The Fundamental k-Form and Global Relations

• Veröffentlicht in: Symmetry, Integrability and Geometry: Methods and Applications
• Datum: 2008
• 1. Verfasser: Anthony C.L. Ashton
• Format: Artikel
• Sprache: English
• Veröffentlicht: Інститут математики НАН України 2008
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/148979
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Zitieren: The Fundamental k-Form and Global Relations / Anthony C.L. Ashton // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 5 назв. — англ.

In [Proc. Roy. Soc. London Ser. A 453 (1997), no. 1962, 1411-1443] A.S. Fokas introduced a novel method for solving a large class of boundary value problems associated with evolution equations. This approach relies on the construction of a so-called global relation: an integral expression that couples initial and boundary data. The global relation can be found by constructing a differential form dependent on some spectral parameter, that is closed on the condition that a given partial differential equation is satisfied. Such a differential form is said to be fundamental [Quart. J. Mech. Appl. Math. 55 (2002), 457-479]. We give an algorithmic approach in constructing a fundamental k-form associated with a given boundary value problem, and address issues of uniqueness. Also, we extend a result of Fokas and Zyskin to give an integral representation to the solution of a class of boundary value problems, in an arbitrary number of dimensions. We present an extended example using these results in which we construct a global relation for the linearised Navier-Stokes equations.