Higher-order averaging of Fokker-Planck equations for nonlinear fiber lay-down processes
We investigate stochastic models for the lay-down of flexible fibers on a moving conveyor belt for the case of small stochasticity and small belt velocity. Using a suitable scaling we obtain a singularly perturbed problem for a stochastic Hamiltonian system and the corresponding Fokker-Planck equation. These equations are investigated using averaging methods. In the present case the impact of the deterministic forcing, i.e., the belt velocity is not captured by the stochastic averaging theorem. To overcome this problem, a formal energy projection method previously introduced by the authors is used, which allows the computation of higher-order stochastic averages for the present highly nonlinear system. The resulting second-order coefficients are computed numerically and a suitably precise scheme is developed. Finally, we illustrate the results of our method with several examples.