Exact numerical solutions and high order nonstandard difference schemes for a second order delay differential equation

In this work, we obtain an expression, given in terms of some special functions, for the exact numerical solution of the second order delay differential equation x"(t)=ax(t)+bx(t-tau), with general initial condition x(t)=f(t) for -tau <= t <= 0. Based on this exact solution, we propose a family of nonstandard finite difference methods that combine high order accuracy and efficient computational properties. We also show that the numerical solutions obtained with the new schemes are consistent with asymptotic stability properties of the exact solutions.