This paper deals with numerical treatment of singularly perturbed parabolic differential equations having large time delay. The highest order derivative term in the equation is multiplied by a perturbation parameter , taking arbitrary value in the interval . For small values of , solution of the problem exhibits an exponential boundary layer on the right side of the spatial domain. The properties and bounds of the solution and its derivatives are discussed. The considered singularly perturbed time delay problem is solved using the Crank-Nicolson method in temporal discretization and exponentially fitted operator finite difference method in spatial discretization. The stability of the scheme is investigated and analysed using comparison principle and solution bound. The uniform convergence of the scheme is discussed and proven. The formulated scheme converges uniformly with linear order of convergence. The theoretical analysis of the scheme is validated by considering numerical test examples for different values of .

中文翻译：

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奇摄动时滞对流扩散方程的新型数值格式

本文研究了具有大时滞的奇摄动抛物型微分方程的数值处理。方程中的最高阶导数项与扰动参数相乘，在区间中取任意值。对于的较小值，问题的解决方案在空间域的右侧显示一个指数边界层。讨论了溶液及其导数的性质和范围。在时间离散化中使用Crank-Nicolson方法，在空间离散化中使用指数拟合算子有限差分法求解所考虑的奇异摄动时滞问题。利用比较原理和解界对方案的稳定性进行了研究和分析。讨论并证明了该方案的统一收敛性。制定的方案以线性收敛顺序均匀收敛。考虑数值实例验证了该方案的理论分析的正确性。