An analytic solution to LO coupled DGLAP evolution equations for $F_2^{\gamma p}(x,Q^2)$ and $G(x,Q^2)$

We have analytically solved the leading-order Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations for the $Q^2$ evolution of the structure function $F_2^{\gamma p}(x,Q^2)$ and the gluon distribution $G(x,Q^2)$ using very efficient Laplace transform techniques. Employing the initial distributions $F_2^{\gamma p}(x,Q_0^2)$ and $G(x,Q_0^2)$ fixed by a global fit to the HERA combined data on deep inelastic $ep$ scattering and the very small non-singlet MSTW2008LO distributions at the initial energy $Q_0^2$, we have calculated the evolved proton structure function $F_2^{\gamma p}(x,Q^2)$ and compared our results with the HERA combined data.