On the second-order neutral Volterra integro-differential equation and its numerical solution

In this paper, we consider an initial-value problem for a second-order neutral Volterra integro-differential equation. First, we give the stability inequality indicating the stability of the problem with respect to the right-side and initial conditions. Further, we develop a finite difference method that uses for differential part second difference derivative, for the integral part appropriate composite trapezoidal and midpoint rectangle rules followed by second-order accurate difference quantities at intermediate points. Error estimate for the approximate solution is established, which shows the second-order accuracy. Finally, the numerical experiments are presented confirming the accuracy of the proposed scheme. •In this paper, a new approach to solve numerically neutral Volterra integro-differential equation (NVIDE) has been considered.•The last works related to numerical analysis of NVIDE were concerned with collocation method, Legendre spectral method and etc. But the most important feature of our study is a finite difference scheme has been constructed on a uniform mesh.•Using composite trapezoidal and midpoint rectangle rules followed by second-order accurate difference quantities at intermediate points, we construct the finite difference scheme which have a second-order accuracy.•Therefore we obtain numerical methods which are as high accuracy as possible.