Decoupling the NLO coupled QED $\otimes$ QCD, DGLAP evolution equations,Using Laplace Transform Method

We analytically solved the QED $\otimes$ QCD coupled DGLAP evolution equations at leading order (LO) quantum electrodynamics (QED) and next to leading order (NLO) quantum chromodynamics (QCD) approximations, using the Laplace transform method and then computed the proton structure function in terms of the unpolarized parton distributions functions. Our analyitical solutions for parton densities are in good agreement with those from APFEL (A PDF Evolution Library) (Computer Physics Communications 185, 1647-1668 (2014)) and CT14QED (Phys. Rev. D 93, 114015 (2016)) global parameterizations. We also compared the proton structure function, $F_{2}^{p}(x,Q^{2})$, with experimental data released by the ZEUS and H1 collaborations at HERA. There is a nice agreement between them in the range of low and high x and $Q^{2}$.