Journal Article

Fitted Operator Method for Singularly Perturbed Delay Parabolic Problems With Boundary Turning Points

Tiruneh, Awoke Andargie, Tsega, Endalew Getnet, Kebede, Yimesgen Mehari

2025

Overview

In this paper, a numerical scheme for time‐delay singularly perturbed parabolic convection‐diffusion problems with boundary turning points is presented. The solution of the problem shows a steep gradient or rapid variation at the left region of the spatial domain as the perturbation parameter approaches zero. The combination of the singular perturbation parameter, time lag and turning points complicates the problem’s theoretical analysis and numerical solution. Classical numerical methods are inefficient and inaccurate when it comes to solving such complex problems. To overcome these difficulties, first, we discretize the time variable using the Crank–Nicolson method on special uniform mesh such that its discrepancy with time lag lies on the nodal points. Then, we introduce the fitting factor at the diffusion part of the differential equation in order to achieve a uniformly valid solution over the entire region of the domain. The stability and parameter uniform convergence of the scheme are analysed using the minimum principle and solution bounds. It is shown that the scheme is stable and parameter uniform convergence with second‐order accuracy in time and first order in space. Two model examples are presented to demonstrate the scheme’s applicability. Their numerical results reinforce the theoretical analysis.

• Publisher: John Wiley & Sons, Inc,Wiley
• Subject: Accuracy / Convection-diffusion equation / Convergence / Crank-Nicholson method / Differential equations / Mathematical analysis / Methods / Numerical analysis / Numerical methods / Parameters / Partial differential equations / Singular perturbation methods / Time lag