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Zbl 0587.30007
Reichel, Lothar
A fast method for solving certain integral equations of the first kind with application to conformal mapping.
(English)
[J] J. Comput. Appl. Math. 14, 125-142 (1986). ISSN 0377-0427

A general approach due to {\it 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 $\Vert B\Vert \gg \Vert C\Vert$, computing the Cholesky decomposition $B=R\sp TR$, and applying Jacobi iteration preconditioned by the inverse of B. \par The method is here in particular applied to {\it 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 {\it 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.
[M.Gutknecht]
MSC 2000:
*30C30 Numerical methods in conformal mapping theory
45B05 Fredholm integral equations
65R20 Integral equations (numerical methods)

Keywords: Fourier-Galerkin method; Fredholm integral equations

Citations: Zbl 0404.65062; Zbl 0156.169; Zbl 0416.65022

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