Exponential methods for singularly perturbed ordinary differential–difference equations

摘要

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.