  The non-linear corrections (NLC) to the longitudinal structure function in a
limited approach is derived at low values of the Bjorken variable $x$ by using
the Laplace transforms technique. The non-linear behavior of the longitudinal
structure function is determined with respect to the Gribov-Levin-Ryskin
Mueller-Qiu (GLR-MQ) and Altarelli-Martinelli (AM) equations. These results
show that the non-linear longitudinal structure function can be determined
directly in terms of the parametrization of $F_{2}(x,Q^{2})$ and the derivative
of the proton structure function with respect to $\ln{Q^{2}}$. These
corrections improve the behavior of the longitudinal structure function at low
values of $Q^{2}$ in comparison with other parametrization methods.
