The role of periodicity in the solution of third order boundary value problems

In this short paper, we elucidate how the solution of certain illustrative boundary value problems for the Airy equation $u_t+u_{xxx}=0$ on $[0,1]$ can be expressed as a perturbation of the solution of the purely periodic problem. The motivation is to understand the role boundary conditions play in the properties of the solution. This is particularly important in related work on the solution of linear dispersive problems with discontinuous initial data and the phenomena of revivals and fractalization.