In this paper, we concentrate on solving second-order singularly perturbed Fredholm integro-differential equations (SPFIDEs). It is well known that solving these equations analytically is a challenging endeavor because of the presence of boundary and interior layers within the domain. To overcome these challenges, we develop a fitted second-order difference scheme that can capture the layer behavior of the solution accurately and efficiently, which is again, based on the integral identities with exponential basis functions, the composite trapezoidal rule, and an appropriate interpolating quadrature rules with the remainder terms in the integral form on a piecewise uniform mesh. Hence, our numerical method acts as a superior alternative to the existing methods in the literature. Further, using appropriate techniques in error analysis the scheme's convergence and stability have been studied in the discrete max norm. We have provided necessary experimental evidence that corroborates the theoretical results with a high degree of accuracy.

翻译：本文关注二阶奇异摄动Fredholm积分微分方程（SPFIDEs）的求解。众所周知，由于区域内存在边界层和内层，这类方程的解析求解极具挑战性。为应对这些挑战，我们基于积分恒等式与指数基函数、复合梯形法则，以及在分段均匀网格上带有积分形式余项的适当插值求积法则，开发了一种能够精确高效捕捉解层行为的拟合二阶差分格式。因此，我们的数值方法可作为现有文献方法的更优替代方案。进一步地，利用误差分析中的适当技术，在离散最大范数下研究了该格式的收敛性和稳定性。我们提供了必要的实验证据，以高精度验证了理论结果。