On optimization of direct methods of solving weakly singular integral equations (Q1260662)

The optimization of (adaptive) direct methods for Fredholm-type integral equations with smooth kernels was studied by, e.g., \textit{S. G. Solodky} [Ukr. Mat. Zh. 42, No. 1, 95-102 (1990; M.R. 91f: 65115)] and by \textit{S. V. Perewersew} and \textit{S. G. Solodki} [Optimierung direkter Verfahren für Gleichungen 2. Art mit Glättungsoperatoren, Math. Nachr. 153, 101-108 (1991)]. In the present paper the authors extend their results to adaptive direct methods for \(z= Hz+ f\), where the integral operator \(H: C\to C^{1- \alpha}\) \((0<\alpha<1)\) is given by \(Hz(t):=\int^ 1_ 0| t- \tau|^{-\alpha} h(t,\tau)z(\tau)d\tau\), with smooth \(h(t,\tau)\). The last section of the paper is devoted to a class of singular integral equations whose kernels have logarithmic singularities (Peierls integral equation of transport theory). There are no numerical results.