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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Pinnola, F. , 2016, “ Statistical Correlation of Fractional Oscillator Response by Complex Spectral Moments and State Variable Expansion,” Commun. Nonlinear Sci. Numer. Simul., 39, pp. 343–359. [CrossRef]
Bonilla, B. , Rivero, M. , and Trujillo, J. J. , 2007, Linear Differential Equations of Fractional Order, Springer, Dordrecht, The Netherlands, pp. 77–91.
Cottone, G. , and Di Paola, M. , 2010, “ A New Representation of Power Spectral Density and Correlation Function by Means of Fractional Spectral Moments,” Probab. Eng. Mech., 25(3), pp. 348–353. [CrossRef]
Cottone, G. , Di Paola, M. , and Santoro, R. , 2010, “ A Novel Exact Representation of Stationary Colored Gaussian Processes (Fractional Differential Approach),” J. Phys. A: Math. Theor., 43(8), p. 085002. [CrossRef]
Cottone, G. , and Di Paola, M. , 2011, “ Fractional Spectral Moments for Digital Simulation of Multivariate Wind Velocity Fields,” J. Wind Eng. Ind. Aerodyn., 99(6–7), pp. 741–747. [CrossRef]
Moore, R. E. , 1966, Interval Analysis, Prentice-Hall, Englewood Cliffs, NJ.
Alefeld, G. , and Herzberger, J. , 1983, Introduction to Interval Computations, Academic Press, New York.
Neumaier, A. , 1990, Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, MA.
Muhanna, R. L. , and Mullen, R. L. , 2001, “ Uncertainty in Mechanics Problems-Interval-Based Approach,” ASCE J. Eng. Mech., 127(6), pp. 557–566. [CrossRef]
Moens, D. , and Vandepitte, D. , 2005, “ A Survey of Non-Probabilistic Uncertainty Treatment in Finite Element Analysis,” Comput. Methods Appl. Mech. Eng., 194(12–16), pp. 1527–1555. [CrossRef]
Hansen, E. R. , 1975, “A Generalized Interval Arithmetic,” Interval Mathematics (Lecture Notes on Computational Science), Vol. 29, K. Nicket , ed., Springer, Berlin, pp. 7–18.
Comba, J. L. D. , and Stolfi, J. , 1993, “ Affine Arithmetic and Its Applications to Computer Graphics,” Anais do VI Simposio Brasileiro de Computaao Grafica e Processamento de Imagens (SIBGRAPI’93), pp. 9–18.
Stolfi, J. , and De Figueiredo, L. H. , 2003, “ An Introduction to Affine Arithmetic,” TEMA Tend Math. Appl. Comput., 4(3), pp. 297–312.
Nedialkov, N. S. , Kreinovich, V. , and Starks, S. A. , 2004, “ Interval Arithmetic, Affine Arithmetic, Taylor Series Methods: Why, What Next?” Numer. Algorithms, 37(1), pp. 325–336. [CrossRef]
Muscolino, G. , and Sofi, A. , 2012, “ Stochastic Analysis of Structures With Uncertain-But-Bounded Parameters Via Improved Interval Analysis,” Probab. Eng. Mech., 28, pp. 152–163. [CrossRef]
Muscolino, G. , and Sofi, A. , 2013, “ Bounds for the Stationary Stochastic Response of Truss Structures With Uncertain-But-Bounded Parameters,” Mech. Syst. Signal Process., 37(1–2), pp. 163–182. [CrossRef]
Muscolino, G. , Santoro, R. , and Sofi, A. , 2014, “ Explicit Frequency Response Functions of Discretized Structures With Uncertain Parameters,” Comput. Struct., 133, pp. 64–78. [CrossRef]
Elishakoff, I. , and Miglis, Y. , 2012, “ Novel Parameterized Intervals May Lead to Sharp Bounds,” Mech. Res. Commun., 44, pp. 1–8. [CrossRef]
Elishakoff, I. , and Miglis, Y. , 2012, “ Overestimation-Free Computational Version of Interval Analysis,” Int. J. Comput. Methods Eng. Sci. Mech., 13(5), pp. 319–328. [CrossRef]
Santoro, R. , Elishakoff, I. , and Muscolino, G. , 2015, “ Optimization and Anti-Optimization Solution of Combined Parameterized and Improved Interval Analyses for Structures With Uncertainties,” Comput. Struct., 149, pp. 31–42. [CrossRef]
Kilbas, A. A. , Srivastava, H. M. , and Trujillo, J. , 2006, Theory and Application of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands.
Samko, S. G. , Kilbas, A. A. , and Marichev, O. I. , 1993, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York (in Russian).
Di Paola, M. , Pinnola, F. P. , and Spanos, P. D. , 2014, “ Analysis of Multi-Degree-of-Freedom Systems With Fractional Derivative Elements of Rational Order,” International Conference on Fractional Differentiation and Its Applications (ICFDA), Catania, Italy, June 23–25, pp. 1–6.


Grahic Jump Location
Fig. 1

SDOF with fractional damping under stochastic force and uncertain stiffness

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
Fig. 4

Variance of the displacement as a function of βI

Grahic Jump Location
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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