Some results presented in the paper ?Modeling fractional stochastic systems as non-random fractional dynamics driven Brownian motions? ?I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999? are discussed in this paper. The slightly modified Grünwald-Letnikov derivative proposed there is used to deduce some interesting results that are in contradiction with those proposed in the referred paper. Keywords: Fractional calculus; Grünwald-Letnikov derivative; Fractional Brownian motion

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Fractional central differences and derivatives are studied in this article. These are generalisations to real orders of the ordinary positive (even and odd) integer order differences and derivatives, and also coincide with the well known Riesz potentials. The coherence of these definitions is studied by applying the definitions to functions with Fourier transformable functions. Some properties of these derivatives are presented and particular cases studied.

The modelling of fractional linear systems through ARMA models is addressed. To perform this study, a new recursive algorithm for impulse response ARMA modelling is presented. This is a general algorithm that allows the recursive construction of ARMA models from the impulse response sequence. This algorithm does not need an exact order specification, as it gives some insights into the correct orders. It is applied to modelling fractional linear systems described by fractional powers of the backward difference and the bilinear transformations. The analysis of the results leads to propose suitable models for those systems.

The alternative system initial conditions versus the derivative initial conditions is focused in this paper. It is shown that Riemann?Liouville and Caputo initial conditions result from the corresponding derivative and not necessarily from the system at hand. To setup the correct system initialization, a formulation generalizing the integer order approach is presented. This is based on a generalization to the fractional environment of the well known jump formula. The obtained scheme is very general and does not depend on any transform. Besides, it can also be used in the time variant case. The Riemann?Liouville and Caputo initial conditions are interpreted in terms of this general framework and deduced equations where they are correct.

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The normal way of introducing the notion of derivative is by means of the limit of an incremental ratio that can assume three forms, depending the used translations as we saw in Chaps. 1 and 4. On the other hand, in those derivatives the limit operation is done over a set of points uniformly spaced: a linear scale was used. Here we present an alternative derivative, that is valid only for t {\ensuremath{>}} 0 or t {\ensuremath{<}} 0 and uses an exponential scale

In this paper a new least-squares (LS) approach is used to model the discrete-time fractional differintegrator. This approach is based on a mismatch error between the required response and the one obtained by the difference equation defining the auto-regressive, moving-average (ARMA) model. In minimizing the error power we obtain a set of suitable normal equations that allow us to obtain the ARMA parameters. This new LS is then applied to the same examples as in ?R.S. Barbosa, J.A. Tenreiro Machado, I.M. Ferreira, Least-squares design of digital fractional-order operators, FDA'2004 First IFAC Workshop on Fractional Differentiation and Its Applications, Bordeaux, France, July 19-21, 2004, P. Ostalczyk, Fundamental properties of the fractional-order discrete-time integrator, Signal Processing 83 (2003) 2367-2376? so performance comparisons can be drawn. Simulation results show that both magnitude frequency responses are essentially identical. Concerning the modeling stability,both algorithms present similar limitations, although for different ARMA model orders.

The well-known Liouville, Riemann?Liouville and Caputo derivatives are extended to the complex functions space, in a natural way, and it is established interesting connections between them and the Grünwald?Letnikov derivative. Particularly, starting from a complex formulation of the Grünwald?Letnikov derivative we establishes a bridge with existing integral formulations and obtained regularised integrals for Liouville, Riemann?Liouville, and Caputo derivatives. Moreover, it is shown that we can combine the procedures followed in the computation of Riemann?Liouville and Caputo derivatives with the Grünwald?Letnikov to obtain a new way of computing them. The theory we present here will surely open a new way into the fractional derivatives computation.

The quantum fractional derivative is defined using formulations analogue to the common Grünwald?Letnikov derivatives. While these use a linear variable scale, the quantum derivative uses an exponential scale and is defined in R? or R?. Two integral formulations similar to the usual Liouville derivatives are deduced with the help of the Mellin transform.

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