Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-23T11:45:14.818Z Has data issue: false hasContentIssue false
  • English
  • Français

A Galerkin-Petrov method for singular integral equations

Published online by Cambridge University Press:  17 February 2009

David Elliott
David Elliott
Affiliation:
Mathematics Department, University of Tasmania, Box 252C, G.P.O., Hobart, Tasmania 7001.
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.

A Galerkin-Petrov method for the approximate solution of the complete singular integral equation with Cauchy kernel, based upon the use of two sets of orthogonal polynomials, is considered. The principal result of this paper proves convergence of the approximate solutions to the exact solution making use of a convergence theorem previously given by the author. In conclusion, some related topics such as a first iterate of the approximate solution and a discretized Galerkin-Petrov method are considered. The paper extends to a much more general equation many results obtained by other authors in particular cases.

Type
Research Article
Information
The ANZIAM Journal , Volume 25 , Issue 2 , October 1983 , pp. 261 - 275
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Dzhishkariani, A. V.. “The solution of singular integral equations by approximate projection methods”. U.S.S.R. Computational Math. and Math. Phys. 19 (1980), 61–74.CrossRefGoogle Scholar
[2]David, Elliott. “A convergence theorem for singular integral equations”, J. Austral. Math. Soc. Ser. B 22 (1981), 539–552.Google Scholar
[3]David, Elliott. “The classical collocation method for singular integral equations”, SIAM J. Numer. Anal. 19 (1982), 816–832.Google Scholar
[4]David, Elliott, “Orthogonal polynomials associated with singular integral equations having a Cauchy kernel”. SIAM J Math Anal. 13 (1982), 1041–1052.Google Scholar
[5]Erdogan, F. and Gupta, G. D., “On the numerical solution of singular integral equations”. Quart. Appl. Math. 30 (1972), 525–534.CrossRefGoogle Scholar
[6]Erdogan, F., “Approximate solutions of systems of singular integral equations”. SIAM J. Appl. Math. 17 (1969), 1041–1051.CrossRefGoogle Scholar
[7]Golberg, Michael A. (ed.), Solution methods for integral equations: theory and applications (Plenum, New York, 1979).CrossRefGoogle Scholar
[8]Michael, A. Golberg, “Galerkin's method for operator equations with non-negative index—with application to Cauchy singular integral equations”, J. Math. Anal. Appl. (to be published).Google Scholar
[9]Ioakimidis, N. I. and 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 (1980), 115–118.CrossRefGoogle Scholar
[10]Ioakimidis, N. I., “On the weighted Galerkin method of numerical solution of Cauchy type singular integral equations”, SIAM J. Numer. Anal. 18 (1981), 1120–1127.CrossRefGoogle Scholar
[11]Junghanns, P. and Silbermann, B., “Zur Theorie der Näherungsverfahren für singuläre Integralgleichungen auf Intervallen”, Math. Nachr. 103 (1981), 199–244.CrossRefGoogle Scholar
[12]Karpenko, L. N.. “Approximate solution of a singular integral equation by means of Jacobi polynomials”, J. Appl. Math. Mech. 30 (1966), 668–675.CrossRefGoogle Scholar
[13]Krasnosel'skii, M. A., Vainikko, G. M., Zabreiko, P. P., Rutitskii, Ya. B. and Stetsenko, Ya. V., Approximate solution of operator equations (Wolters-Noorahoff, Gröningen, 1972).CrossRefGoogle Scholar
[14]Krenk, Steen, “On the use of the interpolation polynomial for solutions of singular integral equations”, Quart. Appl. Math. 32 (1975), 479–484.CrossRefGoogle Scholar
[15]Peter, Linz, “A general theory for the approximate solution of operator equations of the second kind”, SIAM J. Numer. Anal. 14 (1977), 543–554.Google Scholar
[16]Peter, Linz, ‘An analysis of a method for solving singular integral equations”, BIT 17 (1977), 329–337.Google Scholar
[17]Muslthelishvili, N. I., Singular integral equations (P. Noordhoff, Gröningen, 1953).Google Scholar
[18]Ben, Noble, “Error analysis of collocation methods for solving Fredholm integral equations”, In Topics in numerical analysis (ed. Miller, J. J. H.), (Academic Press, New York, 1973), 211–232.Google Scholar
[19]Sloan, Ian H., “Improvement by iteration for compact operator equations”, Math. Comp. 30 (1976), 758–764.CrossRefGoogle Scholar
[20]Thomas, K. S., “On the approximate solution of operator equations”, Numer. Math. 23 (1975), 231–239.CrossRefGoogle Scholar
[21]Thomas, K. S., “Galerkin methods for singular integral equations”, Math. Camp. 36 (1981), 193–205.CrossRefGoogle Scholar
[22]Tricomi, F. G., Integral equations (Interscience, New York, 1957).Google Scholar