Stochastic response of linear and non-linear systems to α-stable Lévy white noises

Abstract

The stochastic response of linear and non-linear systems to external α-stable Lévy white noises is investigated. In the literature, a differential equation in the characteristic function (CF) of the response has been recently derived for scalar systems only, within the theory of the so-called fractional Einstein–Smoluchowsky equations (FESEs). Herein, it is shown that the same equation may be built by rules of stochastic differential calculus, previously applied by one of the authors to systems driven by arbitrary delta-correlated processes. In this context, a straightforward formulation for multi-degree-of-freedom (MDOF) systems is also developed. Approximate CF solutions to the derived equation are sought for polynomial non-linearities, in stationary conditions. To this aim a wavelet representation is used, in conjunction with a weighted residual method. Numerical results prove in excellent agreement with exact solutions, when available, and digital simulation data.