Exponentially fitted numerical method for solving singularly perturbed delay reaction-diffusion problem with nonlocal...

Area of Science:

• Numerical Analysis
• Partial Differential Equations
• Computational Mathematics

Background:

• Singularly perturbed delay reaction-diffusion equations present significant challenges due to boundary and interior layers.
• Nonlocal boundary conditions add complexity to the analysis and numerical treatment of these problems.

Purpose of the Study:

• To develop and analyze a robust numerical method for a specific class of singularly perturbed delay reaction-diffusion problems.
• To address the challenges posed by strong boundary layers and interior layers using an exponential fitting factor.

Main Methods:

• An exponentially fitted finite difference method is proposed.
• The nonlocal boundary condition is handled using the Composite Simpson's rule.
• Stability and uniform convergence analysis are rigorously established.

Main Results:

• The developed method achieves second-order uniform convergence.
• Theoretical error estimations are validated through numerical experiments.
• The method accurately captures solutions within boundary layers.

Conclusions:

• The proposed exponentially fitted finite difference method is a reliable and accurate approach for solving the considered problems.
• The numerical results confirm the theoretical findings on stability and convergence.
• This work contributes an effective computational tool for problems with nonlocal boundary conditions and dual layers.