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The Fundamental Differential Form and Boundary Value Problems

Fokas, A.S.; Zyskin, Maxim


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Abstract: Recently a new method for solving linear and integrable nonlinear PDE's in 2 dimensions has been introduced. This method is based on the construction of appropriate integral representations for both the solution and the spectral function. Here we present an alternative approach for constructing these integral representations. This approach is based on the introducing of what we call a fundamental differential form, and on a generalization of Green's formula. The new approach is illustrated for the Laplace and the modified Helmholtz equations in a convex polygon, for a general dispersive equation on the half-line, and for the heat equation on a domain involving a moving boundary.

23 Pages

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