  We examine the analytic extension of solutions of linear,
constant-coefficient initial-boundary value problems outside their spatial
domain of definition. We use the Unified Transform Method or Method of Fokas,
which gives a representation for solutions to half-line and finite-interval
initial-boundary value problems as integrals of kernels with explicit spatial
and temporal dependence. These solution representations are defined within the
spatial domain of the problem. We obtain the extension of these representation
formulae via Taylor series outside these spatial domains and find the extension
of the initial condition that gives rise to a whole-line initial-value problem
solved by the extended solution. In general, the extended initial condition is
not differentiable or continuous unless the boundary and initial conditions
satisfy compatibility conditions. We analyze dissipative and dispersive
problems, and problems with continuous and discrete spatial variables.
