Derivation and physical interpretation of general boundary conditions

Abstract

In this paper we give new derivations of the heat and wave equation which incorporate the boundary conditions into the formulation of the problems. The principle of least action and the inclusion of a kinetic energy contribution on the boundary are used to derive the wave equation together with kinetic boundary conditions. The methods described for both equations admit all of the standard boundary conditions as well as general Wentzell and dynamic boundary conditions; in addition the boundary conditions arise naturally as part of the formulation of the problems. The physical interpretation for general Wentzell boundary conditions is given for both the heat and wave equations.