  We describe the determination of the heavy quarks structure functions
$F_{2,L}^{\mathcal{Q}\bar{\mathcal{Q}}}$ with help of the scaling properties.
We observe that the structure functions for inclusive charm and bottom
production exhibits geometric scaling at low $x$. The geometrical scaling means
that the heavy quark structure function is a function of only one dimensionless
variable $\tau{\equiv}Q^{2}/Q^{2}_{sat}(x)$ including quark mass. These results
are valid for any value of $\delta$, being $x^{-\delta}$ the behavior of the
parton densities at low $x$. The determination of the heavy quark structure
function is presented as a parameterization of the proton structure function
$F_{2}(x,Q^{2})$ and its derivative. Analytical expressions for
$F_{2,L}^{\mathcal{Q}\bar{\mathcal{Q}}}$ in terms of the effective parameters
of the parameterization of $F_{2}(x,Q^{2})$ ,with respect to the BDH and ASW
models, are presented. To study the heavy quark production processes, we use
the collinear results in DAS approach. Numerical calculations and comparison
with HERA data demonstrate that the suggested method provides reliable
$F_{2}^{\mathcal{Q}\bar{\mathcal{Q}}}$, $R^{c\bar{c}}$ and
$\sigma^{\mathcal{Q}\bar{\mathcal{Q}}}$ at low $x$ in a wide range of the low
absolute four-momentum transfers squared
($4~\mathrm{GeV}^{2}<Q^{2}<2000~\mathrm{GeV}^{2}$). Also, in the HERA kinematic
range, the ratio of $F_{2}^{c\bar{c}}/F_{2}^{b\bar{b}}$,
$F_{2}^{c\bar{c}}/F_{2}^{BDHM}$ and $F_{2}^{b\bar{b}}/F_{2}^{BDHM}$ are
obtained. Expanding the method to low and ultra low values of $x$ can be
considered in the process analysis of the LHeC and FCC-eh colliders.
