Hostname: page-component-74656f9884-rdgjk Total loading time: 0 Render date: 2026-11-08T17:40:27.522Z Has data issue: false hasContentIssue false
  • English
  • Français

A discrete method for the logarithmic-kernel integral equation on an open arc

Published online by Cambridge University Press:  17 February 2009

S. Prössdorf ,
J. Saranen  and
I. H. Sloan
S. Prössdorf
Affiliation:
Karl-Weierstrass-Institut für Mathematik, Mohrenstrasse 39, D-O 1086 Berlin, Germany.
J. Saranen
Affiliation:
Section of Mathematics, Faculty of Technology, University of Oulu, 90570 Oulu, Finland.
I. H. Sloan
Affiliation:
School of Mathematics, University of New South Wales, Sydney, N.S.W. 2033, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Here we discuss the stability and convergence of a quadrature method for Symm's integral equation on an open smooth arc. The method is an adaptation of an approach considered by Sloan and Burn for closed curves. Before applying the quadrature scheme, we use a cosine substitution to remove the endpoint singularity of the solution. The family of methods includes schemes with any order O(hp) of convergence.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 34 , Issue 4 , April 1993 , pp. 401 - 418
Copyright
Copyright © Australian Mathematical Society 1993

References

[1] Atkinson, K. E. and Sloan, I. H., “The numerical solution of first-kind integral equations on smooth open arcs”, Math. Comp. 56 (1991) 119–139.CrossRefGoogle Scholar
[2] Costabel, M., Ervin, V. J. and Stephan, E. P., “On the convergence of collocation methods for Symm's integral equations on open curves”, Math. Comp. 51 (1988) 167–179.Google Scholar
[3] Joe, S. and Yan, Y., “A collocation method using cosine mesh grading for Symm's equation on the interval (−1, 1)”, Preprint, 04 1990, University of the New South Wales, Sydney.Google Scholar
[4] Multhopp, H., “Die Berechnung der Auftriebsverteilung von Tragflögeln”, Luftfahrt-Forschung XV (4) (1938) 153–169.Google Scholar
[5] Prössdorf, S. and Rathsfeld, A., “Quadrature methods for strongly elliptic Cauchy singular integral equations on an interval”, Operator Theory: Advanced and Applications, 41 (Birkhäuser Verlag, Basel, 1989) 435–471.Google Scholar
[6] Prössdorf, S. and Silbermann, B., Projektionsverfahren und die näherungsweise Lösung singulärer Gleichungen, Teubner-Texte fur Mathematik (Teubner Verlagsges., Leipzig, 1977).Google Scholar
[7] Prössdorf, S. and Sloan, I. H., “Quadrature method for singular integral equations on closed curves”, Numer. Math. 61 (1992) 543–559.CrossRefGoogle Scholar
[8] Rathsfeld, A., “Quadraturformelverfahren für Mellin-Operatoren nullter Ordnung”, Math. Nachr. 137 (1988) 321–354.CrossRefGoogle Scholar
[9] Saranen, J., “The modified quadrature method for logarithmic-kernel integral equations on closed curves”, J. Integral Equations Appl. 3 (4) (1991) 575–600.CrossRefGoogle Scholar
[10] Saranen, J. and Sloan, I. H., “Quadrature methods for logarithmic-kernel integral equations on closed curves”, IMA J. Numer. Anal. 12 (1992) 167–187.CrossRefGoogle Scholar
[11] Schleiff, M., “Untersuchung einer linearen singulären Integrodifferentialgleichung der Tragflügeltheorie”, Wiss. Z. der Univ. Halle 17 68 M, (6) (1968) 981–1000.Google Scholar
[12] Schleiff, M., “Über Näherungsverfahren zur Lösung einer singulären linearen Integrodifferentialgleichung”, ZAMM 48 (1968) 477–483.CrossRefGoogle Scholar
[13] Sloan, I. H. and Burn, B. J., “An unconventional quadrature method for logarithmic-kernel integral equations on closed curves”, J. Integral Equations Appl. 4 (1) (1992) 117–151.CrossRefGoogle Scholar
[14] Sloan, I. H. and Spence, A., “The Galerkin method for integral equations of the first kind with logarithmic kernel: Theory”, IMA J. Numer. Anal. 8 (1988) 105–122.CrossRefGoogle Scholar
[15] Sloan, I. H. and Stephan, E. P., “Collocation with Chebyshev polynomials for Symm's integral equation on an interval”, J. Austral. Math. Soc. Ser. B (to appear).Google Scholar
[16] Weissinger, J., “Über Integrodifferentialgleichungen vom Typ der Prandtlschen Tragflügelgleichung”, Math. Nachr. 3 (1950) 316–326.CrossRefGoogle Scholar
[17] Yan, Y. and Sloan, I. H., “On integral equations of the first kind with logarithmic kernel”, J. Integral Equations Appl. 1 (1988) 549–579.CrossRefGoogle Scholar