  We study initial boundary value problems for linear evolution partial
differential equations (PDEs) posed on a time-dependent interval
$l_1(t)<x<l_2(t)$, $0<t<T$, where $l_1(t)$ and $l_2(t)$ are given, real,
differentiable functions, and $T$ is an arbitrary constant. For such problems,
we show how to characterise the unknown boundary values in terms of the given
initial and boundary conditions. As illustrative examples we consider the heat
equation and the linear Schr\"{o}dinger equation. In the first case, the
unknown Neumann boundary values are expressed in terms of the Dirichlet
boundary values and of the initial value through the unique solution of a
system of two linear integral equations with explicit kernels. In the second
case, a similar result can be proved but only for a more restrictive class of
boundary curves.}
