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

Junghanns, Peter ; Kaiser, Robert ; Mastroianni, Giuseppe

Electronic transactions on numerical analysis, Tome 41 (2014), pp. 190-248

Résumé

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 $\pm1$ 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.

Détail

Classification : 65R20, 45E05
Keywords: collocation method, stability, $C^*$-algebra, notched half plane problem