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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 79
PROCEEDINGS OF THE SEVENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping and C.A. Mota Soares
Paper 108
Non-Linear Systems Driven by White Noise Processes and Handled by the Characteristic Function Equations M. Di Paola and G. Cottone
Department of Structural and Geotechnical Engineering, University of Studies of Palermo, Italy Full Bibliographic Reference for this paper
M. Di Paola, G. Cottone, "Non-Linear Systems Driven by White Noise Processes and Handled by the Characteristic Function Equations", in B.H.V. Topping, C.A. Mota Soares, (Editors), "Proceedings of the Seventh International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 108, 2004. doi:10.4203/ccp.79.108
Keywords: non-linear systems, Einstein-Smoluchowski equation, Fokker-Planck-Kolmogorov equation, poissonian and alpha-stable white noise processes, characteristic function, Laguerre polynomial.
Summary
The paper deals with non linear systems driven by white noise processes. The
main feature of the white noises is that they may be thought as derivative of
processes having orthogonal increments. Such processes are able to model a large
variety of real phenomena, and application may be found in almost every branch of
science such as in physics, mechanics, chemistry and so on [1,2,3]. White noise
processes belong to two classes: the Lévy (
![]() ![]() ![]() ![]() ![]() ![]() Linear and non linear systems driven by normal white noise are normally handled by the celebrated Itô stochastic calculus [4,5,6]. It mainly consists in finding the probability density function of the response by solving the parabolic differential equation known as Fokker-Planck-Kolmogorov (FPK) equation. For the case of Poisson white noise the extension of the FPK equation may be found in literature involving an integro-differential equation known as Kolmogorov-Feller equation. At least for the case of Lévy white noise input the Einsten-Smoluchowski (ES) equation involving fractional derivative may be found in literature. The main problem in the numerical treatment of such equations is the very severe limitation that the PDF must satisfy, i.e. the positivity of this function must be always guaranteed. This limitation makes unfeasible every kind of method based on series expansion of the PDF.
In order to overcome this difficulty in this paper a different procedure is
proposed: the equations in terms of PDF (FPK, KF and ES) are transformed in
differential equations governing the evolution of the Characteristic Function. The
latter function is defined in the range
Numerical applications are proposed for the normal, Poissonian and Lévy white noise. References
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