Initial value problems for forced linear and nonlinear partial differe
ntial equations are considered where the forcing is assumed to be rapi
d compared to the unforced dynamics. A multiscale perturbation method
is used to derive solutions in the form of asymptotic expansions. For
linear problems, these expansions are equivalent to the integral formu
las based on Green's function solutions and thus give a method of expa
nding these integral formulas when the kernel is rapidly varying. For
nonlinear problems, the method yields new results for which there am n
o alternative analytical methods. Several model problems are considere
d, including the forced Burgers equation for shock-pulse interactions,
the Korteweg-deVries (KdV) equation for forced interacting solitons,
and the time-dependent Schrodinger equation with rapidly varying poten
tial.