Solutions of Independent DGLAP Evolution Equations for the Gluon Distribution and Singlet Structure Functions in the Next-to-Leading Order at Low x

We present a set of independent formulae to extract the gluon distribution and singlet structure function from its derivatives with respect to $lnQ^{2}$ in the next- to- leading order of perturbation theory at low-$x$ based on a hard pomeron exchange. In this approach, both singlet quarks and gluons have the same high-energy behavior at small $x$. This approach needs the QCD input parameterizations for a independent DGLAP evolutions that we calculated numerically and compared with MRST, GRV and DL model- the Pomeron has a hard nature. Its evolution gives a good fit to the experimental data. The obtained values are in the range $10^{-4}\hspace{0.1cm}{\leq}\hspace{0.1cm}x\hspace{0.1cm}{\leq}10^{-2}$ at $Q^{2}=20 \hspace{0.1cm}GeV^{2}$.