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All colloquia (unless otherwise stated) will take place on Tuesdays at 3.00pm in Drayton B20 Jevons LT which is located in Drayton House (30 Gordon Street). See the link to the map for Drayton House for further details. There will be tea afterwards in Room 606 in the Mathematics Department (25 Gordon Street, across the street from Drayton House) - see how to find us for further details. If you require any more information on the Departmental Colloquia please contact Prof Dima Vassiliev e-mail: d.vassiliev AT ucl.ac.uk or tel: 020-7679-2442.
25 November 2014
Prof Beatrice Pelloni - University of Reading
Title: Boundary Value Problems and the Unified Transform - Advances and Applications
Abstract:
The so-called Inverse Scattering Transform was developed starting from the 1970¹s as a powerful method to attack nonlinear integrable PDEs in one or two space dimensions. Since many important evolution PDEs of mathematical physics fall into this category, a whole industry was set in motion by the pioneering work of Kruskal and co, Lax, Zakharov and Shabat. There are dimensional limitations to this approach, as well as the fundamental limitation that only the pure initial value problem or spatially periodic problems could be analysed.
The efforts to extend this approach to the case when the problem is posed on a bounded, nonperiodic domain, led by Fokas, achieved not only a deeper understanding of integral transforms and integrability, but also unexpected significant developments in the theoretical and computational study of linear, constant coefficients PDEs.
The resulting transform method is now know as the Unified Transform, or Fokas Transform. I will give a broad review of the approach and of the results it achieved in the last ten years. I will concentrate in particular on linear and nonlinear elliptic problems in two dimensions, and on some aspects of the spectral theory of non self-adjoint linear differential operators.