On the numerical solution of fractional differential equations under white noise processes

Abstract

The aim of this paper is to elucidate some relevant aspects concerning the numerical solution of stochastic differential equations in structural and mechanical applications. Specifically, the attention is focused on those differential problems involving fractional operators to model the viscoelastic behavior of the structural/mechanical components and involving a white noise process as stochastic input. Starting from the consideration that the Grünwald–Letnikov based integration scheme, that is a step-by-step procedure often invoked in literature to discretize and integrate the aforementioned differential equations, is not properly employed due to the discontinuous nature of the input, an alternative numerical integration scheme is proposed. The latter is based on the Riemann–Liouville fractional integral and relies on the parabolic piecewise approximation of the response function to be integrated, leading to a more effective and more advantageous solution than that provided by the Grünwald–Letnikov based integration scheme. This is demonstrated analyzing the case study of a fractional Euler–Bernoulli beam and comparing the numerical results with those obtained by an analytical solution available in literature.