A novel approach to analysing initial-boundary value problems for integrable partial differential equations (PDEs) in two dimensions, based on ideas of the inverse scattering transform that the author introduced in 1997. This method is unique in also yielding novel integral representations for linear PDEs. Several new developments are addressed in the book, including a new transform method for linear evolution equations on the half-line and on the finite interval; analytical inversion of certain integrals such as the attenuated Radon transform and the Dirichlet-to-Neumann map for a moving boundary; integral representations for linear boundary value problems; analytical and numerical methods for elliptic PDEs in a convex polygon; and integrable nonlinear PDEs. An epilogue provides a list of problems on which the author's new approach has been used, offers open problems, and gives a glimpse into how the method might be applied to problems in three dimensions.

Athanassios S. Fokas is Professor of Nonlinear Mathematical Science in the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge. In 2000 he was awarded the Naylor Prize for his work on which this book is based. In 2006 he received the Excellence Prize of the Bodossaki Foundation.

Preface; Introduction; 1. Evolution equations on the half-line; 2. Evolution equations on the finite interval; 3. Asymptotics and a novel numerical technique; 4. From PDEs to classical transforms; 5. Riemann-Hilbert and d-Bar problems; 6. The Fourier transform and its variations; 7. The inversion of the attenuated Radon transform and medical imaging; 8. The Dirichlet to Neumann map for a moving boundary; 9. Divergence formulation, the global relation, and Lax pairs; 10. Rederivation of the integral representations on the half-line and the finite interval; 11. The basic elliptic PDEs in a polygonal domain; 12. The new transform method for elliptic PDEs in simple polygonal domains; 13. Formulation of Riemann-Hilbert problems; 14. A collocation method in the Fourier plane; 15. From linear to integrable nonlinear PDEs; 16. Nonlinear integrable PDEs on the half-line; 17. Linearizable boundary conditions; 18. The generalized Dirichlet to Neumann map; 19. Asymptotics of oscillatory Riemann-Hilbert problems; Epilogue; Bibliography; Index.