A Nyström method for Cauchy singular integral equations with negative index

This talk deals with the numerical treatment of the following type of Cauchy singular integral equations with constant coefficients \begin{equation} \label{Cauchyequ} a F(y)+\frac{b}{\pi}\int_{-1}^1 \frac{F(x)}{x-y}dx+ \mu\int_{-1}^1k(x,y)F(x)dx=g(y), \quad |y|<1, \end{equation} where $a,b\in \RR$ are constants such that $a^2+b^2=1$, $b \neq 0$, $\mu \in \RR$ and $k$ and $g$ are given functions on $(-1,1)^2$ and $(-1,1)$, respectively. The function $F$ is the unknown of the equation and it is usually represented in the following form \[F(x)=f(x)v^{alpha,\beta}(x),\] where $f(x)$ is a smooth function and $v^{alpha,\beta}(x)=(1-x)^\alpha(1+x)^\beta$ is a Jacobi weight. The exponents $-1<\alpha, \beta<1$ are given by \[\alpha=M -\frac 1{2\pi i}\log\left(\frac{a+ib}{a-ib}\right), \quad \beta=N +\frac 1{2\pi i}\log\left(\frac{a+ib}{a-ib}\right),\] with $M$ and $N$ integers chosen so that $\chi=-(\alpha+\beta)=-(M+N)=-1$. In this paper we consider equations of the form (\ref{Cauchyequ}) with $\chi=-1$ in spaces of continuous functions with uniform norm. The procedure, we propose here, consists in reducing (\ref{Cauchyequ}), under sui-table assumptions on $k$ and $g$, to an equivalent regularized Fredholm integral equation and in solving the latter by a numerical Nyström-type method. This approach permits to solve a determined and well conditioned linear system and to construct an interpolating function convergent to the exact solution of the original problem.