Conformal Mapping in the Applications of Mathematics
Frank Stenger
JTB 320 3:20pm, Monday, May 5, 1997
Abstract
Conformal maps appear in many forms, in applications, indeed, almost
every time the two dimensional Laplacian operator appears, and in such
cases, a knowledge of the conformal map of the region onto the unit disc
will frequently enable a simple and elegant solution to the problem. It
is also a useful tool in numerical compuations, e.g., enabling the
correct distribution of the elements in finite element (and even Sinc)
methods.
Given a simply connected domain B with rectifiable boundary, a conformal
map F of the unit disc U to B can be explicitly expressed via integrals
(see e.g., Gakhov's book, pp. 266-269). On the other hand, this is not
the case for the construction of an inverse map f from B to U, the
latter being usually carried out via the solution of Symm's integral
equation. In this talk, I will present a novel explicit solution to
Symm's equation, i.e., an explicit expression for f, in terms of
integrals. Thus, we are now able to efficiently and accurately construct
BOTH f and F, via Sinc methods, whenever the boundary of B consists of a
finite number of analytic arcs with well defined angles of intersection
at the junctions of the arcs.
In the process, we obtain a novel method of obtaining accurate solutions
to Dirichlet and Neumann problems over such regions B using only
quadrature formulae, i.e., without solving any algebraic equations.
Reference: F.D. Gakhov, "Boundary Value Problems",
translated from Russian by I.N. Sneddon,
Pergamon Press, Oxford. (1966).
Order reprints via email to stenger@sinc.cs.utah.edu
This source can be found at http://www.math.utah.edu/research/