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A tension spline fitted numerical scheme for singularly perturbed reaction-diffusion problem with negative shift.

Ababi Hailu Ejere1, Tekle Gemechu Dinka2, Mesfin Mekuria Woldaregay2

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

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This study presents a new numerical method for a complex reaction-diffusion problem with boundary and interior layers. The developed scheme achieves accurate results with uniform convergence in both time and space.

Keywords:
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Area of Science:

  • Numerical Analysis
  • Partial Differential Equations
  • Computational Mathematics

Background:

  • Singularly perturbed reaction-diffusion problems present challenges due to boundary and interior layers.
  • Analytical solutions are often intractable for these problems owing to rapid solution variations.

Purpose of the Study:

  • To develop and analyze a uniformly convergent numerical scheme for a singularly perturbed reaction-diffusion problem with a negative shift.
  • To address the difficulties posed by boundary and interior layers in the problem's solution.

Main Methods:

  • A numerical scheme combining the implicit Euler method (temporal) and a fitted tension spline method (spatial) was employed.
  • Uniform meshes were utilized for the spatial discretization.
  • Stability and uniform error estimates were rigorously investigated.

Main Results:

  • The developed numerical scheme demonstrates uniform convergence.
  • The scheme achieves a convergence rate of order one in time and order two in space.
  • Numerical examples validated the theoretical findings on stability and error estimates.

Conclusions:

  • The proposed numerical scheme effectively handles singularly perturbed reaction-diffusion problems with negative shifts.
  • The method provides accurate and stable solutions, particularly in regions with layers.
  • The findings contribute to the numerical treatment of complex differential equations.