Analytical response and Markovianity of systems governed by fractional differential equations driven by α-stable white noise processes

This paper investigates the probabilistic response of systems governed by fractional differential equations and forced by a-stable white noise processes. The response in terms of scale of the response process is built starting from a finite differences version of the governing equation. An efficient formulation in terms of scale of the response process is obtained for both non-stationary and stationary white noise input process. Remarkably, the proposed numerical scheme has an analytical counterpart in terms of time evolution of the scale that is a generalization of the well known Duhamel integral. This analytical formulation permits to find closed form solutions in terms of the time evolution of the scale of the response process. Further, when the input white noise process is stationary, it is demonstrated that the system remains Markovian of order one also in presence of fractional operators in the governing equation. This is a very interesting result that makes the proposed numerical procedure very efficient from a computational point of view and that can be very useful for future developments. Numerical applications demonstrate the accuracy of the proposed numerical scheme and the validity of the analytical solution for some systems of interest in engineering.