On the transport equations of cosmic neutrinos passing through Earth and secondary nu_mu fluxes

The convergence of the iterative solutions of the transport equations of cosmic muon and tau neutrinos propagating through Earth is studied and analyzed. For achieving a fast convergence of the iterative solutions of the coupled transport equations of nu_tau, nubar_tau and the associated tau^{\pm} fluxes, a new semi-analytic input algorithm is presented where the peculiar tau-decay contributions are implemented already in the initial zeroth order input. Furthermore, the common single transport equation for muon neutrinos is generalized by taking into account the contributions of secondary nu_mu and nubar_mu fluxes due to the prompt tau-decay tau -> nu_mu initiated by the associated tau flux. Differential and total nadir angle integrated upward-going mu^- + mu^+ event rates are presented for underground neutrino telescopes and compared with the muon rates initiated by the primary nu_mu, nu_tau and tau fluxes.