Ortigueira, MD.
2010.
On the Fractional Linear Scale Invariant Systems. IEEE Transactions on Signal Processing. 58:6406–6410., Number 12
AbstractThe linear scale invariant systems are introduced for both integer and fractional orders. They are defined by the generalized Euler-Cauchy differential equation. The quantum fractional derivatives are suitable for dealing with this kind of systems, allowing us to define impulse response and transfer function with the help of the Mellin transform. It is shown how to compute the impulse responses corresponding to the two half plane regions of convergence of the transfer function.
Ortigueira, MD, Coito F.
2004.
From Differences to Derivatives. Fractional Calculus and Applied Analysis. 7:459., Number 4: INSTITUTE OF MATHEMATICS AND INFORMATICS BULGARIAN ACADEMY OF SCIENCES
Abstract
Ortigueira, MD.
2008.
An introduction to the fractional continuous-time linear systems: the 21st century systems. IEEE Circuits and Systems Magazine. 8:19–26., Number 3: IEEE
AbstractA brief introduction to the fractional continuous-time linear systems is presented. It will be done without needing a deep study of the fractional derivatives. We will show that the computation of the impulse and step responses is very similar to the classic. The main difference lies in the substitution of the exponential by the Mittag-Leffler function. We will present also the main formulae defining the fractional derivatives.
Ortigueira, M.
2006.
A coherent approach to non-integer order derivatives. Signal Processing. 86:2505–2515., Number 10: Elsevier
AbstractThe relation showing that the Grunwald-Letnikov and generalised Cauchy derivatives are equal is presented. This establishes a bridge between two different formulations and simultaneously between the classic integer order derivatives and the fractional ones. Starting from the generalised Cauchy derivative formula, new relations are obtained, namely a regularised version that makes the concept of pseudo-function appear naturally without the need for a rejection of any infinite part. From the regularised derivative, new formulations are deduced and specialised first for the real functions and afterwards for functions with Laplace transforms obtaining the definitions proposed by Lionville. With these tools suitable definitions of fractional linear systems are obtained.
Ortigueira, MD, Trujillo JJ.
2011.
Generalized Gru?nwald?Letnikov Fractional Derivative and Its Laplace and Fourier Transforms Journal of Computational and Nonlinear Dynamics. 6:034501., Number 3
AbstractThe generalized Grünwald?Letnikov fractional derivative is analyzed in this paper. Its Laplace and Fourier transforms are computed, and some current results are criticized. It is shown that only the forward derivative of a sinusoid exists. This result is used to define the frequency response of a fractional linear system.
Ortigueira, M.
2009.
Comments on ?Modeling fractional stochastic systems as non-random fractional dynamics driven Brownian motions? Applied Mathematical Modelling. 33:2534–2537., Number 5: Elsevier Inc.
AbstractSome 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.
Ortigueira, MD.
2007.
Riesz Potentials as Centred Derivatives. Advances in Fractional Calculus. :93–112.: Springer Netherlands
AbstractGeneralised fractional centred differences and derivatives are studied in this Chapter. These generalise to real orders the existing ones valid for even and odd positive integer orders. For each one, suitable integral formulations are presented. The limit computation inside the integrals leads to generalisations of the Cauchy derivative. Their computations using a special path lead to the well known Riesz potentials. A study for coherence is done by applying the definitions to functions with Fourier transform. The existence of inverse Riesz potentials is also studied.
Ortigueira, MD, Batista AG.
2004.
A Fractional Linear System View of the Fractional Brownian Motion, December. Nonlinear Dynamics. 38:295–303., Number 1-4: Springer
AbstractA definition of the fractional Brownian motion based on the fractional differintegrator characteristics is proposed and studied. It is shown that the model enjoys the usually required properties. A discrete-time version based in the backward difference and in the bilinear transformation is considered. Some results are presented.
Ortigueira, MD.
2007.
Riesz potentials as centred derivatives, September. 2nd Symposium on Fractional Derivatives and Their Applications. (
J Sabatier, OP Agrawal, Machado, J.A.T., Eds.).:93–112.: Springer
AbstractGeneralised fractional centred differences and derivatives are studied in this chapter. These generalise to real orders the existing ones valid for even and odd positive integer orders. For each one, suitable integral formulations are presented. The limit computation inside the integrals leads to generalisations of the Cauchy derivative. Their computations using a special path lead to the well known Riesz potentials. A study for coherence is done by applying the definitions to functions with Fourier transform. The existence of inverse Riesz potentials is also studied.
Ortigueira, M, Tenreiro-Machado JA, da Costa JSá.
2005.
Which Differintegration?, July IEE Proceedings Vision, Image & Signal Processing. 152:846–850., Number 6: IET
Abstractn/a
Ortigueira, M.
2006.
Riesz potential operators and inverses via fractional centred derivatives, May. International Journal of Mathematics and Mathematical Sciences. 2006:1–12.: Hindawi
AbstractFractional centred differences and derivatives definitions are proposed, generalizing to real orders the existing ones valid for even and odd positive integer orders. For each one, suitable integral formulations are obtained. The computations of the involved integrals lead to new generalizations of the Cauchy integral derivative. To compute this integral, a special two-straight-line path was used. With this the referred integrals lead to the well-known Riesz potential operators and their inverses that emerge as true fractional centred derivatives, but that can be computed through summations similar to the well-known Grünwald-Letnikov derivatives.