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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
Summary
A new numerical method solves complex neuroscience equations with small shifts accurately. This ε-uniform method ensures reliable results for computational neuroscience models.
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.
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