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Unified Transform for Boundary Value Problems: Applications and Advances

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by A. S. Fokas and B. Pelloni


2015 / viii + 310 pages / Softcover / ISBN 978-1-611973-81-5 / List Price $100.00 / SIAM Member Price $70.00 / Order Code OT141

Keywords: linear PDEs, integrable PDEs, global relation, Dirichlet to Neumann map, spectral methods


This book describes state-of-the-art advances and applications of the unified transform and its relation to the boundary element method. The authors present the solution of boundary value problems from several different perspectives, in particular the type of problems modeled by partial differential equations (PDEs). They discuss recent applications of the unified transform to the analysis and numerical modeling of boundary value problems for linear and integrable nonlinear PDEs and the closely related boundary element method, a well-established numerical approach for solving linear elliptic PDEs.

The text is divided into three parts:

  • Part I contains new theoretical results on linear and nonlinear evolutionary and elliptic problems. Novel explicit solution representations for several classes of boundary value problems are constructed and rigorously analyzed.
  • Part II is a detailed overview of variational formulations for elliptic problems. It places the unified transform approach in a classic context alongside the boundary element method and stresses its novelty.
  • Part III presents recent numerical applications based on the boundary element method and on the unified transform.

This book is intended for applied and numerical research mathematicians and scientists working on the solution of boundary value problems in physics and engineering.

About the Authors
A. S. Fokas is the Chair of Nonlinear Mathematical Science in the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge. Among his many awards are the Naylor Prize, the Excellence Prize of the Bodossaki Foundation, and a Guggenheim Fellowship. He is a member of the Academy of Athens and is included in the list of most highly cited researchers in mathematics. He is the originator of the unified transform.

B. Pelloni is a Professor of Mathematics at the University of Reading. She has published over 30 papers on the unified transform and lectured on it as the Olga Taussky-Todd prize lecturer at ICIAM 2011. She is a fellow of the Institute for Mathematics and Its Applications and a member of the London Mathematical Society.

ISBN 9781611973815

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