  We present non-linear corrections (NLC) to the distribution functions at low
values of $x$ and $Q^{2}$ using the parametrization $F_{2}(x,Q^{2})$ and
$F_{L}(x,Q^{2})$. We use a direct method to extract non-linear corrections to
the ratio of structure functions and the reduced cross section in the
next-to-next-to-leading order (NNLO) approximation with respect to the
parametrization method (PM). Comparison between the non-linear results with the
bounds in color dipole model (CDM) and HERA data indicate the consistency of
the non-linear behavior of the gluon distribution function at low-$x$ and
low-$Q^{2}$. The non-linear longitudinal structure functions are comparable
with the H1 Collaboration data in a wide range of $Q^{2}$ values. Consequently,
the non-linear corrections at NNLO approximation to the reduced cross sections
at low and moderate $Q^{2}$ values show good agreement with the HERA combined
data. These results at low $x$ and low $Q^{2}$ can be applied to the LHeC
region for analyses of ultra-high-energy processes.
