Condition number of collocation method on a quasiuniform grid for the integral equation of the 1st kind with logarithmic singularity

The paper is devoted to the substantiation of the direct numerical method for the integral equation of the first kind with logarithmic singularity on the closed curve; it is based on the piecewise-linear approximation of unknown function from its values at quasiuniform grid and collocation condition in the same gridpoints. As it is shown, if the initial integral equation is uniquely solvable and the quasiuniform parameter does not exceed 23.3, then for a step small enough the condition number of a matrix obtained in this method is not less than a constant multiplied by the step.