An implicit scheme for singularly perturbed parabolic problem with retarded terms arising in computational neuroscience

A class of time-dependent singularly perturbed convection diffusion problems with retarded terms arising in computational neuroscience is considered. In particular, a numerical scheme for the parabolic convection-diffusion problem where the second-order derivative with respect to the spatial direction is multiplied by a small perturbation parameter epsilon(2); epsilon is an element of (0, 1] and the shifts delta, eta are of o(epsilon) is constructed. The Taylor series expansion is used to tackle the shift terms. The continuous problem is semidiscretized using the Crank-Nicolson finite difference method in the temporal direction and the resulting set of ordinary differential equations is discretized using a midpoint upwind finite difference scheme on an appropriate piecewise uniform mesh, which is dense in the boundary layer region. It is shown that the proposed numerical scheme is second-order accurate in time and almost first-order accurate in space with respect to the perturbation parameter epsilon. To validate the computational results and efficiency of the method some numerical examples are encountered and the numerical results are compared with some existing results. It is observed that the numerical approximations are fairly good irrespective of the size of the delay and the advance till they are of o(epsilon). The effect of the shifts on the boundary layer has also been observed.