Fractional Derivatives in Interval Analysis

[+] Author and Article Information
Giulio Cottone

Università degli Studi di Palermo Palermo, Italy Technische Universität München Munich, Germany

Roberta Santoro

Università degli Studi di Messina Messina, Italy

1Corresponding author.

ASME doi:10.1115/1.4036705 History: Received November 29, 2016; Revised March 10, 2017


In this paper, interval fractional derivative are presented. We consider uncertainty in both the order and the argument of the fractional differentiation. 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 transform. 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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