A nonstandard finite difference scheme for singularly perturbed reaction–diffusion differential equations with large negative and advance shifts

Singularly perturbed reaction–diffusion differential equations with large negative and advanced shifts are significant challenges in mathematical modeling across various scientific disciplines, particularly in control theory and neuroscience. This study addresses the numerical treatment of these complex equations where the presence of a small perturbation parameter combined with large shift terms creates boundary and interior layers that standard numerical approaches fail to capture accurately. We develop a robust nonstandard finite difference scheme (NSFDS) specifically designed to handle the difficulties arising from the singular perturbation parameter and mixed large shift arguments. Our approach incorporates a carefully constructed denominator function that maintains the essential qualitative properties of a continuous model. Through rigorous convergence analysis, we established that the proposed scheme is uniformly convergent with respect to the perturbation parameter, achieving second-order accuracy. Extensive numerical experiments are performed to validate the theoretical predictions, demonstrating the scheme’s superior convergence rate, efficiency, and ability to resolve layer phenomena without spurious oscillations, even for extremely small perturbation parameters. The results demonstrate that our method effectively resolves layer phenomena and maintains stability and accuracy even for extremely small values of the perturbation parameter, outperforming existing standard approaches as well as other existing previous results. This study provides a reliable computational framework for this challenging class of equations, significantly advancing state-of-the-art numerical methods for singularly perturbed problems with shift terms.