Piecewise polynomial numerical method for Volterra integral equations of the fourth-kind with constant delay

This work studies the fourth-kind integral equation as a mixed system of first and secondkind Volterra integral equations (VIEs) with constant delay. Regularity, smoothing properties and uniqueness of the solution of this mixed system are obtained by using theorems which give the relevant conditions related to first and second-kind VIEs with delays. A numerical collocation algorithm making use of piecewise polynomials is designed and the global convergence of the proposed numerical method is established. Some typical numerical experiments are also performed which confirm our theoretical result.