A Unified Approach to Boundary Value Problems
A Unified Approach to Boundary Value Problems
AUD$159.09
exc GSTA 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
Product details
November 2008Paperback
9780898716511
356 pages
250 × 171 × 15 mm
0.56kg
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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.
