A convolution test equation for double delay integral equations

In this paper we consider Volterra integral equations with two constant delays and we carry out the stability analysis of direct quadrature methods with respect to the linear convolution test equation y ( t ) = 1 + ∫ t − τ 2 t − τ 1 ( λ + μ ( t − s ) ) y ( s ) d s , t ∈ [ τ 2 , T ] . We investigate the analytical behavior of the solution of the test equation and derive the qualitative and quantitative properties of the numerical solution. The numerical experiments show that the stability conditions obtained represent sufficient conditions for the stability of the numerical method applied to a more general equation.