Abstract: A set of coupled nonlinear differential equations of the FEL amplifier are derived in three dimensions for a configuration consisting of a coaxial waveguide and a coaxial hybrid wiggler. These equations include the effects of electron momentum spread, space-charge effects and wiggler taper governing the self-consistent evolution of TE mode and trajectories of an ensemble of electrons. The initial conditions are chosen to model the adiabatic injection of a annular electron beam. Computer program based on these equations is made and nonlinear including saturation efficiency and bandwidth is analyzed.