Abstract
A Cauchy type singular integral equation of the first kind can be numerically solved either directly, through the use of a Gaussian numerical integration rule, or by reduction to an equivalent Fredholm integral equation of the second kind, where the Nyström method is applicable. In this note it is proved that under appropriate but reasonable conditions the expressions of the unknown function of the integral equation, resulting from the natural interpolation formulae of the direct method, as well as of the Nyström method, are identical along the whole integration interval.
Zusammenfassung
Eine singuläre Integralgleichung erster Art vom Cauchy-Typ kann entweder direkt, mittels einer Gaußschen numerischen Integrationsformel, oder durch Reduktion auf eine äquivalente Fredholmsche Integralgleichung zweiter Art, wo die Nyström-Methode anwendbar ist, gelöst werden. In dieser Arbeit wird bewiesen, daß unter geeigneten und sinnvollen Bedingungen die Ausdrücke der unbekannten Funktion der Integralgleichung, die einerseits bei den natürlichen Integrationsformeln der direkten Methode und anderseits bei der Nyström-Methode entstehen, im ganzen Integrationsintervall gleich sind.
Article PDF
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Baker, C. T. H.: The numerical treatment of integral equations, 1st ed., pp. 356–375. Oxford: Clarendon Press 1977.
Erdogan, F., Gupta, G. D.: On the numerical solution of singular integral equations. Quart. Appl. Math.29, 525–534 (1972).
Gakhov, F. D.: Boundary value problems, 1st English ed., pp. 472–479. Oxford: Pergamon Press and Addison-Wesley 1966. [Translation of the 2nd Russian ed., Moscow: Fizmatgiz 1963.]
Ioakimidis, N. I., Theocaris, P. S.: On the numerical evaluation of Cauchy principal value integrals. Rev. Roumaine Sci. Tech. Sér Méc. Appl.22, 803–818 (1977).
Ioakimidis, N. I., Theocaris, P. S.: A comparison between the direct and the classical numerical methods for the solution of Cauchy type singular integral equations. SIAM J. Numer. Anal.17, 115–118 (1980).
Theocaris, P. S., Ioakimidis, N. I.: Numerical integration methods for the solution of singular integral equations. Quart. Appl. Math.35, 173–183 (1977).
Theocaris, P. S., Ioakimidis, N. I.: A remark on the numerical solution of singular integral equations and the determination of stress-intensity factors. J. Engrg. Math.13, 213–222 (1979).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ioakimidis, N.I. On the natural interpolation formula for cauchy type singular integral equations of the first kind. Computing 26, 73–77 (1981). https://doi.org/10.1007/BF02243425
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02243425
Key words and phrases
- Cauchy type singular integral equations
- natural interpolation formulae
- Gauss-Chebyshev quadrature rule
- Nyström (quadrature) method for integral equations
- direct numerical solution of Cauchy type singular integral equations