A parameter uniform difference scheme for singularly perturbed parabolic delay differential equation with Robin type boundary condition

Abstract

A Robin type boundary value problem for a singularly perturbed parabolic delay differential equation is studied on a rectangular domain in the x - t plane. The second-order space derivative is multiplied by a small parameter, which gives rise to parabolic boundary layers on the two lateral sides of the rectangle. A numerical method comprising a standard finite difference scheme on a rectangular piecewise uniform fitted mesh of Nx × Nt elements condensing in the boundary layers is suggested and it is proved to be parameter-uniform. More specifically, it is shown that the errors are bounded in the maximum norm by C(Nx−2ln2Nx+Nt−1), where C is a constant independent of Nx, Nt and the small parameter. To validate the theoretical result an example is provided.