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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September 2017
Research-Article
Path Integral Method for Nonlinear Systems Under Levy White Noise
Alberto Di Matteo,
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
Ambientale Aerospaziale e dei Materiali (DICAM),
Università degli Studi di Palermo,
Viale delle Scienze,
Palermo 90128, Italy
e-mail: alberto.dimatteo@unipa.it
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Antonina Pirrotta
Antonina Pirrotta
Dipartimento di Ingegneria Civile,
Ambientale Aerospaziale e dei Materiali (DICAM),
Università degli Studi di Palermo,
Viale delle Scienze,
Palermo 90128, Italy;
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
University of Liverpool,
Liverpool L69 7ZL, UK
e-mails: antonina.pirrotta@unipa.it;
Antonina.Pirrotta@liverpool.ac.uk
Search for other works by this author on:
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
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;
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
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. Sep 2017, 3(3): 030905 (7 pages)
Published Online: June 12, 2017
Article history
Received:
June 22, 2016
Revised:
September 21, 2016
Citation
Di Matteo, A., and Pirrotta, A. (June 12, 2017). "Path Integral Method for Nonlinear Systems Under Levy White Noise." ASME. ASME J. Risk Uncertainty Part B. September 2017; 3(3): 030905. https://doi.org/10.1115/1.4036703
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