Modifications of some first kind integral equations with logarithmic kernel to improve numerical conditioning (Q800708)

We consider a pair of integral equations derived by \textit{G. T. Symm} [Numer. Math. 10, 437-445 (1967; Zbl 0155.215)] and by \textit{G. Hsiao} and \textit{R. C. MacCamy} [SIAM Rev. 15, 687-705 (1973; Zbl 0235.45006)] for use in various boundary value problems. We investigate the matrix which results when some collocation methods are applied to these equations. We find that the \(\ell_ 2\)-condition number of the matrix can be reduced by performing three minor modifications of the integral equations.