Title: The diffusion equation with nonlocal data,
Abstract: We study the diffusion (or heat) equation on a finite 1-dimensional spatial
domain, but we replace one of the boundary conditions with a "nonlocal
condition", through which we specify a weighted average of the solution over
the spatial interval. We provide conditions on the regularity of both the data
and weight for the problem to admit a unique solution, and also provide a
solution representation in terms of contour integrals. The solution and
well-posedness results rely upon an extension of the Fokas (or unified)
transform method to initial-nonlocal value problems for linear equations; the
necessary extensions are described in detail. Despite arising naturally from
the Fokas transform method, the uniqueness argument appears to be novel even
for initial-boundary value problems.