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We analytically solved the QED $\otimes$ QCD coupled DGLAP evolutionequations at leading order (LO) quantum electrodynamics (QED) and next toleading order (NLO) quantum chromodynamics (QCD) approximations, using theLaplace transform method and then computed the proton structure function interms of the unpolarized parton distributions functions. Our analyiticalsolutions for parton densities are in good agreement with those from APFEL (APDF Evolution Library) (Computer Physics Communications 185, 1647-1668 (2014))and CT14QED (Phys. Rev. D 93, 114015 (2016)) global parameterizations. We alsocompared the proton structure function, $F_{2}^{p}(x,Q^{2})$, with experimentaldata released by the ZEUS and H1 collaborations at HERA. There is a niceagreement between them in the range of low and high x and $Q^{2}$.

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