The unified transform for evolution equations on the half‐line with time‐periodic boundary conditions*


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Abstract

jats:titleAbstract</jats:title>jats:pThis paper elaborates on a new approach for solving the generalized Dirichlet‐to‐Neumann map, in the large time limit, for linear evolution PDEs formulated on the half‐line with time‐periodic boundary conditions. First, by employing the unified transform (also known as the Fokas method) it can be shown that the solution becomes time‐periodic for large <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/sapm12452-math-0001.png" xlink:title="urn:x-wiley:00222526:media:sapm12452:sapm12452-math-0001" />. Second, it is shown that the coefficients of the Fourier series of the unknown boundary values can be determined explicitly in terms of the coefficients of the Fourier series of the given boundary data in a very simple, algebraic way. This approach is illustrated for second‐order linear evolution equations and also for linear evolution equations containing spatial derivatives of arbitrary order. The simple and explicit determination of the unknown boundary values is based on the “<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/sapm12452-math-0002.png" xlink:title="urn:x-wiley:00222526:media:sapm12452:sapm12452-math-0002" />‐equation”, which for the linearized nonlinear Schrödinger equation is the linear limit of the quadratic <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/sapm12452-math-0003.png" xlink:title="urn:x-wiley:00222526:media:sapm12452:sapm12452-math-0003" />‐equation introduced by Lenells and Fokas [jats:italicProc. R. Soc. A</jats:italic>, 471, 2015]. Regarding the latter equation, it is also shown here that it provides a very simple, algebraic way for rederiving the remarkable results of Boutet de Monvel, Kotlyarov, and Shepelsky [jats:italicInt. Math. Res. Not</jats:italic>. issue 3, 2009] for the particular boundary condition of a single exponential.</jats:p>

Description

Funder: Engineering and Physical Sciences Research Council; Id: http://dx.doi.org/10.13039/501100000266


Funder: Foundation for Education and European Culture; Id: http://dx.doi.org/10.13039/501100005411


Funder: Cambridge Trust; Id: http://dx.doi.org/10.13039/501100003343


Funder: Christ's College, University of Cambridge; Id: http://dx.doi.org/10.13039/501100000590


Funder: A.G. Leventis Foundation; Id: http://dx.doi.org/10.13039/501100004117

Keywords
4901 Applied Mathematics, 49 Mathematical Sciences
Journal Title
Studies in Applied Mathematics
Conference Name
Journal ISSN
0022-2526
1467-9590
Volume Title
Publisher
Wiley
Sponsorship
Alexander S. Onassis Public Benefit Foundation (F ZQ 004‐1/2020‐2021)