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A uniformly convergent numerical scheme for solving singularly perturbed differential equations with large spatial

Ababi Hailu Ejere1, Gemechis File Duressa2, Mesfin Mekuria Woldaregay1

  • 1Department of Applied Mathematics, Adama Science and Technology University, 1888 Adama, Ethiopia.

SN Applied Sciences
|November 21, 2022
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Summary

This study introduces a new numerical method for singularly perturbed parabolic differential equations with large spatial delays. The developed scheme offers accurate solutions for problems with complex boundary and interior layers.

Keywords:
Boundary layersLarge delaySingularly perturbationUniform convergence

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Area of Science:

  • Numerical analysis
  • Computational mathematics
  • Differential equations

Background:

  • Singularly perturbed parabolic differential equations with large spatial delays present significant analytical challenges.
  • The presence of boundary and interior layers complicates traditional numerical approaches.
  • Efficient and accurate numerical methods are crucial for solving these complex problems.

Purpose of the Study:

  • To develop and analyze a parameter-uniform numerical scheme for a specific class of differential equations.
  • To address the difficulties arising from boundary and interior layers caused by perturbation parameters and spatial delays.
  • To establish the stability and convergence properties of the proposed numerical method.

Main Methods:

  • A numerical scheme was developed using the weighted average (-method) difference approximation on a uniform time mesh.
  • A piece-wise uniform spatial mesh with central difference method was employed.
  • Stability and convergence analyses were rigorously established for the proposed scheme.

Main Results:

  • The numerical scheme demonstrated uniform convergence of order two in the temporal direction.
  • The scheme achieved almost second-order convergence in the spatial direction.
  • The method's applicability was validated through two model examples, confirming theoretical findings.

Conclusions:

  • The proposed parameter-uniform numerical scheme effectively handles singularly perturbed parabolic differential equations with large spatial delays.
  • The method provides accurate results, even in the presence of strong boundary and interior layers.
  • The established convergence rates confirm the scheme's efficiency and reliability for such problems.