The Fundamental $k$-Form and Global Relations

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 diferential 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.