Time-periodic linear boundary value problems on a finite interval

We study the large time behaviour of the solution of a linear dispersive PDEs posed on a finite interval, when the prescribed boundary conditions are time periodic. We use the approach pioneered by A. S. Fokas and J. Lenells in The unified method: II. NLS on the half-line with t-periodic boundary conditions, J. Phys. A 45 (2012) for nonlinear integrable PDEs and then applied to linear problems on the halfline in A. S. Fokas and M. C. van derWeele, The unified transform for evolution equations on the half-line with time-periodic boundary conditions, Stud. Appl. Math. 147 (2021) to characterise necessary conditions for the solution of such a 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 ones for the third order one.