Numerical solutions of delay volterra integro-differential equations

In this study, we consider an initial value problem for a linear first order Volterra integro differential equation with delay. We construct a new difference scheme for numerical solution of this problem by the finite difference method. The method is based on difference scheme on a uniform mesh which is achieved by using the method of integral identities which includes the exponential basis functions and applying interpolating quadrature formulas which contain the remainder term in integral form. Also, the method is proved to be first-order convergent in the discrete maximum norm. Moreover, a numerical experiment is performed to verify the theoretical results. Finally, the proposed scheme is compared with the implicit Euler scheme.