If mathematical developments in physics count as "applied mathematics" there are many examples --- as requested by the OP here is a recent one (< 30 years old) and an older one:

 - Gauge theory spawned fundamental resuls in
   differential geometry, see for example <A
   HREF="https://arxiv.org/abs/1207.6271">A proof via the Seiberg-Witten
   moduli space of Donaldson's theorem on smooth 4-manifolds with
   definite intersection forms</A> (1995)
 - Dirac delta functions were introduced to simplify the normalization of quantum mechanical wave functions, and were formalized by Laurent Schwartz in the theory of distributions.