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A fast method for solving certain integral equations of the first kind with application to conformal mapping. (English) Zbl 0587.30007

A general approach due to J. Delves [J. Inst. Math. Appl. 20, 173- 182 (1977; Zbl 0404.65062)] for solving operator equations iteratively by Galerkin methods is elaborated for Fredholm integral equations of the first kind whose kernels have a logarithmic principle part. Essentially, the method consists of splitting the matrix A resulting from the Fourier- Galerkin approach, $A=B+C$ with $\parallel B\parallel \gg \parallel C\parallel$, computing the Cholesky decomposition $B={R}^{T}R$, and applying Jacobi iteration preconditioned by the inverse of B.

The method is here in particular applied to G. T. Symm’s integral equation for computing the conformal mapping of a simply connected region onto the unit disk [Numer. Math. 9, 250-258 (1966; Zbl 0156.169)] and to corresponding integral equations for doubly and multiply connected regions. In the simply connected case the method generalizes one proposed by P. Henrici [SIAM Rev. 21, 481-527 (1979; Zbl 0416.65022)], where B is the diagonal matrix corresponding to the logarithmic principle part of the kernel.

Reviewer: M.Gutknecht
##### MSC:
 30C30 Numerical methods in conformal mapping theory 45B05 Fredholm integral equations 65R20 Integral equations (numerical methods)
##### Keywords:
Fourier-Galerkin method; Fredholm integral equations