13. Formulation of Riemann—Hilbert Problems
ATHANASSIOS S. FOKAS · Society for Industrial and Applied Mathematics eBooks · 2008
The novel integral representations derived in Chapter 11 express the solution q(x, y) of the basic elliptic PDEs in terms of certain transforms of the boundary values denoted by {q^j (k)}1n. These functions are coupled by the two global relations. For the case of simple polygonal domains, the algebraic manipulation of the global relations and of the equations obtained from the global relations via certain transformations in the complex k-plane immediately yield a Riemann—Hilbert (RH) problem for the characterization of {q^j(k)}n. In what follows we illustrate this approach for the simple cases of the quarter plane and the semi-infinite strip.