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We prove optimality conditions for different variational functionals containing left and right Caputo fractional derivatives. A sufficient condition of minimization under an appropriate convexity assumption is given. An Euler-Lagrange…
Neural Ordinary Differential Equations model dynamical systems with ODEs learned by neural networks. However, ODEs are fundamentally inadequate to model systems with long-range dependencies or discontinuities, which are common in…
In this paper, we propose a fractional time extension of the Quan tum Master Equation. We introduce a Caputo-type fractional derivative in time as an extension of the exponential decay of the Lindblad framework through the incorporation of…
In this article, we study the continuous and discrete fractional persistence problem which looks for the persistence of properties of a given classical ($\alpha=1$) differential equation in the fractional case (here using fractional…
In recent years, the use of variable-order differential operators has emerged as a powerful tool in the analysis of nonlinear fractional differential equations and chaotic systems. In finance, the accurate prediction of market trends and…
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…
An implicit finite difference method with non-uniform timesteps for solving the fractional diffusion equation in the Caputo form is proposed. The method allows one to build adaptive methods where the size of the timesteps is adjusted to the…
We study controllability and observability concepts of tempered fractional linear systems in the Caputo sense. First, we formulate a solution for the class of tempered systems under investigation by means of the Laplace transform method.…
We consider a discretization of Caputo derivatives resulted from deconvolving a scheme for the corresponding Volterra integral. Properties of this discretization, including signs of the coefficients, comparison principles, and stability of…
This paper establishes the equivalence between systems described by a single first-order hyperbolic partial differential equation and systems described by integral delay equations. System-theoretic results are provided for both classes of…
In a smooth flow, the leading-order response of trajectories to infinitesimal perturbations in their initial conditions is described by the finite-time Lyapunov exponents and associated characteristic directions of stretching. We give a…
In this article, we present a new second order finite difference discrete scheme for fractal mobile/immobile transport model based on equivalent transformative Caputo formulation. The new transformative formulation takes the singular kernel…
We present a delay-compensating control method that transforms exponentially stabilizing controllers for an undelayed system into a sample-based predictive controller with numerical integration. Our method handles both first-order and…
In this paper, a modified nonlinear Schr\"{o}dinger equation with spatio-temporal dispersion is formulated in the senses of Caputo fractional derivative and conformable derivative. A new generalized double Laplace transform coupled with…
In this paper, stability and synchronization of a Caputo fractional BAM neural network system of high-order type and neutral delays are examined. A mixture of properties of fractional calculus, Laplace transform, and analytical techniques…
In this paper we develop a fractional Hamiltonian formulation for dynamic systems defined in terms of fractional Caputo derivatives. Expressions for fractional canonical momenta and fractional canonical Hamiltonian are given, and a set of…
In this paper, we are interested in the study of a problem with fractional derivatives having boundary conditions of integral types. The problem represents a Caputo type advection-diffusion equation where the fractional order derivative…
In the present paper, we address a class of the fractional derivatives of constant and variable orders for the first time. Fractional-order relaxation equations of constants and variable orders in the sense of Caputo type are modeled from…
The fractional calculus is useful to model non-local phenomena. We construct a method to evaluate the fractional Caputo derivative by means of a simple explicit quadratic segmentary interpolation. This method yields to numerical resolution…
We derive computationally tractable methods to select a small subset of experiment settings from a large pool of given design points. The primary focus is on linear regression models, while the technique extends to generalized linear models…