Boundary Value Problems for Linear Elliptic PDEs

First and foremost I thank my supervisor Thanasis Fokas. It has been a pleasure and a privilege to be his student. His energy and enthusiasm are infectious (and I hope his mathematical talent is too!). I was supported by an EPSRC studentship, I hope they are happy with their investment. George Dassios, Arieh Iserles and Stephen Cowley shared their knowledge and advice with me, both mathematical and non-mathematical, and I am very grateful. I thank the members of “the Fokas group”: Kostas Kalimeris, Anthony Ashton, Sheila Smitheman, Jonatan Lenells, Paolo Barbano, and in particular my office mates Michail Dimakos, Eleftherios Garyfallidis, and Dionyssios Mantzavinos. I have enjoyed all our interactions and adventures! I frequently burst into the offices of Chris Taylor, Alexander Frolkin, Darren Kenneally, and especially Ben Adcock, demanding answers left, right and centre and they had the patience to send me out each time a little more enlightened than when I came in. If this thesis, or any of my papers, is easy to read and understand then Marj Batchelor deserves much credit (if not, then I should have listened to her more). I thank my fellow students and colleagues in the ACA “super-group”. During my time in Cambridge the Applied and Computational Analysis community has become much more vibrant, and I hope this will continue. I thank my friends Claude Warnick, Kate Marvel, Clare Burrage, Chris Taylor, Toby Wood, Mark Rosin and John Fletcher, for good times in the central core, Cambridge and elsewhere. Many thanks are due to Alfredo Deaño and Mojo Rathbone for their last minute assistance the day before my submission deadline. I thank Graeme Spence and particularly Maxine von Eye for travelling with me on the rollercoaster rides of both my research and life in general. Finally I thank my parents: undertaking this PhD is the latest in a long list of opportu-nities that I would not have had without their many sacrifices on my behalf