Markovian embedding of nonlocal equations using spectral representation

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.