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Extended cubic B-spline collocation method for singularly perturbed parabolic differential-difference equation

Imiru Takele Daba1, Gemechis File Duressa2

  • 1Department of Mathematics, Wollega University, Nekemte, Oromia, Ethiopia.

International Journal for Numerical Methods in Biomedical Engineering
|November 22, 2020
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A new numerical method solves complex neuroscience equations with small shifts accurately. This ε-uniform method ensures reliable results for computational neuroscience models.

Keywords:
extended cubic B-splinesparabolic differential-difference equationsingular perturbation problem

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Area of Science:

  • Computational Neuroscience
  • Applied Mathematics
  • Numerical Analysis

Background:

  • Singularly perturbed parabolic differential-difference equations with small shifts are crucial in computational neuroscience.
  • Accurate numerical solutions are essential for understanding complex neural dynamics.

Purpose of the Study:

  • To develop a parameter uniform numerical method for singularly perturbed parabolic differential-difference equations with small shift arguments.
  • To ensure ε-uniform convergence for improved accuracy in computational neuroscience models.

Main Methods:

  • Taylor's series expansion is used to approximate terms with shift arguments.
  • The implicit Euler method is applied in the temporal direction.
  • Extended cubic B-spline basis functions with a free parameter λ are used for the spatial direction.

Main Results:

  • The proposed method achieves an accuracy of order .
  • The method demonstrates ε-uniform convergence, preserving accuracy across different parameter values.
  • Numerical results for two test examples show excellent agreement with theoretical predictions and existing methods.

Conclusions:

  • The developed numerical method is effective and accurate for solving a specific class of differential-difference equations relevant to neuroscience.
  • The ε-uniform convergence property makes the method robust for a wide range of parameters.
  • This approach provides a reliable tool for computational neuroscience research.