The Fundamental k-Form and Global Relations

TL;DR

This paper introduces an algorithmic method to construct fundamental k-forms for boundary value problems, extending Fokas' global relation approach to multiple dimensions and applying it to linearized Navier-Stokes equations.

Contribution

It provides a systematic way to construct fundamental forms and global relations, extending Fokas' method to higher dimensions and addressing uniqueness issues.

Findings

Developed an algorithm for constructing fundamental k-forms.

Extended Fokas and Zyskin's results to arbitrary dimensions.

Applied the method to linearized Navier-Stokes equations.

Abstract

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.