Extensions of the d’Alembert formulae to the half line and the finite interval obtained via the unified transform

Abstract

Abstract We derive the solution of the one-dimensional wave equation for the Dirichlet and Robin initial-boundary value problems (IBVPs) formulated on the half line and the finite interval, with non-homogeneous boundary conditions. Although explicit formulas already exist for these problems, the unified transform method provides a convenient framework for deriving different representations of the solutions for these and other types of IBVPs. Specifically, it provides solution formulas in the Fourier space or solutions which constitute the extension of the classical formula of d’Alembert of the initial value problem on the full line. We also derive the solution of the forced wave equation on the half line.