A transform method for linear evolution PDEs on a finite interval

We study initial boundary value problems for linear scalar evolution partial differential equations, with spatial derivatives of arbitrary order, posed on the domain {t > 0, 0 < x < L}. We 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. The derivation of this integral representation relies on the analysis of the global relation, which is an algebraic relation defined in the complex k-plane coupling all boundary values of the solution. For particular cases, such as the case of periodic boundary conditions, or the case of boundary value problems for even-order PDEs, it is possible to obtain directly from the global relation an alternative representation for the solution, in the form of an infinite series. We stress, however, that there exist initial boundary value problems for which the only representation is an integral which cannot be written as an infinite series. An example of such a problem is provided by the linearized version of the KdV equation. Similarly, in general the solution of odd-order linear initial boundary value problems on a finite interval cannot be expressed in terms of an infinite series.