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.
2011.
The Fractional Quantum Derivative and the Fractional Linear Scale Invariant Systems. Fractional Calculus for Scientists and Engineers. 84:123–144.: Springer-Verlag
AbstractThe 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
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.
Magin, R, Ortigueira MD, Podlubny I, Trujillo J.
2011.
On the fractional signals and systems. Signal Processing. 91:350–371., Number 3: Elsevier
AbstractA look into fractional calculus and its applications from the signal processing point of view is done in this paper. A coherent approach to the fractional derivative is presented, leading to notions that are not only compatible with the classic but also constitute a true generalization. This means that the classic are recovered when the fractional domain is left. This happens in particular with the impulse response and transfer function. An interesting feature of the systems is the causality that the fractional derivative imposes. The main properties of the derivatives and their representations are presented. A brief and general study of the fractional linear systems is done, by showing how to compute the impulse, step and frequency responses, how to test the stability and how to insert the initial conditions. The practical realization problem is focussed and it is shown how to perform the input?ouput computations. Some biomedical applications are described.
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