Title: Linear diffusion under the presence of a boundary
Abstract: The linear diffusion (heat) equation is one of the most well-known models in applied mathematics. At the same time, thanks to its simplicity, it also serves as a standard textbook model in any introductory partial differential equations course. Indeed, concepts like separation of variables, Fourier series and the Fourier transform are usually introduced to the students through examples involving the linear diffusion equation. Despite its popularity, not much is discussed about the equation in settings outside those few standard textbook examples. Indicatively, one might ask: (1) What if the boundary conditions are non-zero? (2) What if the boundary conditions are non-separable (a pretty common scenario in physical applications)? (3) What about the lack of uniform convergence of the standard solution representations at the boundary? This talk aims to shed some light on the above questions by discussing certain non-separable boundary conditions for the linear diffusion equation on the half-line and the finite interval via the so-called unified transform method (a.k.a. the Fokas method) for the solution of initial-boundary value problems.