Numerical solution of singularly perturbed parabolic differential difference equations

• Published in: Қарағанды университетінің хабаршысы. Математика сериясы Vol. 119; no. 3; pp. 107 - 124
• Main Authors: Huka, D.B., Melesse, W.G., Gelu, F.W.
• Format: Journal Article
• Language: English
• Published: Academician Ye.A. Buketov Karaganda University 30.09.2025
• Subjects: differential difference equation; implicit Euler technique; parameter-uniform convergence; Shishkin mesh; singular perturbation problem; upwind method
• ISSN: 2518-7929; 2663-5011; 2663-5011
• DOI: 10.31489/2025m3/107-124

This study presents a computational method for the singularly perturbed parabolic differential difference equations with small negative shifts in convection and reaction terms. To handle the small negative shifts, the Taylor series expansion is applied. Then, the resulting asymptotically equivalent singularly perturbed parabolic convection-diffusion-reaction problem is discretized in the time variable using the implicit Euler technique on a uniform mesh, while the upwind method on a Shishkin mesh is used to discretize the space variable. Almost first-order convergence was achieved by establishing the stability and parameter-uniform convergence of the method. The Richardson extrapolation approach improved the rate of convergence to nearly second-order. Numerical experiments have been carried out in order to support the findings from the theory. The numerical results are presented in tables in terms of maximum absolute errors and graphs. The present results improve the existing methods in the literature. This finding highlights the efficiency of the method, paving the way for its application in other types of singularly perturbed parabolic problems. This method is capable of greatly improving computing performance in a variety of scenarios, which researchers can further explore.