Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 033, 15 pages      arXiv:0711.4707      http://dx.doi.org/10.3842/SIGMA.2008.033

The Fundamental k-Form and Global Relations

Anthony C.L. Ashton
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, UK

Received December 20, 2007, in final form March 03, 2008; Published online March 20, 2008

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.

Key words: fundamental k-form; global relation; boundary value problems.

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References

  1. Dickey L.A., Soliton equations and Hamiltonian systems, 2nd ed., Advanced Series in Mathematical Physics, Vol. 26, World Scientific, River Edge, NJ, 2003.
  2. Ehrenpreis L., Fourier analysis in several complex variables, Wiley-Interscience Publishers, New York, 1970.
  3. Fokas A.S., A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London Ser. A 453 (1997), no. 1962, 1411-1443.
  4. Fokas A.S., Pelloni B., Generalized Dirichlet to Neumann map for moving initial-boundary value problems, J. Math. Phys. 48 (2007), 013502, 14 pages, math-ph/0611009.
  5. Fokas A.S., Zyskin M., The fundamental differential form and boundary-value problems, Quart. J. Mech. Appl. Math. 55 (2002), 457-479.

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