  We propose the numerical methods for solution of the weakly regular linear
and nonlinear evolutionary (Volterra) integral equation of the first kind. The
kernels of such equations have jump discontinuities along the continuous curves
(endogenous delays) which starts at the origin. In order to linearize these
equations we use the modified Newton-Kantorovich iterative process. Then for
linear equations we propose two direct quadrature methods based on the
piecewise constant and piecewise linear approximation of the exact solution.
The accuracy of proposed numerical methods is $\mathcal{O}(1/N)$ and
$\mathcal{O}(1/N^2)$ respectively. We also suggest a certain iterative
numerical scheme enjoying the regularization properties. Furthermore, we adduce
generalized numerical method for nonlinear equations. We employ the midpoint
quadrature rule in all the cases. In conclusion we include several numerical
examples in order to demonstrate the efficiency of proposed numerical methods
