Quadratic B‐spline collocation method for time dependent singularly perturbed differential‐difference equation arising in the modeling of neuronalactivity.

Abstract

[EN]In this paper, we consider a time-dependent singularly perturbed differential-difference equation with small shifts arising in the field of neuroscience. The terms containing the delay and advance parameters are approximated by using the Taylor’s series expansion. The continuous problem is semi-discretized using the Crank–Nicolson finite difference method in the time direction on uniform mesh and quadratic B-spline collocation method in the space direction on exponentially graded mesh. The method is shown to be second-order uniformly convergent in space and time direction. Theoretical estimates are carried out which support the obtained numerical experiments.