  The postburst evolution of fireballs that produce $\gamma$-ray bursts is
studied, assuming the expansion of fireballs to be adiabatic and relativistic.
Numerical results as well as an approximate analytic solution for the evolution
are presented. Due to adoption of a new relation among $t$, $R$ and $\gamma$
(see the text), our results differ markedly from the previous studies.
Synchrotron radiation from the shocked interstellar medium is attentively
calculated, using a convenient set of equations. The observed X-ray flux of GRB
afterglows can be reproduced easily. Although the optical afterglows seem much
more complicated, our results can still present a rather satisfactory approach
to observations. It is also found that the expansion will no longer be highly
relativistic about 4 days after the main GRB. We thus suggest that the
marginally relativistic phase of the expansion should be investigated so as to
check the afterglows observed a week or more later.
