Exponential methods for singularly perturbed ordinary differential-difference equations

A variety of exponential methods based on piecewise analytical solutions of advection-reaction-diffusion operators is proposed for the numerical solution of linear ordinary differential-difference equations with small delay. These methods are shown to be exact (in exact arithmetic) for linear ordinary differential equations with constant coefficients and right-hand side, are non-standard and include three-point non-local approximations for the coefficients, right-hand side and dependent variable. Also, a variety of exponential methods based on the piecewise analytical solution of advection-diffusion operators with non-local approximations is presented. It is shown that exponential methods based on the advection-diffusion-reaction operator provide more accurate results than those based on the analytical solution of the advection-diffusion operator. It is also shown that, at least, for the problems considered here, non-local approximations for either the coefficients or the right-hand side of the differential equations do not result in an increase of accuracy over those based on local approximations.