A numerical comparison of integral equations of the first and second kind for conformal mapping

Authors:
John K. Hayes, David K. Kahaner and Richard G. Kellner

Journal:
Math. Comp. **29** (1975), 512-521

MSC:
Primary 65E05; Secondary 30A28

DOI:
https://doi.org/10.1090/S0025-5718-1975-0371036-8

MathSciNet review:
0371036

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Two methods for computing numerical conformal mappings are compared. The first, due to Symm, uses a Fredholm integral equation of the first kind while the other, due to Lichtenstein, uses a Fredholm integral equation of the second kind. The two methods are tested on ellipses with different ratios of major to minor axes. The method based on the integral equation of the second kind is superior if the ratio is less than or equal to 2.5. The opposite is true if the ratio is greater than or equal to 10. Similar results are obtained for other regions.

**[1]**G. Birkhoff, D. M. Young, and E. H. Zarantonello,*Numerical methods in conformal mapping*, Proceedings of Symposia in Applied Mathematics, vol. IV, Fluid dynamics, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953, pp. 117–140. MR**0057637****[2]**Dieter Gaier,*Konstruktive Methoden der konformen Abbildung*, Springer Tracts in Natural Philosophy, Vol. 3, Springer-Verlag, Berlin, 1964 (German). MR**0199360****[3]***Selected numerical methods for linear equations, polynomial equations, partial differential equations, conformal mappings*, Report on studies sponsored by The Carlsberg Foundation, Regnecentralen, Copenhagen, 1962. MR**0149638****[4]**J. HAYES,*Four Computer Programs Using Green's Third Formula to Numerically Solve Laplace's Equation in Inhomogeneous Media*, Los Alamos Scientific Laboratory Report, LA-4423 (April 1970).**[5]**John K. Hayes, David K. Kahaner, and Richard G. Kellner,*An improved method for numerical conformal mapping*, Math. Comp.**26**(1972), 327–334; suppl., ibid. 26 (1972), no. 118, loose microfiche suppl. A1–B14. MR**0301176**, https://doi.org/10.1090/S0025-5718-1972-0301176-8**[6]**John Hayes and Richard Kellner,*The eigenvalue problem for a pair of coupled integral equations arising in the numerical solution of Laplace’s equation*, SIAM J. Appl. Math.**22**(1972), 503–513. MR**0305634**, https://doi.org/10.1137/0122044**[7]**M. A. Jaswon,*Integral equation methods in potential theory. I*, Proc. Roy. Soc. Ser. A**275**(1963), 23–32. MR**0154075****[8]**L. LICHTENSTEIN, "Zur konformen Abbildung einfach zusammenhängender schlichter Gebiete,"*Arch. Math. Phys.*, v. 25, 1917, pp. 179-180.**[9]**N. I. Muskhelishvili,*Singular integral equations*, Wolters-Noordhoff Publishing, Groningen, 1972. Boundary problems of functions theory and their applications to mathematical physics; Revised translation from the Russian, edited by J. R. M. Radok; Reprinted. MR**0355494****[10]**Ben Noble,*Some applications of the numerical solution of integral equations to boundary value problems*, Conference on Applications of Numerical Analysis (Univ. Dundee, Dundee, 1971), Springer, Berlin, 1971, pp. 137–154. Lecture Notes in Math., Vol. 228. MR**0353711****[11]**S. N. Karp and S. E. Shamma,*A generalization of separability in boundary value problems*, SIAM J. Appl. Math.**20**(1971), 536–546. MR**0289977**, https://doi.org/10.1137/0120056**[12]**George T. Symm,*An integral equation method in conformal mapping*, Numer. Math.**9**(1966), 250–258. MR**0207240**, https://doi.org/10.1007/BF02162088**[13]**S. E. Warschawski,*Recent results in numerical methods of conformal mapping*, Proceedings of Symposia in Applied Mathematics. Vol. VI. Numerical analysis, McGraw-Hill Book Company, Inc., New York, for the American Mathematical Society, Providence, R. I., 1956, pp. 219–250. MR**0086403**

Retrieve articles in *Mathematics of Computation*
with MSC:
65E05,
30A28

Retrieve articles in all journals with MSC: 65E05, 30A28

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1975-0371036-8

Keywords:
Numerical conformal mapping,
numerical solution of integral equations of the first kind

Article copyright:
© Copyright 1975
American Mathematical Society