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A convergence theorem 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
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Abstract

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The principal result of this paper states sufficient conditions for the convergence of the solutions of certain linear algebraic equations to the solution of a (linear) singular integral equation with Cauchy kernel. The motivation for this study has been the need to provide a convergence theory for a collocation method applied to the singular integral equation taken over the arc (−1, 1). However, much of the analysis will be applicable both to other approximation methods and to singular integral equations taken over other arcs or contours. An estimate for the rate of convergence is also given.

Type
Research Article
Information
The ANZIAM Journal , Volume 22 , Issue 4 , April 1981 , pp. 539 - 552
Copyright
Copyright © Australian Mathematical Society 1981

References

[1]Dow, M. L. and Elliott, David, “The numerical solution of singular integral equations over (−1, 1)”, SIAM J. Numer.Anal. 16 (1979), 115–134.CrossRefGoogle Scholar
[2]Elliott, David, “Orthogonal polynomials associated with singular integral equations having a Cauchy kernel”, Univ. of Tasmania Maths. Dept. Tech. Rep. No. 144 (1980).Google Scholar
[3]Elliott, David, “The classical collocation method for singular integral equations”, Univ. of Tasmania Maths. Dept. Tech. Rep. No. 145 (1980).Google Scholar
[4]Gakhov, F. D., Boundary value problems (Oxford: Pergamon Press, 1966).CrossRefGoogle Scholar
[5]Linz, Peter, “A general theory for the approximate solution of operator equations of the second kind”, SIAM J. Numer. Anal. 14 (1977), 543–554.CrossRefGoogle Scholar
[6]Linz, Peter, “An analysis of a method for solving singular integral equations”, BIT 17 (1977), 329–337.CrossRefGoogle Scholar
[7]Muskhelishvili, N. I., Singular integral equations (Gröningen: P. Noordhoff, 1953).Google Scholar
[8]Noble, B., Applied linear algebra (Englewood Cliffs NJ.: Prentice-Hall, 1969).Google Scholar
[9]Noble, B., “Error analysis of collocation methods for solving Fredholm integral equations”, in Topics in numerical analysis (ed. Miller, J. J. H.) (New York: Academic Press, 1973), 211–232.Google Scholar
[10]Stummel, Friedrich, “Diskrete konvergenz linearer operatoren I”, Math. Ann. 190 (1970), 45–92.CrossRefGoogle Scholar