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 $ln{Q}^2$ 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 ${10}^{-4}\le x\le {10}^{-2}$ at $Q^2=20Ge{V}^2$.