Computation Method for a Differential-Difference Equation with Boundary Layer in Neuronal Variability Modelling using a Mixed Nonpolynomial Spline

Abstract

This article introduces a computational difference scheme designed to solve singularly perturbed differential-difference equations (SPDDE) exhibiting boundary layer behaviour, utilizing a mixed nonpolynomial spline. To effectively manage boundary layer oscillations, the SPDDE is transformed into an equivalent two-point boundary layer problem. The adjustment of the fitting factor within the difference scheme is crucial for controlling these oscillations. The Thomas algorithm is employed to illustrate the discrete system of the difference scheme. This computational strategy demonstrates a second-order convergence rate, with a brief discussion on convergence analysis. The effectiveness of this approach is validated through numerical examples, with comprehensive comparisons confirming its reliability and consistency. Graphical representations of layer profiles are provided for various delay and advance parameter values