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Full Description
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.
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
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