We describe a new form of diagonalization for linear two point constant coefficient differential operators with arbitrary linear boundary conditions. Although the diagonalization is in a weaker sense than that usually employed to solve initial boundary value problems (IBVP), we show that it is sufficient to solve IBVP whose spatial parts are described by such operators. We argue that the method described may be viewed as a reimplementation of the Fokas transform method for linear evolution equations on the finite interval. The results are extended to multipoint and interface operators, including operators defined on networks of finite intervals, in which the coefficients of the differential operator may vary between subintervals, and arbitrary interface and boundary conditions may be imposed; differential operators with piecewise constant coefficients are thus included.
BIO: Dave Smith is an Applied Mathematician working at Yale-NUS College, Singapore since 2016. He completed his PhD at University of Reading UK, and held postdoctoral positions at the Universities of Crete, Cincinnati and Michigan. Much of his work in the fields of Partial Differential Equations and Spectral Theory is coauthored with undergraduate students.