The post-burst evolution of fireballs that produce gamma-ray bursts (G
RBs) is studied, assuming the expansion of fireballs to be adiabatic a
nd relativistic. Numerical results as well as an approximate analytic
solution for the evolution are presented. Owing to the adoption of a n
ew relation between t, R and gamma, our results differ markedly from p
revious studies. Synchrotron radiation from the shocked interstellar m
edium is carefully calculated, using a conventional set of equations,
The observed X-ray flux of GRB afterglows can be reproduced easily. Al
though the optical afterglows seem much more complicated, our results
can still present a rather satisfactory agreement with observations. W
e also find that the expansion will no longer be highly relativistic a
bout 4 d after the main GRB. We thus suggest that the marginally relat
ivistic phase of the expansion should be investigated so as to check t
he afterglows observed a week or more later.