  We introduce templates for exponential asymptotic expansions that, in
contrast to matched asymptotic approaches, enable the simultaneous satisfaction
of both boundary values in classes of linear and nonlinear equations that are
singularly perturbed with an asymptotic parameter epsilon \to 0+ and have a
single boundary layer at one end of the interval. For linear equations, the
template is a transseries that takes the form of a sliding ladder of
exponential scales. For nonlinear equations, the transseries template is a
two-dimensional array of exponential scales that tilts and realigns asymptotic
balances as the interval is traversed. An exponential asymptotic approach also
reveals how boundary value problems force the surprising presence of
transseries in the linear case and negative powers of epsilon terms in the
series beyond all orders in the nonlinear case. We also demonstrate how these
transseries can be resummed to generate multiple-scales-type approximations
that can generate uniformly better approximations to the exact solution out to
larger values of the perturbation parameter. Finally we show for a specific
example how a reordering of the terms in the exponential asymptotics can lead
to an acceleration of the accuracy of a truncated expansion.
