Solvability and numerical solutions of systems of nonlinear Volterra integral equations of the first kind with piecewise continuous kernels

The existence theorem for systems of nonlinear Volterra integral equations kernels of the first kind with piecewise continuous is proved. Such equations model evolving dynamical systems. A numerical method for solving nonlinear Volterra integral equations of the first kind with piecewise continuous kernels is proposed using midpoint quadrature rule. Also numerical method for solution of systems of linear Volterra equations of the first kind is described. The examples demonstrate efficiency of proposed algorithms. The accuracy of proposed numerical methods is $\mathcal{O}(N^{-1})$.