Improved Filon-Type Asymptotic Methods for Highly Oscillatory Differential Equations
This chapter presents an effective improvement on the existing Filon-type asymptotic methods, so that the improved methods can numerically solve a class of multi-frequency highly oscillatory second-order differential equations with a positive semi-definite singular matrix which implicitly contains and preserves the oscillatory frequencies of the underlying problem.
We study non-equilibrium velocity fluctuations in a model for the sedimentation of non-Brownian particles experiencing long-range hydrodynamic interactions. The complex behavior of these fluctuations, the outcome of the collective dynamics of the particles, exhibits many of the features observed in sedimentation experiments. In addition, our model predicts a final relaxation to an anisotropic (hydrodynamic) diffusive state that could be observed in experiments performed over longer time ranges.