Keywords: non-linear systems, Einstein-Smoluchowski equation, Fokker-Planck-Kolmogorov equation, poissonian and alpha-stable white noise processes, characteristic function, Laguerre polynomial.

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 (-stable; ) white noise and the Poissonian white noise. The class of Lévy white noise is continuous and characterized by long tails in the probability distribution. The Poissonian white noise is a purely discontinuous process, and it is constituted by a train of Dirac's deltas distributed in time according to Poisson law. Each delta occurrence has random amplitude independent on the random time. The Gaussian white noise is a common point for the two aforementioned processes. In fact the Gaussian white noise may be obtained either by setting in the Lévy white noise, or as the limit of Poisson white noise when , the mean number of impulses per unit time, goes to infinity, and, at the same time, keeps a constant value.

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 , and this allows the use of classical approach based on the weighted residual method using classical series expansions. In this paper it is shown that by expanding the unknown characteristic function in terms of orthogonal polynomials and residual weighted method, the CF of the response may be easily obtained if properly polynomials are properly selected. In particular for the case of Poisson white noise the best choice is a modification of Hermite polynomials, while for -stable Lévy white noise the best choice is the Laguerre polynomials expansion of the characteristic function.

Numerical applications are proposed for the normal, Poissonian and Lévy white noise.

1

Grigoriu, M., "White noise processes", Journal Engineering Mechanics, ASCE, 113, 757-765, 1978. doi:10.1061/(ASCE)0733-9399(1987)113:5(757)

2

Grigoriu, M., "Linear and nonlinear systems with non-Gaussian white noise input", Probabilistic Engineering Mechanics, 10, 171-179, 1995. doi:10.1016/0266-8920(95)00014-P

3

Grigoriu, M., "Applied Non-Gaussian Processes: Example Theory, Simulation, Linear Random Vibration and Method solution", Prentice-Hall, Engewood Cliffs, NS, 1995.

4

Di Paola, M., "Stochastic differential calculus, in Casciati (Ed.), Dynamic Motion: Chaotic and Stochastic Behavior", Springer-Verlag, Vienna, 29-92, 1993.

5

Sobzick, K., "Stochastic differential equation", Kluwer Academic Publishers, 1991.

6

Itô, K., "On a formula concerning stochastic differentials", Nagoya Mathematical Journal, 3, 55-65, 1951.

7

Di Paola M., Pirrotta A., Zingales M., "Non-Stationary Probabilistic Analysis of Dynamical Systems Under Delta-Correlated White Noise". Proceedings of the 4th International Conference on Structural Dynamics, EURODYN2002, Munich, Germany, 767-772, 2002.

8

Wen, Y.K., "Approximate method for non linear random vibration". Journal of Engineering Division ASCE 101, 389-401, 1975.

9

Von Wagner, U. and Wedig, W.V., "On the calculation of stationary solutions of multi-dimensional Fokker-Plank equations by orthogonal functions", Non linear Dynamics, 21, 289-306, 2000. doi:10.1023/A:1008389909132

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £135 +P&P)