  The Fick--Jacobs equation has been widely studied, because of its
applications in the diffusion and transport of non-interacting particles in
narrow channels. It is also known that a modified version of this equation can
be used to describe the same system with particles interacting through a
hard-core potential. In this work we present a system that can be exactly
solved using the Fick--Jacobs equation. The exact results of the particle
concentration profile along the channel $n$, the current, $J$, and the
mobility, $\mu$, of particles as a function of an external force are contrasted
with Monte Carlo simulations results of non-interacting particles. For
interacting particles the behavior of $n$, $J$ and $\mu$, obtained from the
modified Fick--Jacobs equation are in agreement with numerical simulations,
where the hard-core interaction is taken into account. Even more, for
interacting particles the modified Fick--Jacobs equation gives comparatively
more accurate results of the current difference (when a force is applied in
opposite directions) than the exact result for the non-interacting ones.
