An efficient numerical approach for singularly perturbed time delayed parabolic problems with two-parameters.

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Differential Equations

Background:

• Singularly perturbed problems often exhibit boundary layers, posing numerical challenges.
• Time delays introduce further complexity in solving parabolic partial differential equations.
• Efficient and stable numerical methods are crucial for analyzing such problems.

Purpose of the Study:

• To design an efficient numerical scheme for singularly perturbed time-delayed parabolic problems with two parameters.
• To enhance the accuracy and order of convergence of the numerical method.
• To analyze the parameter-uniform convergence properties of the proposed scheme.

Main Methods:

• Approximation of time derivatives using the implicit Euler method.
• Approximation of space derivatives using a non-standard finite difference method.
• Enhancement of accuracy and convergence order via Richardson extrapolation.

Main Results:

• The proposed scheme accurately captures the layer behavior of solutions, consistent with theoretical predictions.
• Numerical experiments on model examples demonstrate the effectiveness of the method.
• Error analysis confirms parameter-uniform convergence with a specific order.

Conclusions:

• The developed numerical scheme is efficient and accurate for the targeted problem class.
• The method's parameter-uniform convergence is a significant advantage for practical applications.
• The findings contribute to the numerical solution of complex differential equations with delays and perturbations.