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Solving singularly perturbed fredholm integro-differential equation using exact finite difference method.
Solomon Regasa Badeye1, Mesfin Mekuria Woldaregay2, Tekle Gemechu Dinka1
1Department of Applied Mathematics, Adama Science and Technology University, Adama, Ethiopia.
BMC Research Notes
|September 28, 2023
Summary
This study introduces a novel numerical scheme for singularly perturbed Fredholm integro-differential equations. The method achieves second-order uniform convergence, validated by theoretical analysis and computational results.
Area of Science:
- Numerical Analysis
- Computational Mathematics
- Differential Equations
Background:
- Singularly perturbed Fredholm integro-differential equations present significant numerical challenges.
- Existing methods may struggle with uniform convergence across various parameter values.
Purpose of the Study:
- To design and analyze a novel numerical scheme for singularly perturbed Fredholm integro-differential equations.
- To ensure stability and uniform convergence of the proposed method.
Main Methods:
- The scheme combines the exact (non-standard) finite difference method for the differential part.
- The composite Simpson's 1/3 rule is employed for the integral part.
Main Results:
- Stability and uniform convergence are rigorously analyzed using solution and truncation error bounds.
- Computational results demonstrate second-order uniform convergence for model examples.
- The method shows excellent agreement between theoretical predictions and numerical outcomes.
Conclusions:
- The proposed numerical scheme is effective and accurate for solving the target equations.
- The method exhibits robust performance across different perturbation parameters and mesh sizes.
- This work contributes a valuable tool for the numerical solution of these complex equations.
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