A Legendre–Petrov–Galerkin method for solving Volterra integro-differential equations with proportional delays

In this paper, a fully discrete Legendre–Petrov–Galerkin method is presented for solving functional Volterra integro-differential equations with vanishing delays. This method produces a fully discrete linear system. We prove that this system has a unique solution for sufficiently large n, where n+1 denotes the order of the system. Moreover, we prove that the approximate solution and its corresponding derivative function arrive at an optimal...