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 HERA data from the inclusive structure function F${}_2$(x, Q${}^2$) for x ≤ 10${}^{-2}$ and 5 ≤ Q${}^2$ ≤ 250 GeV${}^2$.