In this paper, we further discuss the properties of three kinds of fractional derivatives: the Grünwald–Letnikov derivative, the Riemann–Liouville derivative and the Caputo derivative. Especially, we compare the Riemann–Liouville derivative with the Caputo derivative. And sequential property of the Caputo derivative is also derived, which is helpful in translating the higher fractional-order di...

and Applied Analysis 3 Definition 2. The Grünwald-Letnikov fractional derivative with order α of function x(t) is defined below:

The formulations of Riemann-Liouville and Caputo derivatives in the complex plane are presented. Two versions corresponding to the whole or half plane. It is shown that they can be obtained from the Grünwald-Letnikov derivative.

the aim of this paper is to study the high order difference scheme for the solution of a fractional partial differential equation (pde) in the electroanalytical chemistry. the space fractional derivative is described in the riemann-liouville sense. in the proposed scheme we discretize the space derivative with a fourth-order compact scheme and use the grunwald- letnikov discretization of the ri...

The aim of this paper is to study the high order difference scheme for the solution of a fractional partial differential equation (PDE) in the electroanalytical chemistry. The space fractional derivative is described in the Riemann-Liouville sense. In the proposed scheme we discretize the space derivative with a fourth-order compact scheme and use the Grunwald- Letnikov discretization of the Ri...

The previous work in [1], Tseng et al. have designed a fractional order differentiator using radial basis function by directly truncating the coefficients to approximate the fractional order derivative Dα of the given digital signal. This paper presents the designing of fractional order differentiator using radial basis function and window. Three design examples are given to illustrate that the...

Stochastic differential equations (SDEs) have been applied by engineers and economists because it can express the behavior of stochastic processes in compact expressions. In this paper, by using Grunwald-Letnikov fractional derivative, the stochastic differential model is improved. Two numerical examples are presented to show efficiency of the proposed model. A numerical optimization approach b...

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 time domain is divided into several subdomains, and a fractional derivative (FDs) at a time node point is approximated using a modified Grünwald-Letnikov ap...

In this paper, we introduce the nabla fractional derivative and integral on time scales in Riemann-Liouville sense. We also Gr\"unwald-Letnikov Some of basic properties theorems related to calculus are discussed.

The fractional derivative of order α, with 1 < α ≤ 2 appears in several diffusion problems used in physical and engineering applications. Therefore to obtain highly accurate approximations for this derivative is of great importance. Here, we describe and compare different numerical approximations for the fractional derivative of order 1 < α ≤ 2. These approximations arise mainly from the Grünwa...