Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-28T23:00:34.424Z Has data issue: false hasContentIssue false
  • English
  • Français

A quadrature method for constant-coefficient Cauchy singular integral equations on an interval

Published online by Cambridge University Press:  17 February 2009

Youngmok Jeon
Youngmok Jeon
Affiliation:
Department of Mathematics, Ajou University, Suwon 442-749, Korea.
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.

We consider a mesh grading quadrature method for real constant-coefficient Cauchy singular integral equations of index 0. The quadrature method is based on the trapezoidal rule. A complete stability and convergence analysis is given by the use of the noncompact perturbation analysis as in Jeon [10] and Elschner and Stephan [7]. The order of convergence can be arbitrarily high if the order of mesh grading is high enough. We also provide an efficient way of evaluating asymptotics of the solution at the end points. Experimentally, we observe that the method also works well for Cauchy singular integral equations with variable coefficients.

Type
Research Article
Information
The ANZIAM Journal , Volume 42 , Issue 2 , October 2000 , pp. 287 - 311
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Anselone, P. M., Collectively Compact Operator Approximation Theory and Applications to Integral Equations (Prentice-Hall, Eaglewood Cliffs, 1971).Google Scholar
[2]Atkinson, K. E., An Introduction to Numerical Analysis, 2nd ed. (John Wiley Pub, New York, 1989).Google Scholar
[3]Atkinson, K. E., A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind (SIAM Pub., 1976), Chapter 2.1, Part II.Google Scholar
[4]Belotserkovski, S. M. and Lifanov, I. K., Method of discrete vortices (CRC Press, Boca Raton, 1993).Google Scholar
[5]Elliott, D., “Convergence theorems for singular integral equations”, in Numerical Solution of Integral Equations, Mathematical Concepts and Methods in Science and Engineering, 42 (ed. Golberg, M. A.), (Plenum Press, New York, 1990) 309–361.Google Scholar
[6]Elschner, J. and Graham, I. G., “An optimal order collocation method for first kind boundary integral equations on polygons”, Numer. Math. 70 (1995) 1–31.Google Scholar
[7]Elschner, J. and Stephan, E. P., “A discrete collocation method for Symm's integral equation on curves with corners”, submitted (1995).Google Scholar
[8]Golberg, M., “Perturbed projection methods for various classes of operator and integral equations”, in Numerical Solution of Integral Equations. Mathematical Concepts and Methods in Science and Engineering, 42 (ed. Golberg, M. A.) (Plenum Press, New York, 1990) 71–129.Google Scholar
[9]Golberg, M., “Introduction to the numerical solutions of Cauchy singular integral equations”, in Numerical Solution of Integral Equations. Mathematical concepts and methods in science and engineering, 42 (ed. Golberg, M. A.) (Plenum Press, New York, 1990) 183–308.Google Scholar
[10]Jeon, Y. “A quadrature method for the Cauchy singular integral equations”, J. Int. Eqns. Appl. 7, (1995) 425–461.Google Scholar
[11]Kress, R., “A Nyström method for boundary integral equations in domains with corners”, Numer. Math. 58 (1990) 145–161.Google Scholar
[12]Lyness, J. N., “The Euler Maclaurin expansion for the Cauchy principal value integral”, Numer. Math. 46 (1985) 611–622.CrossRefGoogle Scholar
[13]Lyness, J. N. and Ninham, B. W., “Numerical quadrature and asymptotic expansions”, Math. Comp. 21 (1967) 162–178.Google Scholar
[14]Muskhelishvili, N. I., Singular Integral Equations (Dover Pub., 1992).Google Scholar
[15]Prössdorf, S. and Rathsfeld, A., “Quadrature methods for strongly elliptic Cauchy singular integral equation on an interval”, in Operator Theory: Advances and Applications, 41 (Birkhäuser Verlag, Basel, 1989) 435–471.Google Scholar
[16]Prössdorf, S. and Silbermann, B., Numerical Analysis for Integral and Related Operator Equations (Birkhäuser, Basel, Boston, 1991).Google Scholar
[17]Rivlin, T., The Chebyshev Polynomials (Wiley, New York, 1974).Google Scholar
[18]Saranen, J. and Sloan, I. H., “Quadrature methods for logarithmic-kernel equations on closed curves”, IMA J. Numer. Anal. 12 (1992) 537–567.Google Scholar
[19]Sneddon, I. N., The Use of Integral Transforms, (McGraw-Hill, New York, 1972).Google Scholar
[20]Venturino, E., “Unconventional solution of singular integral equation”, J. Int. Eqns. Appl. 4 (1992) 574–587.Google Scholar