NUMERICAL SIMULATION OF TIME-DEPENDENT PARABOLIC DIFFERENTIAL$-$DIFFERENCE EQUATIONS WITH SINGULAR PERTURBATION USING AN ADAPTIVE SPLINE

Abstract

The work introduces a numerical scheme tailored for time‑dependent parabolic singularly perturbed differential–difference equations (SPPDDEs), specifically those containing small delay terms in both the convection and diffusion components. When the delay or advance parameters are much smaller than the perturbation parameter, the delay terms are approximated using a Taylor series expansion. The approach applies the backward Euler method for time discretization and employs an adaptive spline technique on a uniform spatial grid. This combination yields first‑order accuracy in time and second‑order accuracy in space. A series of numerical experiments is carried out to support the theoretical analysis and to benchmark the method against several established techniques. The results indicate that the proposed scheme offers enhanced precision and improved convergence relative to existing methods.