Two–dimensional linear partial differential equations in a convex polygon

Abstract

A method is introduced for solving boundary‐value problems for linear partial differential equations (PDEs) in convex polygons. It consists of three algorithmic steps. (1) Given a PDE , construct two compatible eigenvalue equations. (2) Given a polygon , perform the simultaneous spectral analysis of these two equations. This yields an integral representation in the complex k ‐plane of the solution q (x1,x2) in terms of a function q ( k ), and an integral representation in the (x1, x2)‐plane of q( k ) in terms of the values of q and of its derivatives on the boundary of the polygon. These boundary values are in general related, thus only some of them can be prescribed. (3) Given appropriate boundary conditions , express the part of q ( k ) involving the unknown boundary values in terms of the boundary conditions. This is based on the existence of a simple global relation formulated in the complex k ‐plane, and on the invariant properties of this relation. As an illustration, the following integral representations are obtained: (a) q (x, t ) for a general dispersive evolution equation of order n in a domain bounded by a linearly moving boundary; (b) q (x,y) for the Laplace, modified Helmholtz and Helmholtz equations in a convex polygon. These general formulae and the analysis of the associated global relations are used to discuss typical boundary‐value problems for evolution equations and for elliptic equations.