2008 / xx + 336 pages / Softcover / ISBN: 978-0-898716-51-1 / List Price $82.00 / SIAM Member Price $57.40/ Order Code CB78
Keywords: Riemann-Hilbert, integrable, boundary value problems, Fourier transform, asymptotics
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.
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.
About the Author
Athanassios S. Fokas has the chair of Nonlinear Mathematical Sciences in the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge, UK. In 2000 he was awarded the Naylor Prize for his work on which this book is based. He is co-author or co-editor of nine additional books and author or co-author of over 200 papers.
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