A stable numerical method for Volterra integral equations with discontinuous kernel (Q2473849)

The authors consider Volterra integral equations with two constant delays \(\tau_2>\tau_1>0\) \[ \begin{multlined} y(t)=f(t)+\int^t_{t-\tau_1}k_1(t-s)g(y(s))\,ds + \int^{t-\tau_1}_{t-\tau_2}k_2(t-s)g(y(s))\,ds \\ + \int^{t-\tau_2}_0 k_3(t-s)g(y(s))\,ds, \quad t \in I:[\tau_2,T]\end{multlined} \tag{1} \] with \(y(t)=u(t),\) \(t \in I:[0,\tau_2].\) Numerical methods for Volterra integral equations with discontinuous kernel as (1) need to be tuned to their peculiar form. The authors propose a version of the trapezoidal direct quadrature method adapted to such a type of equations. In order to delineate its stability properties, they investigate about the behavior of the solution of a suitable (basic) test equation and then find out under which hypotheses the trapezoidal direct quadrature method provides numerical solutions which inherit the properties of the continuous problem.