This report addresses the boundary value problem for a second-order linear singularly perturbed FIDE. Traditional methods for solving these equations often face stability issues when dealing with small perturbation parameters. We propose an exact finite difference method to solve these equations and provide a detailed stability and $\varepsilon$-uniform convergence analysis. Our approach is validated with an example, demonstrating its uniform convergence and applicability, with a convergence order of 1. The results illustrate the method's robustness in handling perturbation effects efficiently.

翻译：本报告针对一类二阶线性奇异摄动Fredholm积分微分方程的边值问题展开研究。传统方法在求解此类方程时，面对小摄动参数常面临稳定性问题。我们提出一种精确有限差分方法来求解这些方程，并给出详细的稳定性及$\varepsilon$-一致收敛性分析。通过算例验证了所提方法的有效性，证明了该方法具有一致收敛性及适用性，其收敛阶为1。结果表明该方法能有效处理摄动效应，具有较强的鲁棒性。