A discretization of Volterra integral equations of the third kind with weakly singular kernels

In this paper we propose a method of piecewise constant approximation for the solution of ill-posed third kind Volterra equations *(T) dr = /(), ί € [0, 1], 0 < ο < 1. Q \l — Τ ) Here p(t) vanishes on some subset of [£1,^2] C [0,1] and \p(t)\ < δ for t 6 [ίι,^], where δ is a sufficiently small positive number. The proposed method gives the accuracy O^"/^"*)) with respect to the JD2-norm, where i/ is the parameter of sourcewise representation of the exact solution on [^1,^2]» d uses no more than O(6~(~V\ogf δ~) values of Galerkin functional, where λ G (0, 1/2) is determined in the act of choosing the regularization parameter within the framework of Morozov's discrepancy principle.