Analytic Approach to the Approximate Solution of the Independent DGLAP Evolution Equations with Respect to the Hard-Pomeron Behavior

We show that it is possible to use hard-Pomeron behavior to the gluon distribution and singlet structure function at low $x$. We derive a second-order independent differential equation for the gluon distribution and the singlet structure function. In this approach, both singlet quarks and gluons have the same high-energy behavior at small $x$. These equations are derived from the next-to-leading order DGLAP evolution equations. All results can be consistently described in the framework of perturbative QCD, which shows an increase of gluon distribution and singlet structure functions as $x$ decreases.