The fundamental differential form and boundary-value problems

AS Fokas, M Zyskin

Research output: Contribution to journalArticle (Academic Journal)peer-review

27 Citations (Scopus)


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.
Translated title of the contributionThe fundamental differential form and boundary-value problems
Original languageEnglish
Pages (from-to)457 - 479
Number of pages23
JournalQuarterly Journal of Mechanics and Applied Mathematics
Volume55 (3)
Publication statusPublished - Aug 2002

Bibliographical note

Publisher: Oxford University Press


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