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

Imiru Takele Daba¹, Wondewosen Gebeyaw Melesse², Fasika Wondimu Gelu²

¹Mathematics, Dilla University, Dilla, 419, Ethiopia. imirutakele@gmail.com.

BMC Research Notes

|June 6, 2024

Summary

This study introduces an efficient numerical scheme for singularly perturbed time-delayed parabolic problems. The method ensures parameter-uniform convergence, enhancing accuracy for complex mathematical models.

Keywords:

Non-standard finite difference method Richardson extrapolation Time delay Two-parameter

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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.