Numerical solutions of second order linear volterra integro-differential equations with delay

In this study, an initial value problem for second-order linear delay Volterra integro-differential equations is considered. Both the analytical and numerical solutions of the considered problem are investigated. The step method is used to find the analytical solution of the problem, and a difference scheme is established using the finite difference method to find the numerical solution. This difference scheme is based on interpolation quadrature formulas that contain a weight function and have the remainder term in integral form. Additionally, the convergence of the method is examined, and it is determined to be first-order convergent in the discrete maximum norm. By examining two examples, it is shown that the numerical results support the theoretical results. Finally, the presented scheme is compared with the classical scheme.