  We study the synchronous stochastic dynamics of the random field and random
bond Ising chain. For this model the generating functional analysis methods of
De Dominicis leads to a formalism with transfer operators, similar to transfer
matrices in equilibrium studies, but with dynamical paths of spins and
(conjugate) fields as arguments, as opposed to replicated spins. In the
thermodynamic limit the macroscopic dynamics is captured by the dominant
eigenspace of the transfer operator, leading to a relative simple and
transparent set of equations that are easy to solve numerically. Our results
are supported excellently by numerical simulations.
