  The gluon distribution is obtained from the
Golec-Biernat-W$\ddot{\mathrm{u}}$sthoff (GBW) and Bartels, Golec-Biernat and
Kowalski (BGK) models at low $x$. We derive analytical results for the
unintegrated color dipole gluon distribution function at small transverse
momentum, which provides useful information to constrain the $k_{t}$-shape of
the unintegrated gluon distribution in comparison with the unintegrated gluon
distribution (UGD) models. The longitudinal proton structure function
$F_{L}(x,Q^2)$ from the $k_{t}$ factorization scheme, using the unintegrated
gluon density is computed. We compare the predictions for the on-shell and
twist-2 corrections with the HERA data and the CJ15 parametrization method for
$F_{L}$. We show that this method is very well described the experimental data
within the on-shell and twist-2 framework. Effects of parameters on $F_{L}$,
where charm contribution is taken into account, are investigated. These results
are in good agreement with the data at fixed $W$.
