Volume 41, pp. 190-248, 2014.

Collocation for singular integral equations with fixed singularities of particular Mellin type

Peter Junghanns, Robert Kaiser, and Giuseppe Mastroianni

Abstract

This paper is concerned with the stability of collocation methods for Cauchy singular integral equations with fixed singularities on the interval $[-1,1]$. The operator in these equations is supposed to be of the form $a\mathcal{I}+b\mathcal{S}+{\mathcal{B}}^{\pm }$ with piecewise continuous functions $a$ and $b$. The operator $\mathcal{S}$ is the Cauchy singular integral operator and ${\mathcal{B}}^{\pm }$ is a finite sum of integral operators with fixed singularities at the points $\pm 1$ of special kind. The collocation methods search for approximate solutions of the form $\nu (x)p_n(x)$ or $\mu (x)p_n(x)$ with Chebyshev weights $\nu (x)=\sqrt{\frac{1+x}{1-x}}$ or $\mu (x)=\sqrt{\frac{1-x}{1+x}}$, respectively, and collocation with respect to Chebyshev nodes of first and third or fourth kind is considered. For the stability of collocation methods in a weighted ${\mathbf{L}}^2$-space, we derive necessary and sufficient conditions.

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Key words

collocation method, stability, $C^{\ast }$-algebra, notched half plane problem

AMS subject classifications

65R20, 45E05

Links to the cited ETNA articles

[12]	Vol. 14 (2002), pp. 79-126 P. Junghanns and A. Rathsfeld: A polynomial collocation method for Cauchy singular integral equations over the interval
[13]	Vol. 17 (2004), pp. 11-75 P. Junghanns and A. Rogozhin: Collocation methods for Cauchy singular integral equations on the interval