Initial value problem for delay differential equation and its solutions

In this study, we deal with an initial value problem for a linear second order delay differential equation. We investigate both the analytical and numerical solution of this problem. We use the steps method and the Laplace transform for the analytical solution of the problem. We construct an appropriate difference scheme on a uniform using the method of integral identities which contains basis functions and interpolating quadrature rules with weight and remainder term in integral form for the numerical solution. We also proved that the method is first-order convergent in discrete maximum norm. Next, we present an example that confirms the theoretical results. Finally, we compare the proposed method with the implicit Euler method. Keywords: Delay differential equation, Error estimate, Finite difference method, Initial-value problem.