On a Collocation-quadrature Method for the Singular Integral Equation of the Notched Half-plane Problem

A Cauchy singular integral equation describing the notched halfplane problem of two-dimensional elasticity theory is considered. This equation contains an additional fixed singularity represented by a Mellin convolution operator. We study a polynomial collocation-quadrature method for its numerical solution which takes into account the “natural” asymptotic of the solution at the endpoints of the integration interval and for which until now no criterion for stability is known. The present paper closes this gap. For this, we present a new technique of proving that the operator sequence of the respective collocation-quadrature method belongs to a certain C∗-algebra, in which we can study the stability of these sequences. One of the main ingredients of this technique is to show that the part of the operator sequence, associated with the Mellin part of the original equation, is “very close” to the finite section of particular operators belonging to a C∗-algebra of Toeplitz operators. Moreover, basing on these stability results numerical results are presented obtained by an implementation of the proposed method.