Markovian Embedding of Nonlinear Memory via Spectral Representation

Abstract

Differential equations containing memory terms that depend nonlinearly on past states model a variety of non-Markovian processes. In this study, we present a Markovian embedding procedure for such equations with distributed delay by utilising an exact spectral representation of the nonlinear memory function. This allows us to transform the nonlocal system to an equivalent local-in-time system in an abstract extended space. We demonstrate our embedding procedure for two one-dimensional physical models: (i) the walking droplet and (ii) the single-phase Stefan problem. In addition to providing an alternative representation of the underlying physical system, the local representation finds applications in designing efficient time-integrators with time-independent computational costs for memory-dependent systems which typically suffer from growing-in-time costs.

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