Photoproduction of $J/psi$ in p-p ultra-peripheral collisions at LHC energy

$J/psi$ production in p-p ultra-peripheral collisions through the elastic and inelastic photoproduction processes, where the virtual photons emitted from the projectile interact with the target, are studied. The comparisions between the exact treatment results and the ones of equivalent photon approximation are expressed as $Q^{2}$ (virtuality of photon), $z$ and $p_{T}$ distributions, and the total cross sections are also estimated. The method developed by Martin and Ryskin is employed to avoid double counting when the different production mechanism are considered simultaneously. The numerical results indicate that, the equivalent photon approximation can be only applied to the coherent or elastic electromagnetic process, the improper choice of $Q^{2}_{mathrm{max}}$ and $y_{mathrm{max}}$ will cause obvious errors. And the exact treatment is needed to deal accurately with the $J/psi$ photoproduction.