  The study of the phase ordering kinetics of the ferromagnetic one-dimensional
Ising model dates back to 1963 for non conserved order parameter (NCOP) and to
1991 for conserved order parameter (COP). The case of long range interactions
$J(r)$ has been widely studied at equilibrium but their effect on relaxation is
a much less investigated field. Here we make a detailed numerical and
analytical study of both cases, NCOP and COP. Many results are valid for any
positive, decreasing coupling $J(r)$, but we focus specifically on the
exponential case, $J_{exp}(r)=e^{-r/R}$ with varying $R>0$, and on the
integrable power law case, $J_{pow}(r) =1/r^{1+\sigma}$ with $\sigma > 0$. We
find that the {\it asymptotic} growth law $L(t)$ is the usual algebraic one,
$L(t)\sim t^{1/z}$, of the corresponding model with nearest neighborg
interaction ($z_{NCOP}=2$ and $z_{COP}=3$) for all models except $J_{pow}$ for
small $\sigma$: in the non conserved case when $\sigma\le 1$
($z_{NCOP}=\sigma+1$) and in the conserved case when $\sigma\to 0^+$
($z_{COP}=4\beta+3$, where $\beta=1/T$ is the inverse of the absolute
temperature). The models with space decaying interactions also differ markedly
from the ones with nearest neighbors due to the presence of many long-lasting
preasymptotic regimes, such as an exponential mean-field behavior with
$L(t)\sim e^{t}$, a ballistic one with $L(t)\sim t$, a slow (logarithmic)
behavior $L(t)\sim \ln t$ and one with $L(t)\sim t^{1/\sigma+1}$. All these
regimes and their validity ranges have been found analytically and verified in
numerical simulations. Our results show that the main effect of the
conservation law is a strong slowdown of COP dynamics if interactions have an
extended range. Finally, we compare the Ising model at hand with continuum
approaches based on a Ginzburg-Landau free energy.
