A hybrid numerical scheme for singularly perturbed parabolic differential-difference equations arising in the modeling of neuronal variability

This study aims at constructing a robust numerical scheme for solving singularly perturbed parabolic delay differential equations arising in the modeling of neuronal variability. Taylor's series expansion is applied to approximate the shift terms. The obtained result is approximated by using the implicit Euler method in the temporal discretization on a uniform step size with the hybrid numerical scheme consisting of the midpoint upwind method in the outer layer region and the cubic spline in tension method in the inner layer region on a piecewise uniform Shishkin mesh in the spatial discretization. The constructed scheme is shown to be an $\epsilon$-uniformly convergent accuracy of order $O\left(\Lambda t+N^{-2}{\ln }^{3}N\right)$. Two model examples are given to testify the theoretical findings.