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Accurate numerical scheme for singularly perturbed parabolic delay differential equation.

Mesfin Mekuria Woldaregay1, Gemechis File Duressa2

  • 1Department of Applied Mathematics, Adama Science and Technology University, Adama, Ethiopia. msfnmkr02@gmail.com.

BMC Research Notes
|September 16, 2021
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Summary

This study introduces a novel numerical method for singularly perturbed parabolic delay differential equations, achieving accurate solutions for boundary layer problems. The proposed scheme offers improved convergence rates, validated by numerical examples.

Keywords:
Boundary layerNon-standard finite differenceSingularly perturbed

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Area of Science:

  • Numerical Analysis
  • Differential Equations
  • Computational Mathematics

Background:

  • Singularly perturbed parabolic delay differential equations present significant numerical challenges due to boundary layers.
  • Accurate computation of these equations is crucial for various scientific and engineering applications.

Purpose of the Study:

  • To develop and analyze an accurate numerical scheme for singularly perturbed parabolic delay differential equations.
  • To address the difficulties in numerical computation arising from boundary layer behavior.

Main Methods:

  • The study employs a combination of the [Formula: see text]-method for temporal discretization and a non-standard finite difference method for spatial discretization.
  • Stability and uniform convergence of the proposed numerical scheme are rigorously investigated.

Main Results:

  • The proposed scheme demonstrates uniform convergence.
  • A linear order of convergence is achieved before Richardson extrapolation.
  • A second-order convergence rate is obtained after applying Richardson extrapolation.

Conclusions:

  • The developed numerical scheme provides an accurate and stable method for solving these challenging differential equations.
  • The theoretical findings on convergence are validated through numerical experiments.