Constraining a possible time-variation of the speed of light along with
  the fine-structure constant using strong gravitational lensing and Type Ia
  supernovae observations
  The possible time variation of the fundamental constants of nature has been
an active subject of research since the large-number hypothesis was proposed by
Dirac. In this paper, we propose a new method to investigate a possible time
variation of the speed of light ($c$) along with the fine-structure constant
($\alpha$) using Strong Gravitational Lensing (SGL) and Type Ia Supernovae (SNe
Ia) observations. We assume a general approach to describe the mass
distribution of lens-type galaxies, the one in favor of the power-law index
model (PLAW). We also consider the runaway dilaton model to describe a possible
time-variation of $\alpha$. In order to explore the results deeply, we split
the SGL sample into five sub-samples according to the lens stellar velocity
dispersion and three sub-samples according to lens redshift. The results
suggest that it is reasonable to treat the systems separately, but no strong
indication of varying $c$ was found.
