  The properties are considered in detail of a non-local (integral) equation
for the superconducting gap parameter, which is obtained by a coarse-graining
procedure applied to the Bogoliubov-deGennes (BdG) equations over the whole
coupling-vs-temperature phase diagram associated with the superfluid phase. It
is found that the limiting size of the coarse-graining procedure, which is
dictated by the range of the kernel of this integral equation, corresponds to
the size of the Cooper pairs over the whole coupling-vs-temperature phase
diagram up to the critical temperature, even when Cooper pairs turn into
composite bosons on the BEC side of the BCS-BEC crossover. A practical method
is further implemented to solve numerically this integral equation in an
efficient way, which is based on a novel algorithm for calculating the Fourier
transforms. Application of this method to the case of an isolated vortex,
throughout the BCS-BEC crossover and for all temperatures in the superfluid
phase, helps clarifying the nature of the length scales associated with a
single vortex and the kinds of details that are in practice disposed off by the
coarse-graining procedure on the BdG equations.
