zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 {\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. 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.
Reviewer: M.Gutknecht

30C30Numerical methods in conformal mapping theory
45B05Fredholm integral equations
65R20Integral equations (numerical methods)
Full Text: DOI
[1] Delves, L. M.: The numerical solution of sets of linear equations from Ritz-Galerkin methods. J. inst. Maths. applics. 20, 163-171 (1977) · Zbl 0369.65028
[2] Delves, L. M.: A fast method for the solution of Fredholm integral equations. J. inst. Maths. applics. 20, 173-182 (1977) · Zbl 0404.65062
[3] Fornberg, B.: A numerical method for conformal mapping. SIAM J. Sci. stat. Comput. 1, 386-400 (1980) · Zbl 0451.30003
[4] Gaier, D.: Integralgleichungen erster art and konforme abbildung. Math. Z. 147, 113-129 (1976) · Zbl 0304.30006
[5] Gaier, D.: Das logarithmische potential und die konforme abbildung mehrfach zusammenhängender gebiete. E.B. Christoffel, the influence of his work on mathematics and the physical sciences, 290-303 (1981)
[6] Golub, G. H.; Van Loan, C. F.: Matrix computations. (1983) · Zbl 0559.65011
[7] Gutknecht, M. H.: Solving theodorsen’s integral equation for conformal maps with the fast Fourier transform and various nonlinear iterative methods. Numer. math. 36, 405-429 (1981) · Zbl 0451.65101
[8] Henrici, P.: Fast Fourier methods in computational complex analysis. SIAM rev. 21, 481-527 (1979) · Zbl 0416.65022
[9] Jawson, M. A.; Symm, G. T.: Integral equation methods in potential theory and elastostatics. (1977) · Zbl 0414.45001
[10] Reichel, L.: On the determination of boundary collocation points for solving some problems for the Laplace operator. J. comput. Appl. math. 11, 175-196 (1984) · Zbl 0569.65083
[11] Reichel, L.: On polynomial approximation in the complex plane with application to conformal mapping. Math. comp. 44, 425-433 (1985) · Zbl 0579.41009
[12] Richter, G. R.: Numerical solution of integral equations of the first kind with nonsmooth kernels. SIAM J. Numer. anal. 15, 511-522 (1978) · Zbl 0388.65056
[13] Wegmann, R.: An iterative method for conformal mapping. J. comput. Appl. math. 14, 7-18 (1986) · Zbl 0577.30008