A unified approach to boundary value problems

Athanassios S. Fokas9780898716511, 0898716519

This book 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. Audience: A Unified Approach to Boundary Value Problems is appropriate for courses in boundary value problems at the advanced undergraduate and first-year graduate levels. Applied mathematicians, engineers, theoretical physicists, mathematical biologists, and other scholars who use PDEs will also find the book valuable. Contents: 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.