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The Unified Transform for the modified Helmholtz equation in the exterior of a square

The Unified Transform provides a novel method for analyzing boundary value problems for linear and for integrable nonlinear PDEs. The numerical implementation of this method to linear elliptic PDEs formulated in the {\it interior} of a polygon has been investigated by several authors (see the article by Iserles, Smitheman, and one of the authors in this book). Here, we show that the Unified Transform also yields a novel numerical technique for computing the solution of linear elliptic PDEs in the {\it exterior} of a polygon. One of the advantages of this new technique is that it actually yields directly the scattering amplitude. Details are presented for the modified Helmholtz equation in the exterior of a square.

preprint2014arXivOpen access
A. S. FokasJ. Lenells
math.AP
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A. S. FokasJ. Lenells

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