On Rational Boundary Conditions for Higher-Order Long-Wave Models

Abstract

Higher-order corrections to classical long-wave theories enable simple and efficient modelling of the onset of wave dispersion and size effects produced by underlying micro-structure. Since such models feature higher spatial derivatives, one needs to formulate additional boundary conditions when confined to bounded domains. There is a certain controversy associated with these boundary conditions, because it does not seem possible to justify their choice by purely physical consid- erations. In this paper an asymptotic model for one-dimensional chain of particles is chosen as an exemplary higher-order theory. We demonstrate how the presence of higher-order derivative terms results in the existence of non-physical extraneous boundary layer-type solutions and argue that the additional boundary conditions should generally be formulated to eliminate the contribution of these boundary lay- ers into the averaged solution. Several new methods of deriving additional boundary conditions are presented for essential boundaries. The results are illustrated by nu- merical examples featuring comparisons with an exact solution for the finite chain.