The diffusion equation with nonlocal data

TL;DR

This paper investigates a heat equation on a finite domain with a nonlocal boundary condition, extending the Fokas transform method to ensure well-posedness and providing explicit solution representations.

Contribution

It extends the Fokas transform method to handle initial-nonlocal boundary conditions for linear diffusion equations, establishing conditions for unique solutions and deriving integral solution formulas.

Findings

Conditions for regularity and uniqueness are established.

Solution representation via contour integrals is provided.

The method extends the Fokas transform to nonlocal boundary conditions.

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.