A fitted operator numerical method for singularly perturbed Fredholm integro-differential equation with integral
Aklilu Fufa Oljira1, Mesfin Mekuria Woldaregay2
1Department of Mathematics, College of Natural and Computational Science, Oda Bultum University, Chiro, Ethiopia.
BMC Research Notes
|January 15, 2024
Summary
A new numerical method uniformly converges for a difficult-to-solve singularly perturbed Fredholm integro-differential equation. This fitted operator finite difference scheme offers accurate solutions for problems with boundary layers.
Area of Science:
- Numerical Analysis
- Computational Mathematics
- Applied Mathematics
Background:
- Singularly perturbed Fredholm integro-differential equations present significant numerical challenges due to boundary layers.
- Standard numerical methods often fail to provide accurate solutions for these types of equations.
- Integral initial conditions add further complexity to the problem formulation.
Purpose of the Study:
- To propose a uniformly convergent numerical scheme for a specific class of singularly perturbed Fredholm integro-differential equations.
- To address the difficulties associated with left boundary layers in these equations.
- To provide a reliable method for problems with integral initial conditions.
Main Methods:
- A fitted operator finite difference method is employed to discretize the differential component of the equation.
- The composite Simpson rule is utilized for approximating the integral terms within the equation and the initial condition.
- Stability bounds and error estimations are rigorously analyzed to demonstrate convergence properties.
Main Results:
- The proposed numerical scheme achieves uniform convergence of order one in the maximum norm.
- Theoretical analysis is substantiated by numerical examples, demonstrating the accuracy and efficiency of the method.
- Maximum absolute errors and convergence rates are computed for validation.
Conclusions:
- The developed fitted operator finite difference method provides an effective and accurate solution for singularly perturbed Fredholm integro-differential equations with integral initial conditions.
- The scheme's uniform convergence guarantees reliable results even in the presence of boundary layers.
- This work contributes a valuable numerical tool for researchers and practitioners dealing with complex integro-differential equations.
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