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A Unified Approach to Boundary Value Problems

A Unified Approach to Boundary Value Problems

$92.00

Part of CBMS-NSF Regional Conference Series in Applied Mathematics

  • Date Published: November 2008
  • availability: This item is not supplied by Cambridge University Press in your region. Please contact Soc for Industrial & Applied Mathematics for availability.
  • format: Paperback
  • isbn: 9780898716511

$ 92.00
Paperback

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About the Authors
  • 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.

    • Unifies the most extensively used techniques for solving boundary value problems for linear PDEs
    • Includes a new approach to an important medical imaging technique
    • Unique in presenting an extension of the inverse scattering method from initial value problems to boundary value problems
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    Product details

    • Date Published: November 2008
    • format: Paperback
    • isbn: 9780898716511
    • length: 356 pages
    • dimensions: 250 x 171 x 15 mm
    • weight: 0.56kg
    • availability: This item is not supplied by Cambridge University Press in your region. Please contact Soc for Industrial & Applied Mathematics for availability.
  • Table of Contents

    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.

  • Author

    Athanassios S. Fokas, University of Cambridge
    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.

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