a fitted operator method of line scheme for solving two-parameter singularly perturbed parabolic convection-diffusion problems with time delay

This paper presents a parameter-uniform numerical scheme for the solution of two-parameter singularly perturbed parabolic convection-diffusion problems with a delay in time. the continuous problem is semi-discretized using the crank-nicolson finite difference method in the temporal direction. the resulting differential equation is then discretized on a uniform mesh using the fitted operator finite difference method of line scheme. the method is shown to be accurate in $ o(left(delta t right)^{2}  + n^{-2}) $, where $ n $ is the number of mesh points in spatial discretization and $ delta t $ is the mesh length in temporal discretization. the parameter-uniform convergence of the method is shown by establishing the theoretical error bounds. finally, the numerical results of the test problems validate the theoretical error bounds.