Abstract
In this chapter the solution of Fokker-Planck-Kolmogorov type equations is pursued with the aid of Complex Fractional Moments (CFMs). These quantities are the generalization of the well-known integer-order moments and are obtained as Mellin transform of the Probability Density Function (PDF). From this point of view, the PDF can be seen as inverse Mellin transform of the CFMs, and it can be obtained through a limited number of CFMs. These CFMs’ capability allows to solve the Fokker-Planck-Kolmogorov equation governing the evolutionary PDF of non-linear systems forced by white noise with an elegant and efficient strategy. The main difference between this new approach and the other one based on integer moments lies in the fact that CFMs do not require the closure scheme because a limited number of them is sufficient to accurately describe the evolutionary PDF and no hierarchy problem occurs.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
G. Alotta, M. Di Paola, Probabilistic characterization of nonlinear systems under α-stable white noise via complex fractional moments. Phys. A 420, 265–276 (2015)
G. Alotta, M. Di Paola, F.P. Pinnola, Cross-correlation and cross-power spectral density representation by complex spectral moments. Int. J. Non-Linear Mech. 94, 20–27 (2017)
D.C.C. Bover, Moment equation methods for nonlinear stochastic system. J. Math. Anal. Appl. 65, 306–320 (1978)
S. Butera, M. Di Paola, Fractional differential equations solved by using Mellin transform. Comm. Nonlinear Sci. Num. Simul. 19(7), 2220–2227 (2014)
G.Q. Cai, Y.K. Lin, Exact and approximate solutions for randomly excited non-linear systems. Int. J. Nonlinear Mech. 31, 647–655 (1996)
A. Chechkin, V. Gonchar, J. Klafter, R. Metzler, L. Tanatarov, Stationary state of non-linear oscillator driven by Lévy noise. Chem. Phys. 284, 233–251 (2002)
G. Cottone, M. Di Paola, On the use of fractional calculus for the probabilistic characterization of random variable. Probab. Eng. Mech. 24, 321–330 (2009)
G. Cottone, M. Di Paola, A new representation of power spectral density and correlation function by means of fractional spectral moments. Probab. Eng. Mech. 25(3), 348–353 (2010)
G. Cottone, M. Di Paola, R. Metzler, Fractional calculus approach to the statistical characterization of random variables and vectors. Phys. A 389, 909–920 (2010)
H. Dai, Z. Ma, L. Li, An improved complex fractional moment-based approach for the probabilistic characterization of random variables. Probab. Eng. Mech. 53, 52–58 (2018)
A. Di Matteo, M. Di Paola, A. Pirrotta, Probabilistic characterization of nonlinear systems under Poisson white noise via complex fractional moments. Nonlinear Dyn. 77(3), 729–738 (2014)
A. Di Matteo, M. Di Paola, A. Pirrotta, Poisson white noise parametric input and response by using complex fractional moments. Probab. Eng. Mech. 38, 119–126 (2014)
M. Di Paola, G. Ricciardi, M. Vasta, A method for the probabilistic analysis of nonlinear systems. Probab. Eng. Mech. 10, 1–10 (1995)
M. Di Paola, Fokker Planck equation solved in terms of complex fractional moments. Probab. Eng. Mech. 38, 70–76 (2014)
M. Di Paola, F.P. Pinnola, Riesz fractional integrals and complex fractional moments for the probabilistic characterization of random variables. Probab. Eng. Mech. 29, 149–156 (2012)
G.K. Er, Exponential closure methods for some randomly excited nonlinear systems. Int. J. Nonlinear Mech. 35, 69–78 (2000)
C.W. Gardiner, Handbook of Stochastic Methods for Physics Chemistry and The Natural Science (Springer, Berlin, 1983)
R. Hilfer, Application of Fractional Calculus in Physics (World Scientific, Singapore, 2000)
R.A. Ibrahim, Parametric Random Vibrations (Wiley, New York, 1985)
R. Iwankiewicz, S.R.K. Nielsen, Solution techniques for pulse problems in non-linear stochastic dynamics Original Research Article. Probab. Eng. Mech. 15, 25–36 (2000)
X. Jin, Y. Wang, Z. Huang, M. Di Paola, Constructing transient response probability density of non-linear system through complex fractional moments. Int. J. Non-Linear Mech. 65, 253–259 (2014)
I.A. Kougioumtzoglou, P.D. Spanos, An analytical Wiener path integral technique for non-stationary response determination of nonlinear oscillators. Probab. Eng. Mech. 28, 125–131 (2012)
Y.K. Lin, Probabilistic Theory of Structural Dynamics (McGraw Hill, New York, 1967)
E. Mamontov, A. Naess, An analytical-numerical method for fast evaluation of probability densities for transient solutions of nonlinear Itô’s stochastic differential equations. Int. J. Eng. Sci. 47, 116–130 (2009)
G. Muscolino, G. Ricciardi, M. Vasta, Stationary and non-stationary probability density function for nonlinear oscillator. Int. J. Nonlinear Mech. 32, 1051–1064 (1997)
G. Muscolino, G. Ricciardi, Probability density function of MDOF systems under non-normal delta-correlated inputs. Comput. Methods Appl. Mech. Eng. 168, 121–133 (1999)
A. Naess, V. Moe, Efficient path integration methods for nonlinear dynamic systems. Probab. Eng. Mech. 15, 221–231 (2000)
F.P. Pinnola, Statistical correlation of fractional oscillator response by complex spectral moments and state variable expansion. Commun. Nonlinear Sci. Numer. Simul. 39, 343–359 (2016)
A. Pirrotta, R. Santoro, Probabilistic response of nonlinear systems under combined normal and Poisson white noise via path integral method. Probab. Eng. Mech. 25, 25–32 (2011)
I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)
H. Risken, The Fokker-Planck Equation. Methods of Solution and Applications (Springer, Berlin, 1989)
J.B. Roberts, P.D. Spanos, Stochastic averaging: an approximate method of solving random vibration problems Review Article. Int. J. Non-Linear Mech. 21, 111–134 (1986)
G.S. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach Science Publishers, 1993)
P.D. Spanos, A. Sofi, M. Di Paola, Nonstationary response envelope probability densities of nonlinear oscillators. J. Appl. Mech. Trans. ASME 74(2), 315–324 (2007)
B.F. Spencer, L.A. Bergman, On the numerical solution of the Fokker-Planck equation for nonlinear stochastic systems. Nonlinear Dyn. 4, 357–372 (1993)
A. Vriza, A. Kargioti, P.J. Papakanellos, G. Fikioris, Analytical evaluation of certain integrals occurring in studies of wireless communications systems using the Mellin-transform method. Phys. Comm. 31, 133–140 (2018)
W.Q. Zhu, Stochastic averaging method in random vibration. Appl. Mech. Rev. 41, 189–199 (1988)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Di Paola, M., Pirrotta, A., Alotta, G., Di Matteo, A., Pinnola, F.P. (2019). Complex Fractional Moments for the Characterization of the Probabilistic Response of Non-linear Systems Subjected to White Noises. In: Belhaq, M. (eds) Topics in Nonlinear Mechanics and Physics. Springer Proceedings in Physics, vol 228. Springer, Singapore. https://doi.org/10.1007/978-981-13-9463-8_11
Download citation
DOI: https://doi.org/10.1007/978-981-13-9463-8_11
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-9462-1
Online ISBN: 978-981-13-9463-8
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)