Uniform Convergent Solution of Singularly Perturbed Parabolic Differential Equations with General Temporal-Lag

In this paper, we presented a parameter uniform B-spline collocation nonstandard finite difference scheme for a class of singularly perturbed parabolic one-dimensional convection-diffusion problems with a time delay on a uniform mesh. When the delay parameter is smaller than the perturbation parameter, the delayed term is expanded in Taylor series and a B-spline collocation tridiagonal nonstandard finite difference scheme is developed. Here, the proposed finite difference scheme is unconditionally stable and is first-order convergent in the temporal direction and second-order accurate in the spatial direction. When the delay parameter is larger than the perturbation parameter, a special type of mesh is used for the temporal variable so that the delay lie on the nodal points and B-spline collocation nonstandard finite difference scheme is developed. The scheme is also unconditionally stable and is first-order convergent in the temporal direction and second-order accurate in the spatial direction. We carried out numerical simulations to verify the theoretical results.