Solution of QCD$\otimes$QED coupled DGLAP equations at NLO

Abstract: In this work, we present an analytical solution for QCD$\otimes$QED coupled Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations at the leading order (LO) accuracy in QED and next-to-leading order (NLO) accuracy in perturbative QCD using double Laplace transform. This technique is applied to obtain the singlet, gluon and photon distribution functions and also the proton structure function. We also obtain contribution of photon in proton at LO and NLO at high energy and successfully compare the proton structure function with HERA data \cite{12} and APFEL results \cite{7}. Some comparisons also have been done for the singlet and gluon distribution functions with the MSTW results\cite{9}. In addition, the contribution of photon distribution function inside the proton has been compared with results of MRST \cite{11} and with the contribution of sea quark distribution functions which obtained by MSTW \cite{9} and CTEQ6M \cite{14}.