Time-periodic linear boundary value problems on a finite interval

We study the large time behaviour of the solution of linear dispersive partial differential equations posed on a finite interval, when at least one of the prescribed boundary conditions is time periodic. We use the Q equation approach, pioneered in Fokas & Lenells 2012 and applied to linear problems on the half-line in Fokas & van der Weele 2021, to characterise necessary conditions for the solution of such problem to be periodic, at least in an asymptotic sense. We then fully describe the periodicity properties of the solution in three important illustrative examples, recovering known results for the second-order cases and establishing new results for the third order case.