Calculation of the triple and double cross sections in the proton-hydrogen impact at the ionization channel by removing singularity in the scattering amplitudes

In the present work, the triple and double cross sections of atomic hydrogen ionization in collision with the proton are  calculated. The potential of the interaction is considered as a Coulombic;  also,  the first and second order Born approximations have been used. The singularity of the Green operator in the second order approximation has been removed for the analytical solution of the scattering amplitude and the triple cross section calculation. Finally, the results for triple and double cross sections have been compared with the available theoretical and experimental results. The first and the second Born approximations are usually applied at intermediate and high impact energy ranges. However, it is  shown that it could be successfully applied at lower energies as well.