High-rate convergent multistep collocation techniques to a first-kind Volterra integral equation along with the proportional vanishing delay

In the present study, we construct a considerably fast convergent multistep collocation technique in order to solve Volterra integral equations, especially first-kind ones with variable vanishing delays. Through a robust theoretical analysis, the optimal global convergence of the numerically achieved solutions to their exact counterparts has been demonstrated with the corresponding high orders. The allusion to the strategy of reformulating a first-kind Volterra integral equation into a second-kind Volterra functional integral equation, assists us for the establishment of regularity, existence and uniqueness features of analytical solution over under consideration equation. The existence and uniqueness of numerical solution have also been shown. Eventually, some test problems have been provided to evaluate effectiveness of the proposed multistep collocation technique.