© 2002 by Oxford University Press
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The Fundamental Differential Form and Boundary-Value Problems
( 1 Department of Applied Mathematics and Theoretical Physics, Cambridge University, Cambridge CB3 9EW 2 St Catherine College, Oxford OX1 3UJ )
Recently a new method for solving linear and integrable nonlinear partial differential equations (PDEs) in two dimensions has been introduced. For linear PDEs, this method involves the construction of appropriate integral representations for both the solution and the associated spectral function. Here we present an alternative approach for constructing these integral representations. This approach is based on the introduction of what we call a fundamental differential form, and on a reformulation of Green's formula. The new approach is illustrated for the Laplace and the modified Helmholtz equations in a convex polygon, for a dispersive evolution equation with spatial derivatives of arbitrary order on the half-line, and for the heat equation in a domain involving a moving boundary. In addition we show that the new approach can be used to construct appropriate integral representations for multidimensional linear PDEs.
Received 21 August 2000. Revised 22 March and 11 October 2001.