  We present an analytical approach to the dynamical evolution of fireballs or
axially symmetric jets expanding into an external medium, with application to
Gamma-Ray Burst remnants. This method leads to numerical calculations of
fireball dynamics that are computationally faster than hydrodynamic
simulations. It is also a very flexible approach, that can be easily extended
to include more complex situations, such as a continuous injection of energy at
the reverse shock, and the sideways expansion in non-spherical ejecta.
  Some features of the numerical results for the remnant dynamical evolution
are discussed and compared to the analytical results. We find that the ratio of
the observer times when the jet half-angle reaches twice its initial value and
when the light cone becomes wider than the jet is substantially smaller than
predicted analytically. The effects arising from the remnant curvature and
thickness further reduce this ratio, such that the afterglow light-curve break
due to the remnant finite angular extent overlaps the weaker break due to the
jet's sideways expansion.
  An analysis of the effects of the curvature and thickness of the remnant on
the afterglow light-curves shows that these effects are important and should be
taken into account for accurate calculations of the afterglow emission.
