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.
Heliyon
|October 4, 2022
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.
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.
Keywords:
Cubic-spline in compression methodDelay parabolic differential equationsImplicit Euler methodSingular perturbation problemMore Related Videos
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