  The nature of the progenitors of Type Ia supernovae (SNe Ia) is still
unclear. In this paper, by considering the effect of the instability of
accretion disk on the evolution of white dwarf (WD) binaries, we performed
binary evolution calculations for about 2400 close WD binaries, in which a
carbon--oxygen WD accretes material from a main-sequence star or a slightly
evolved subgiant star (WD + MS channel), or a red-giant star (WD + RG channel)
to increase its mass to the Chandrasekhar (Ch) mass limit. According to these
calculations, we mapped out the initial parameters for SNe Ia in the orbital
period--secondary mass ($\log P^{\rm i}-M^{\rm i}_2$) plane for various WD
masses for these two channels, respectively. We confirm that WDs in the WD + MS
channel with a mass as low as $0.61 M_\odot$ can accrete efficiently and reach
the Ch limit, while the lowest WD mass for the WD + RG channel is $1.0 \rm
M_\odot$. We have implemented these results in a binary population synthesis
study to obtain the SN Ia birthrates and the evolution of SN Ia birthrates with
time for both a constant star formation rate and a single starburst. We find
that the Galactic SN Ia birthrate from the WD + MS channel is $\sim$$1.8\times
10^{-3} {\rm yr}^{-1}$ according to our standard model, which is higher than
previous results. However, similar to previous studies, the birthrate from the
WD + RG channel is still low ($\sim$$3\times 10^{-5} {\rm yr}^{-1}$). We also
find that about one third of SNe Ia from the WD + MS channel and all SNe Ia
from the WD + RG channel can contribute to the old populations ($\ga$1 Gyr) of
SN Ia progenitors.
