  In this work, we present a leptonic, time-dependent model of pulsar wind
nebulae (PWNe). The model seeks a solution for the lepton distribution function
considering the full time-energy dependent diffusion-loss equation. The
time-dependent lepton population is balanced by injection, energy losses, and
escape. We include synchrotron, inverse Compton (IC, with the cosmic-microwave
background as well as with IR/optical photon fields), self-synchrotron Compton
(SSC), and bremsstrahlung processes, all devoid of any radiative
approximations. With this model in place we focus on the Crab nebula as an
example and present its time dependent evolution. Afterwards, we analyze the
impact of different approximations made at the level of the diffusion-loss
equation, as can be found in the literature. Whereas previous models ignored
the escape term, e.g., with the diffusion-loss equation becoming advective,
others approximated the losses as catastrophic, so that the equation has only
time derivatives. Additional approximations are also described and computed. We
show which is the impact of these approaches in the determination of the PWN
evolution. In particular, we find the time-dependent deviation of the
multi-wavelength spectrum and the best-fit parameters obtained with the
complete and the approximate models.
