On constructing boundaries for boundary value problems defined by continuous 2-D autonomous systems

Abstract

Abstract In this paper, we provide a constructive way of specifying initial/boundary data for a given continuous 2-D autonomous system described by a set of linear partial differential equations (PDEs) with real constant coefficients. One of the ways of specifying initial/boundary data is by specifying the values of various derivatives of the solution trajectories at the origin; the derivatives correspond to a standard monomial set obtained using Grobner basis. However, such an initial/boundary data often lacks physical interpretation. In this paper, we consider subsets of the domain having some algebraic structure (in the form of subspaces and strips of finite width around such subspaces) such that trajectories restricted to these subsets, often called characteristic sets, serve as initial/boundary conditions for the given autonomous system of linear PDEs. We provide a systematic way to construct such characteristic sets with the help of Grobner bases and Oberst-Riquier algorithm. Thus we bridge the gap between initial/boundary conditions involving standard monomials and more conventional initial/boundary conditions in the form of restrictions on characteristic sets. We also show that every scalar system of PDEs admits such a characteristic set given by a rectangular strip of finite width around a subspace whose dimension equals the Krull dimension of the system’s quotient ring.