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Fourth-order fitted mesh scheme for semilinear singularly perturbed reaction-diffusion problems.

Birtukan Tebabal Reda1, Tesfaye Aga Bullo2, Gemechis File Duressa1

  • 1Department of Mathematics, College of Natural Science, Jimma University, Jimma, Ethiopia.

BMC Research Notes
|November 30, 2023
PubMed
Summary
This summary is machine-generated.

This study introduces a fourth-order fitted mesh scheme for semilinear singularly perturbed reaction-diffusion problems, offering more accurate solutions than existing methods.

Keywords:
Accurate solutionFourth-orderNon-uniform meshSemilinear singularly perturbed

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Area of Science:

  • Numerical analysis
  • Computational mathematics
  • Applied mathematics

Background:

  • Singularly perturbed reaction-diffusion problems are common in various scientific and engineering fields.
  • Accurate numerical solutions are crucial for understanding these complex phenomena.
  • Existing methods may lack sufficient accuracy for certain problem types.

Purpose of the Study:

  • To develop and present a novel fourth-order fitted mesh scheme.
  • To enhance the accuracy of solutions for semilinear singularly perturbed reaction-diffusion problems.
  • To provide a more reliable numerical tool for researchers.

Main Methods:

  • Application of the quasilinearization technique to handle the semilinear term.
  • Discretization of the solution domain using a piecewise uniform mesh.
  • Finite difference approximations to convert the differential equation into a system of difference algebraic equations.
  • Solution of the resulting system using the Thomas algorithm.

Main Results:

  • The proposed scheme achieves fourth-order accuracy.
  • Convergence analysis confirms the method's stability and error bounds.
  • Numerical examples demonstrate superior accuracy compared to existing methods.
  • The scheme is shown to be applicable to the target problem class.

Conclusions:

  • The developed fourth-order fitted mesh scheme effectively solves semilinear singularly perturbed reaction-diffusion problems.
  • The method offers improved accuracy and reliability.
  • This work contributes a valuable numerical technique to the field.