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Path Integral Method for Nonlinear Systems Under Levy White Noise

[+] Author and Article Information
Alberto Di Matteo

Dipartimento di Ingegneria Civile,
Ambientale Aerospaziale e dei Materiali (DICAM),
Università degli Studi di Palermo,
Viale delle Scienze,
Palermo 90128, Italy
e-mail: alberto.dimatteo@unipa.it

Antonina Pirrotta

Dipartimento di Ingegneria Civile,
Ambientale Aerospaziale e dei Materiali (DICAM),
Università degli Studi di Palermo,
Viale delle Scienze,
Palermo 90128, Italy;
Department of Mathematical Sciences,
University of Liverpool,
Liverpool L69 7ZL, UK
e-mails: antonina.pirrotta@unipa.it;
Antonina.Pirrotta@liverpool.ac.uk

Manuscript received June 22, 2016; final manuscript received September 21, 2016; published online June 12, 2017. Assoc. Editor: Ioannis Kougioumtzoglou.

ASME J. Risk Uncertainty Part B 3(3), 030905 (Jun 12, 2017) (7 pages) Paper No: RISK-16-1095; doi: 10.1115/1.4036703 History: Received June 22, 2016; Revised September 21, 2016

In this paper, the probabilistic response of nonlinear systems driven by alpha-stable Lévy white noises is considered. The path integral solution is adopted for determining the evolution of the probability density function of nonlinear oscillators. Specifically, based on the properties of alpha-stable random variables and processes, the path integral solution is extended to deal with Lévy white noises input with any value of the stability index alpha. It is shown that at the limit when the time increments tend to zero, the Einstein–Smoluchowsky equation, governing the evolution of the response probability density function, is fully restored. Application to linear and nonlinear systems under different values of alpha is reported. Comparisons with pertinent Monte Carlo simulation data and analytical solutions (when available) demonstrate the accuracy of the results.

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Figures

Grahic Jump Location
Fig. 1

Evolution of the PDF for α=0.5: linear case (γ=1)

Grahic Jump Location
Fig. 2

Evolution of the PDF for α=1: (a) linear case (γ=1) and (b) nonlinear case (γ=1, ε=1, ρ=1.5)

Grahic Jump Location
Fig. 3

Evolution of the PDF for α=1.5: (a) linear case (γ=1) and (b) nonlinear case (γ=1, ε=1, ρ=1.5)

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