Furthermore, the nonlocal operators, such as the "fractional order…
In this paper, we investigate the existence and uniqueness of solutions and derive the Ulam--Hyers--Mittag--Leffler stability results for impulsive implicit $\Psi$--Hilfer fractional differential equations with time delay. It is…
The purpose of this paper is to introduce and investigate a new system of global fractional-order interval implicit projection neural networks. An existence and uniqueness theorem of the equilibrium point for such kind of global…
This paper is concerned with the existence and uniqueness, and Ulam--Hyers stabilities of solutions of nonlinear impulsive $\varphi$--Hilfer fractional differential equations. Further, we investigate the dependence of the solution on the…
This paper presents a data-integrated framework for learning the dynamics of fractional-order nonlinear systems in both discrete-time and continuous-time settings. The proposed framework consists of two main steps. In the first step,…
Interval approaches for the reachability analysis of initial value problems for sets of classical ordinary differential equations have been investigated and implemented by many researchers during the last decades. However, there exist…
This work deals with the finite time stability of generalized proportional fractional systems with time delay. First, based on the generalized proportional Gr\"onwall inequality, we derive an explicit criterion that enables the system…
This paper introduces significant advancements in fractional neural operators (FNOs) through the integration of adaptive hybrid kernels and stochastic multiscale analysis. We address several open problems in the existing literature by…
Open quantum systems exhibit dynamics ranging from unitary evolution to irreversible dissipation. While the Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) equation uniquely characterizes Markovian CPTP evolution, many physical platforms…
This paper is devoted to the investigation of the nonnegative solutions and the stability and asymptotic properties of the solutions of fractional differential dynamic systems involving delayed dynamics with point delays. The obtained…
This work investigates the exponential stability of neural networks (NNs) systems with time delays. By considering orthogonal polynomials with weighted terms, a new weighted integral inequality is presented. This inequality extend several…
The linear nonhomogeneous fractional difference system with constant coefficients is introduced. An explicit solution to the system is acquired by proposing a newly discrete retarded perturbation of the nabla Mittag-Leffer-type function…
This paper is about the study of a new class of fractional-order projection neural networks with impulses which capture the desired features of both the variational inequality and the fractional-order impulsive dynamical systems within the…
We investigate the dynamical and analytical consequences of truncating the Gr\"unwald--Letnikov memory term in a fractional Duffing oscillator. The truncated memory is treated not merely as a computational approximation, but as a…
We present a general framework for the rigorous numerical analysis of time-fractional nonlinear parabolic partial differential equations, with a fractional derivative of order $\alpha\in(0,1)$ in time. The framework relies on three…
Many physical, biological, and engineered systems exhibit memory effects that challenge Markovian models. Fractional calculus provides nonlocal operators to capture hereditary dynamics. This survey connects modeling, analysis, and…
In the theoretical research of Hepatitis B virus, mathematical models of its transmission mechanism have been thoroughly investigated, while the dynamics of the immune process in vivo have not. At present, nearly all the existing models are…
The study of systems with memory requires methods which are different from the methods used in regular dynamics. Systems with power-law memory in many cases can be described by fractional differential equations, which are…
This paper proposes and analyzes an efficient difference scheme for the nonlinear complex Ginzburg-Landau equation involving fractional Laplacian. The scheme is based on the implicit midpoint rule for the temporal discretization and a…
This paper establishes new sufficient conditions for Mittag-Leffler stability of Caputo fractional-order nonlinear systems with state-dependent delays. The central analytical tool is a class of Lyapunov-Krasovskii functionals that…