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

•Singularly perturbed time delay partial differential equation is considered.•Boundary condition is of Robin type.•A parameter uniform numerical method is suggested.•An error estimate is derived and the error is of order two.•An illustration is provided.

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.