The behavior of the structure function by using the effective exponent at low-x

An analytical solution of the QCD evolution equations for the singlet and gluon distribution is presented. We decouple DGLAP evolution equations into the initial conditions by using a Laplace transform method at $N^{n}LO$ analysis. The relationship between the nonlinear behavior and color dipole model is considered based on an effective exponent behavior at low-$x$ values. We obtain the effective exponent at NLO analysis from the decoupled behavior of the distribution functions. The proton structure function compared with H1 data from the inclusive structure function $F_{2}(x,Q^{2})$ for $x {\leq}~ 10^{-2}$ and $5 {\leq} Q^{2} {\leq} 250~GeV^{2}$.