An efficient numerical method for a singularly perturbed Fredholm integro-differential equation with integral...

Area of Science:

• Numerical analysis
• Computational mathematics
• Differential equations

Background:

• Singularly perturbed problems present challenges due to rapid solution changes.
• Fredholm integro-differential equations combine differential and integral terms.
• Integral conditions add complexity to initial value problems.

Purpose of the Study:

• To develop and analyze a numerical method for a specific class of singularly perturbed problems.
• To address the challenges posed by integral conditions in initial value problems.
• To achieve a high order of accuracy in numerical solutions.

Main Methods:

• A fitted finite difference method is employed on a Shishkin-type mesh.
• A composite trapezoidal rule is used for discretizing both the integral term and the initial condition.
• The method is designed to handle the boundary layers characteristic of singularly perturbed equations.

Main Results:

• The proposed numerical scheme achieves a uniform second-order convergence rate.
• Convergence is independent of the perturbation parameter, a key advantage.
• Theoretical error estimates are supported by numerical results.

Conclusions:

• The fitted finite difference method is effective for the considered problem class.
• The approach provides a reliable and accurate numerical solution.
• This work contributes to the numerical solution of complex differential equations.