  The search for a space-time variation of the fundamental constants has been
explored over the years to test our physical theories. In this paper, we use
the dispersion measure ($DM$) of fast radio bursts (FRB) combined with type Ia
supernovae (SNe) data to investigate a possible redshift evolution of the
fine-structure constant ($\alpha$), considering the runaway dilaton scenario,
which predicts $\frac{\Delta \alpha}{\alpha} = - \gamma\ln{(1+z)}$, where
$\gamma$ is a constant proportional to the current value of the coupling
between the dilaton field and hadronic matter. We derive all the relevant
expressions for the $DM$ dependence concerning the fine-structure constant and
constrain the parameter $\gamma$ from measurements of 17 well-localized FRBs
and 1048 SNe data from the Pantheon compilation. We also use Monte Carlo
simulations to forecast the constraining power of larger samples of FRB
measurements for data sets with $N = 500$ and $N = 1000$ points. We found that
the uncertainty on $\gamma$ can be improved by one order of magnitude and that
limits on $\frac{\Delta \alpha}{\alpha}$ beyond $\sigma \sim 10^{-2}$ will
depend crucially on better control of statistical and systematic uncertainties
of upcoming FRB data.
