Probabilistic characterization of nonlinear systems under \u3b1-stable white noise via complex fractional moments

The probability density function of the response of a nonlinear system under external \u3b1-stable L\ue9vy white noise is ruled by the so called Fractional Fokker-Planck equation. In such equation the diffusive term is the Riesz fractional derivative of the probability density function of the response. The paper deals with the solution of such equation by using the complex fractional moments. The analysis is performed in terms of probability density for a linear and a non-linear half oscillator forced by L\ue9vy white noise with different stability indexes \u3b1. Numerical results are reported for a wide range of non-linearity of the mechanical system and stability index of the L\ue9vy white noise