Presents a new approach to analyzing initial-boundary value problems for integrable partial differential equations (PDEs) in two dimensions, a method that the author first introduced in 1997 and which is based on ideas of the inverse scattering transform. This method is unique in also yielding novel integral representations for the explicit solution of linear boundary value problems, which include such classical problems as the heat equation on a finite interval and the Helmholtz equation in the interior of an equilateral triangle. The author's thorough introduction allows the interested reader to quickly assimilate the essential results of the book, avoiding many computational details. 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; 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. 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. Integrable nonlinear PDEs.
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 Chapter 1: Evolution Equations on the Half-Line Chapter 2: Evolution Equations on the Finite Interval Chapter 3: Asymptotics and a Novel Numerical Technique Chapter 4: From PDEs to Classical Transforms Chapter 5: Riemann-Hilbert and d-Bar Problems Chapter 6: The Fourier Transform and Its Variations Chapter 7: The Inversion of the Attenuated Radon Transform and Medical Imaging Chapter 8: The Dirichlet to Neumann Map for a Moving Boundary Chapter 9: Divergence Formulation, the Global Relation, and Lax Pairs Chapter 10: Rederivation of the Integral Representations on the Half-Line and the Finite Interval Chapter 11: The Basic Elliptic PDEs in a Polygonal Domain Chapter 12: The New Transform Method for Elliptic PDEs in Simple Polygonal Domains Chapter 13: Formulation of Riemann-Hilbert Problems Chapter 14: A Collocation Method in the Fourier Plane Chapter 15: From Linear to Integrable Nonlinear PDEs Chapter 16: Nonlinear Integrable PDEs on the Half-Line Chapter 17: Linearizable Boundary Conditions Chapter 18: The Generalized Dirichlet to Neumann Map Chapter 19: Asymptotics of Oscillatory Riemann-Hilbert Problems Epilogue Bibliography Index
