A numerical scheme for a class of singularly perturbed delay parabolic partial differential equations which has wide applications in the various branches of science and engineering is suggested. The solution of these problems exhibits a parabolic boundary layer on the lateral side of the rectangular domain which continuously depends on the perturbation parameter. For the small perturbation parameter, the standard numerical schemes for the solution of these problems fail to resolve the boundary layer(s) and the oscillations occur near the boundary layer. Thus, in this paper to resolve the boundary layer the extended cubic B‐spline basis functions consisting of a free parameter λ are used on a fitted‐mesh. The extended B‐splines are the extension of classical B‐splines. To find the best value of λ the optimization technique is adopted. The extended cubic B‐splines are an advantage over the classical B‐splines as for some optimized value of λ the solution obtained by the extended B‐splines is better than the solution obtained by classical B‐splines. The method is shown to be first‐order accurate in t and almost the second‐order accurate in x. It is also shown that this method is better than some existing methods. Several test problems are encountered to validate the theoretical results.

中文翻译：

---

具有时滞的奇摄动偏微分方程的参数一致格式

提出了一类奇摄动时滞抛物型偏微分方程的数值方案，该方案在科学和工程的各个领域都有广泛的应用。这些问题的解决方案在矩形区域的侧面展示了一个抛物线形边界层，该边界层连续取决于扰动参数。对于较小的摄动参数，解决这些问题的标准数值方案无法解析边界层，并且振荡会在边界层附近发生。因此，在本文中，为解决边界层问题，在拟合网格上使用了由自由参数λ组成的扩展三次B样条基函数。扩展的B样条曲线是经典的扩展B样条曲线。为了找到最佳的λ值，采用了优化技术。扩展三次B样条曲线优于经典B样条曲线，因为对于λ的某些优化值，扩展B样条曲线获得的解比经典B样条曲线获得的解更好。该方法在t中显示为一阶精度，而在x中显示为几乎二阶精度。还表明该方法优于某些现有方法。为了验证理论结果，遇到了一些测试问题。