  Nonlocal evolutionary equations containing memory terms model a variety of
non-Markovian processes. We present a Markovian embedding procedure for a class
of nonlocal equations by utilising the spectral representation of the nonlinear
memory kernel. This allows us to transform the nonlocal system to a
local-in-time system in an abstract extended space. We demonstrate our
embedding procedure and its efficacy for two different physical models, namely
the (i) 1D walking droplet and (ii) the 1D single-phase Stefan problem.
