  In the present article, two analytical solutions based on the Laplace
transforms method for the linear and non-linear gluon distribution functions
have been presented at low values of $x$. These linear and non-linear methods
are presented based on the solutions of the Dokshitzer-Gribov-
Lipatov-Altarelli-Parisi (DGLAP) evolution equation and the Gribov-Levin-Ryskin
Mueller-Qiu (GLR-MQ) equation at the leading-order accuracy in perturbative QCD
respectively. The gluon distributions are obtained directly in terms of the
parametrization of structure function $F_{2}(x,Q^{2})$ and its derivative and
compared with the results from the parametrization models. The $n_{f}$ changes
at the threshold are considered in the numerical results. The effects of the
non-linear corrections are visible as $Q^{2}$ decreases and vanish as $Q^{2}$
increases. The nonlinear corrections tame the behavior of the gluon
distribution function at low
  $x$ and $Q^{2}$ in comparison with the parametrization models.
