Fractional Derivatives in Interval Analysis

[+] Author and Article Information
Giulio Cottone

Dipartimento di Ingegneria Civile,
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

Roberta Santoro

Dipartimento di Ingegneria,
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 3(3), 030907 (Jun 12, 2017) (6 pages) Paper No: RISK-16-1142; doi: 10.1115/1.4036705 History: Received November 29, 2016; Revised March 10, 2017

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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Fig. 1

SDOF with fractional damping under stochastic force and uncertain stiffness

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Fig. 2

Comparison between the envelope of all the PSDs obtained by the vertex method, the lower (“x”) and upper (“o”) bound PDSs for β = 0.2

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Fig. 3

Comparison between the envelope of all the PSDs obtained by the vertex method, the lower (“x”) and upper (“o”) bound PDSs for β = 0.75

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Fig. 4

Variance of the displacement as a function of βI

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Fig. 5

In gray, the envelope of all the PSD obtained by varying βI: in red line—the approximated lower bound PSD and in blue line—the approximated upper bound PSD. The continuous thick line is the PSD associated to the minimal variance.




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