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, 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.
2009.
Comments on ?Modeling fractional stochastic systems as non-random fractional dynamics driven Brownian motions? Applied Mathematical Modelling. 33:2534–2537., Number 5
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. Keywords: Fractional calculus; Grünwald-Letnikov derivative; Fractional Brownian motion
Ortigueira, MD, Rodríguez-Germá L, Trujillo JJ.
2011.
Complex Grünwald?Letnikov, Liouville, Riemann?Liouville, and Caputo derivatives for analytic functions Communications in Nonlinear Science and Numerical Simulation.
AbstractThe 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.
Ortigueira, MD.
2008.
Fractional Central Differences and Derivatives. Journal of Vibration and Control. 14:1255–1266., Number 9-10
AbstractFractional 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.
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