  We study the 1D kinetics of diffusion-limited coalescence and annihilation
with back reactions and different kinds of particle input. By considering the
changes in occupation and parity of a given interval, we derive sets of
hierarchical equations from which exact expressions for the lattice coverage
and the particle concentration can be obtained. We compare the mean-field
approximation and the continuum approximation to the exact solutions and we
discuss their regime of validity.
