Numerical approach for differential-difference equations having layer behaviour with small or large delay using non-polynomial spline

A numerical approach is suggested for the layer behaviour differential-difference equations with small and large delays in the differentiated term. Using the non-polynomial spline, the numerical scheme is derived. The discretization equation is constructed using the first-order derivative continuity at non-polynomial spline internal mesh points. A fitting parameter is introduced into the scheme with the help of the singular perturbation theory to minimize the error in the solution. The maximum errors in the solution are tabulated to verify the competence of the numerical method relative to the other methods in literature. We also focussed on the impact of large delays on the layer behaviour or oscillatory behaviour of solutions using a special mesh with and without fitting parameter in the proposed scheme. Graphs show the effect of the fitting parameter on the solution layer.

Automated summary

A numerical approach is proposed for differential-difference equations with small and large delays, utilizing non-polynomial spline and singular perturbation theory. The method is validated by tabulating maximum errors in the solution and focusing on the impact of large delays on solution behavior. The study also explores the effect of fitting parameter on solution layer using special mesh configurations.