Title: The Initial-Boundary Value Problem for the Linearized Classical Boussinesq System with Nonzero Boundary Conditions on the Half-Line
Abstract: Although the initial value problem for linear evolution PDEs is well-understood, several open questions remain when one considers these PDEs in the initial-boundary value problem (IBVP) setting. Moreover, the analysis of IBVPs for systems of linear evolution PDEs is even less explored. In this talk, we solve analytically the linearized classical Boussinesq system, which is an important approximation to the Euler equations of hydrodynamics, on the half-line with nonzero boundary conditions. Our analysis relies on the unified transform of Fokas, which is a method specifically developed for solving linear IBVPs and which was recently extended to apply to systems of linear equations. We will establish an explicit formula that provides a novel representation for the solution which, thanks to being uniformly convergent at the boundary, can be directly shown to satisfy the linearized classical Boussinesq system as well as the prescribed initial and boundary data.