  While the Gross--Pitaevskii equation is well-established as the canonical
dynamical description of atomic Bose-Einstein condensates (BECs) at
zero-temperature, describing the dynamics of BECs at finite temperatures
remains a difficult theoretical problem, particularly when considering
low-temperature, non-equilibrium systems in which depletion of the condensate
occurs dynamically as a result of external driving. In this paper, we describe
a fully time-dependent numerical implementation of a second-order,
number-conserving description of finite-temperature BEC dynamics. This
description consists of equations of motion describing the coupled dynamics of
the condensate and non-condensate fractions in a self-consistent manner, and is
ideally suited for the study of low-temperature, non-equilibrium, driven
systems. The \delta-kicked-rotor BEC provides a prototypical example of such a
system, and we demonstrate the efficacy of our numerical implementation by
investigating its dynamics at finite temperature. We demonstrate that the
qualitative features of the system dynamics at zero temperature are generally
preserved at finite temperatures, and predict a quantitative finite-temperature
shift of resonance frequencies which would be relevant for, and could be
verified by, future experiments.
