A Transform Method for Evolution PDEs on a Finite Interval

Abstract

We study initial boundary value problems for linear scalar partial differential equations with constant coefficients, with spatial derivatives of {\em arbitrary order}, posed on the domain $\{t>0, 0<x<L\}$. We first show that by analysing the so-called {\em global relation}, which is an algebraic relation defined in the complex $k$-plane coupling all boundary values of the solution, it is possible to identify how many boundary conditions must be prescribed at each end of the space interval in order for the problem to be well posed. We then show that the solution can be expressed as an integral in the complex $k$-plane. This integral is defined in terms of an $x$-transform of the initial condition and a $t$-transform of the boundary conditions. For particular cases, such as the case of periodic boundary conditions, or the case of boundary value problems for {\em second} order PDEs, the integral can be rewritten as an infinite series. However, there exist initial boundary value problems for which the only representation is an integral which {\em cannot} be written as an infinite series. An example of such a problem is provided by the linearised version of the KdV equation. Thus, contrary to common belief, the solution of many linear initial boundary value problems on a finite interval {\em cannot} be expressed in terms of an infinite series.