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Computational method for singularly perturbed parabolic differential equations with discontinuous coefficients and

Imiru Takele Daba1, Gemechis File Duressa2

  • 1Department of Mathematics, Dilla university, Dilla, SNNP, P.O. Box 419, Ethiopia.

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Summary

This study introduces a new computational method for solving complex parabolic differential equations. The proposed technique offers improved accuracy and parameter-uniform convergence for these challenging problems.

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Cubic-spline in compression methodDelay parabolic differential equationsImplicit Euler methodSingular perturbation problem

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Area of Science:

  • Numerical analysis
  • Computational mathematics
  • Differential equations

Background:

  • Singularly perturbed parabolic differential equations present significant numerical challenges.
  • Discontinuous coefficients and large negative shifts exacerbate these difficulties.
  • Existing computational methods often struggle with accuracy and stability for such problems.

Purpose of the Study:

  • To develop and analyze a novel computational method for a class of second-order singularly perturbed parabolic differential equations.
  • To address the challenges posed by discontinuous coefficients and large negative shifts.
  • To demonstrate the superior accuracy and convergence properties of the proposed method.

Main Methods:

  • A hybrid approach combining the implicit Euler method for temporal discretization.
  • Utilizing cubic-spline in compression for spatial discretization.
  • Conducting numerical experiments on model examples to validate the method.

Main Results:

  • The proposed method demonstrates higher accuracy compared to existing literature methods.
  • Graphical analysis confirms that the layer behavior of solutions aligns with theoretical predictions.
  • Error analysis confirms parameter-uniform convergence with a specific order.

Conclusions:

  • The developed computational method is effective for solving the targeted class of differential equations.
  • The method exhibits enhanced accuracy and reliable convergence properties.
  • The findings contribute to the advancement of numerical techniques for singularly perturbed problems.