  A spatially independent model that calculates the time evolution of the
electron spectrum in a spherically expanding pulsar wind nebula (PWN) is
presented, allowing one to make broadband predictions for the PWN's non-thermal
radiation. The source spectrum of electrons injected at the termination shock
of the PWN is chosen to be a broken power law. In contrast to previous PWN
models of a similar nature, the source spectrum has a discontinuity in
intensity at the transition between the low and high-energy components. To test
the model, it is applied to the young PWN G21.5--0.9, where it is found that a
discontinuous source spectrum can model the emission at all wavelengths better
than a continuous one. The model is also applied to the unidentified sources
HESS J1427--608 and HESS J1507--622. Parameters are derived for these two
candidate nebulae that are consistent with the values predicted for other PWNe.
For HESS J1427--608 a present-day magnetic field of $B\s{age}=0.4\,\mu\text{G}$
is derived. As a result of the small present-day magnetic field, this source
has a low synchrotron luminosity, while remaining bright at GeV/TeV energies.
It is therefore possible to interpret HESS J1427--608 within the ancient PWN
scenario. For the second candidate PWN HESS J1507--622, a present-day magnetic
field of $B\s{age}=1.7\,\mu\text{G}$ is derived. Furthermore, for this
candidate PWN a scenario is favoured in the present paper in which HESS
J1507--622 has been compressed by the reverse shock of the supernova remnant.
