In this article, we consider a class of singularly perturbed two-parameter parabolic partial differential equations with time delay on a rectangular domain. The solution bounds are derived by asymptotic analysis of the problem. We construct a numerical method using a hybrid monotone finite difference scheme on a rectangular mesh which is a product of uniform mesh in time and a layer-adapted Shishkin mesh in space. The error analysis is given for the proposed numerical method using truncation error and barrier function approach, and it is shown to be almost second- and first-order convergent in space and time variables, respectively, independent of both the perturbation parameters. At the end, we present some numerical results in support of the theory.

中文翻译：

---

两参数奇摄动时滞抛物线问题的鲁棒数值方法

在本文中，我们考虑一类在矩形域上具有时滞的奇摄动两参数抛物型偏微分方程。通过对问题的渐近分析得出解的界线。我们在矩形网格上使用混合单调有限差分方案构造了一种数值方法，该矩形网格是时间上均匀的网格和空间上的层自适应Shishkin网格的乘积。使用截断误差和势垒函数方法对所提出的数值方法进行了误差分析，结果表明，空间和时间变量分别几乎是二阶和一阶收敛的，与扰动参数无关。最后，我们提出了一些数值结果以支持该理论。