The purpose of this paper is to prove some classical estimates for fractional derivatives of functions defined on a Coifman-Weiss space of homogeneous type. In particular the Product Rule and Chain Rule estimates in [KP] and [CW]. The fractional calculus of M. Riesz was extended to these spaces in [GSV]. Our main tools are fractional difference quotients and the square fractional derivative of ...

Fractional calculus, which has almost the same history as classic calculus, did not attract enough attention for a long time. However, in recent decades, fractional calculus and fractional differential equations become more and more popular because of its powerful potential applications. A large number of new differential equations (models) that involve fractional calculus are developed. These ...

Finite differences, as a subclass of direct methods in the calculus of variations, consist in discretizing the objective functional using appropriate approximations for derivatives that appear in the problem. This article generalizes the same idea for fractional variational problems. We consider a minimization problem with a Lagrangian that depends only on the left Riemann–Liouville fractional ...

the aim of this paper is to design a fractional order sliding mode controllers (fosmc)for a class of dc-dc converters such as boost and buck converters. firstly, the control lawis designed with respect to the properties of fractional calculus, the design yields an equiv-alent control term with an addition of discontinuous (attractive) control law. secondly, themathematical proof of the stabilit...

Reaction–diffusion systems have a broad variety of applications, particularly in biology, and it is well known that fractional calculus has been successfully used with this type system. However, analyzing these using discrete novel requires significant research diversity disciplines. Thus, paper, we investigate the discrete-time fractional-order Lengyel–Epstein system as model chlorite iodide m...

This paper deals with the problems of consistence and strong consistence of the maximum likelihood estimators of the mean and variance of the drift fractional Brownian motions observed at discrete time instants. A central limit theorem for these estimators is also obtained by using the Malliavin calculus. 1. Introduction. Long memory processes have been widely applied to various fields, such as...

The purpose of this brief note is to describe a discrete approximation to fractional Brownian motion. The approximation works for all Hurst indices H , but take slightly different forms for H ≤ 1 2 and H > 1 2 . There are already several discrete approximations to fractional Brownian motion in the literature (see, e.g., [11], [1], [3], [10], [4], [2], [5], [8] for this and related topics), and ...