Fractional differential equations under Poissonian white noise input: Transient response and probability density function

Fractional differential equations driven by stochastic excitation are widely used in several engineering and physical applications, as they provide an effective framework for modelling systems characterized by hereditary behavior, memory effects, and nonlocal interactions under random inputs. Although the response probability density function (PDF) of fractional differential equations excited by Gaussian white noise has been extensively investigated, this is not the case for fractional differential equations driven by Poissonian white noise. Indeed, while the analysis of integer-order differential equations subjected to Poissonian white noise has been widely addressed, the few available studies concerning fractional differential equations under Poissonian white noise excitation are mainly restricted to the stationary component of the response PDF. To overcome these limitations, this paper proposes a continuous-time method to obtain the PDF of the response of two-term fractional differential equations subjected to Poissonian white noise excitation. In the proposed method, the response process is first represented by means of the method of iterated kernels. This representation is then employed to derive the characteristic function of the response process, from which the PDF is evaluated through inverse Fourier transform. To assess the accuracy of the proposed method, numerical simulations have been performed on two different structural systems. The results show an excellent agreement between the PDF obtained through the proposed method and that evaluated by Monte Carlo simulation.