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A Riemann–Hilbert Approach to the Laplace Equation

✍ Scribed by A.S. Fokas; A.A. Kapaev

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
207 KB
Volume
251
Category
Article
ISSN
0022-247X

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✦ Synopsis

Let q x, y satisfy the Laplace equation in an arbitrary convex polygon. By performing the spectral analysis of the equation y ik s q y iq , z s x q iy, z x y Ž . which involves solving a scalar Riemann᎐Hilbert RH problem, we construct an Ž . integral representation in the complex k-plane of q x, y in terms of a function Ž . Ž . k . It has been recently shown that the function k can be expressed in terms of the given boundary conditions by solving a matrix RH problem. Here we show that this method is also useful for solving problems in a non-convex polygon.

We also recall that for simple polygons it is possible to bypass the above integral representation and to solve the Laplace equation by formulating a RH problem in the complex z-plane.

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