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This paper has been submitted for publication in J. Math. Anal. Appl. To download a preprint of this paper, just click here.

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1513054. Any opinions, findings and conclusions or recomendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation (NSF).

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The Diffusion Equation With Nonlocal Data

Peter D. Miller

Department of Mathematics, University of Michigan

David A. Smith

Yale-NUS College, Singapore

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.