We consider a nonlinear singularly perturbed Volterra integro-differential equation. The problem is discretized by an implicit finite difference scheme on an arbitrary non-uniform mesh. The scheme comprises of an implicit difference operator for the derivative term and an appropriate quadrature rule for the integral term. We establish both a priori and a posteriori error estimates for the scheme that hold true uniformly in the small perturbation parameter. Numerical experiments are performed and results are reported for validation of the theoretical error estimates.

中文翻译：

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非线性奇摄动的Volterra积分微分方程的分析

我们考虑一个非线性奇摄动的Volterra积分微分方程。通过在任意非均匀网格上的隐式有限差分方案将问题离散化。该方案包括一个用于导数项的隐式差分算子和一个用于积分项的适当正交规则。我们为该方案建立了先验误差和后验误差估计，它们在小扰动参数中均等地保持正确。进行数值实验，并报告结果以验证理论误差估计。