相关论文: The Gr\"unwald-Letnikov fractional-order derivativ…

Using the Mellin transform approach, it is shown that, in contrast with integer-order derivatives, the fractional-order derivative of a periodic function cannot be a function with the same period. The three most widely used definitions of…

The fact that the first variation of a variational functional must vanish along an extremizer is the base of most effective solution schemes to solve problems of the calculus of variations. We generalize the method to variational problems…

Of the many definitions for fractional order differintegral, the Grunwald-Letnikov definition is arguably the most important one. The necessity of this definition for the description and analysis of fractional order systems cannot be…

In this paper an offset-free model predictive control scheme is presented for fractional-order systems using the Gr\"unwald-Letnikov derivative. The infinite-history fractional-order system is approximated by a finite-dimensional…

We introduce a notion of fractional (noninteger order) derivative on an arbitrary nonempty closed subset of the real numbers (on a time scale). Main properties of the new operator are proved and several illustrative examples given.

Long and short memory in economic processes is usually described by the so-called discrete fractional differencing and fractional integration. We prove that the discrete fractional differencing and integration are the Grunwald-Letnikov…

In this paper, by using a characterization of functions having fractional derivative, we propose a rigorous fractional Lyapunov function candidate method to analyze stability of fractional-order nonlinear systems. First, we prove an…

In this work, a second-order approximation of the fractional substantial derivative is presented by considering a modified shifted substantial Gr\"{u}nwald formula and its asymptotic expansion. Moreover, the proposed approximation is…

In this paper, a fractional derivative with short-term memory properties is defined, which can be viewed as an extension of Caputo fractional derivative. Then, some properties of the short memory fractional derivative are discussed. Also, a…

We prove maximum and comparison principles for fractional discrete derivatives in the integers. Regularity results when the space is a mesh of length $h$, and approximation theorems to the continuous fractional derivatives are shown. When…

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…

A fractional generalization of variations is used to define a stability of non-integer order. Fractional variational derivatives are suggested to describe the properties of dynamical systems at fractional perturbations. We formulate…

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…

Fractional order differential and difference equations are used to model systems with memory. Variable order fractional equations are proposed to model systems where the memory changes in time. We investigate stability conditions for linear…

Fractional diffusion equations for three-dimensional lattice models based on fractional-order differences of the Grunwald-Letnikov type are suggested. These lattice fractional diffusion equations contain difference operators that describe…

We consider some possible approaches to the fractional-order generalization of definition of variation (functional) derivative. Some problems of formulation of a fractional-order variational derivative are discussed. To give a consistent…

We present and investigate a new type of implicit fractional linear multistep method of order two for fractional initial value problems. The method is obtained from the second order super convergence of the Gr\"unwald-Letnikov approximation…

We consider a conflict-controlled dynamical system described by a nonlinear ordinary fractional differential equation with the Caputo derivative of an order $\alpha \in (0, 1).$ Basing on the finite-difference Gr\"{u}nwald-Letnikov…

The paper discusses the characteristic properties of fractional derivatives of non-integer order. It is known that derivatives of integer orders are determined by properties of differentiable functions only in an infinitely small…

This paper presents a modified numerical scheme for a class of Fractional Optimal Control Problems (FOCPs) formulated in Agrawal (2004) where a Fractional Derivative (FD) is defined in the Riemann-Liouville sense. In this scheme, the entire…