  The conventional generic model is deemed to explain the dynamics of
$\gamma$-ray burst remnants very well, no matter whether they are adiabatic or
highly radiative. However, we find that for adiabatic expansion, the model
could not reproduce the Sedov solution in the non-relativistic phase, thus the
model needs to be revised. In the present paper, a new differential equation is
derived. The generic model based on this equation has been shown to be correct
for both radiative and adiabatic fireballs, and in both ultra-relativistic and
non-relativistic phase.
