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The Fundamental $k$-Form and Global Relations
Symmetry, integrability and geometry: methods and applications, Tome 4 (2008) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: fundamental $k$-form; global relation; boundary value problems. 
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     author = {Anthony C. L. Ashton},
     title = {The {Fundamental} $k${-Form} and {Global} {Relations}},
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Anthony C. L. Ashton. The Fundamental $k$-Form and Global Relations. Symmetry, integrability and geometry: methods and applications, Tome 4 (2008). http://geodesic.mathdoc.fr/item/SIGMA_2008_4_a32/

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