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The Unified Transform and the Water Wave Problem

  • A. S. Fokas
  • K. KalimerisEmail author
Chapter
Part of the Tutorials, Schools, and Workshops in the Mathematical Sciences book series (TSWMS)

Abstract

The unified transform, also known as the Fokas method, was introduced in 1997 by one of the authors Fokas (Proc R Soc Lond A: Math Phys Eng Sci 453(1962):1411–1443, 1997 ) for the analysis of nonlinear initial-boundary value problems. Later, it was realised that this method also yields novel results for linear problems. In 2006, the classical water wave problem was studied via the Fokas method (Ablowitz et al., J Fluid Mech 562:313–343, 2006), yielding a novel non-local formulation. In this paper we review the unified transform, with particular emphasis on its application in water wave in two spacial dimensions with moving boundaries.

Keywords

Unified transform Non-local formulation Water waves 

Mathematics Subject Classification (2000)

Primary 35Q35; Secondary 76M40 

Notes

Acknowledgements

The authors “A. S. Fokas and K. Kalimeris” were supported by EPSRC. The authors are grateful to the Erwin Schrödinger International Institute for Mathematics and Physics, Vienna, for the support and hospitality during the 2017 Nonlinear Water Waves—an Interdisciplinary Interface workshop.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeUK
  2. 2.Viterbi School of EngineeringUniversity of Southern CaliforniaLos AngelesUSA

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