In this paper, interval fractional derivatives are presented. We consider uncertainty in both the order and the argument of the fractional operator. The approach proposed takes advantage of the property of Fourier and Laplace transforms with respect to the translation operator, in order to first define integral transform of interval functions. Subsequently, the main interval fractional integrals and derivatives, such as the Riemann–Liouville, Caputo, and Riesz, are defined based on their properties with respect to integral transforms. Moreover, uncertain-but-bounded linear fractional dynamical systems, relevant in modeling fractional viscoelasticity, excited by zero-mean stationary Gaussian forces are considered. Within the interval analysis framework, either exact or approximate bounds of the variance of the stationary response are proposed, in case of interval stiffness or interval fractional damping, respectively.
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September 2017
Research-Article
Fractional Derivatives in Interval Analysis
Giulio Cottone,
Giulio Cottone
Dipartimento di Ingegneria Civile,
Ambientale, Aerospaziale, dei Materiali,
Università degli Studi di Palermo,
Palermo 90128, Italy;
Ambientale, Aerospaziale, dei Materiali,
Università degli Studi di Palermo,
Palermo 90128, Italy;
Engineering Risk Analysis Group,
Technische Universität München,
Munich 80333, Germany
e-mail: giulio.cottone@unipa.it
Technische Universität München,
Munich 80333, Germany
e-mail: giulio.cottone@unipa.it
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Roberta Santoro
Roberta Santoro
Dipartimento di Ingegneria,
Università degli Studi di Messina,
Messina 98166, Italy
e-mail: roberta.santoro@unime.it
Università degli Studi di Messina,
Messina 98166, Italy
e-mail: roberta.santoro@unime.it
Search for other works by this author on:
Giulio Cottone
Dipartimento di Ingegneria Civile,
Ambientale, Aerospaziale, dei Materiali,
Università degli Studi di Palermo,
Palermo 90128, Italy;
Ambientale, Aerospaziale, dei Materiali,
Università degli Studi di Palermo,
Palermo 90128, Italy;
Engineering Risk Analysis Group,
Technische Universität München,
Munich 80333, Germany
e-mail: giulio.cottone@unipa.it
Technische Universität München,
Munich 80333, Germany
e-mail: giulio.cottone@unipa.it
Roberta Santoro
Dipartimento di Ingegneria,
Università degli Studi di Messina,
Messina 98166, Italy
e-mail: roberta.santoro@unime.it
Università degli Studi di Messina,
Messina 98166, Italy
e-mail: roberta.santoro@unime.it
Manuscript received November 29, 2016; final manuscript received March 10, 2017; published online June 12, 2017. Assoc. Editor: Francesco Paolo Pinnola.
ASME J. Risk Uncertainty Part B. Sep 2017, 3(3): 030907 (6 pages)
Published Online: June 12, 2017
Article history
Received:
November 29, 2016
Revised:
March 10, 2017
Citation
Cottone, G., and Santoro, R. (June 12, 2017). "Fractional Derivatives in Interval Analysis." ASME. ASME J. Risk Uncertainty Part B. September 2017; 3(3): 030907. https://doi.org/10.1115/1.4036705
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