The present paper aims to carry out a new scheme for solving a type of singularly perturbed boundary value problem with a second order delay differential equation. Getting through the solution, we used Reproducing Kernel Hilbert Space (RKHS) method as an efficient approach to obtain the analytical solution for ordinary or partial differential equations that appear in vast areas of science and engineering. A key of this method is to keeping the continuous form of problems. Indeed, without discretizing the continuous problem, we change it to an equivalent iterative form and proving its convergence. Also, we will present a construction of the reproducing kernel in Hilbert space that satisfying the homogeneous nonlinear boundary conditions of the considered problem. Accuracy amount of absolute error with respect to different parameters of singularity has been studied for the performance of this method by solving several hardly nonlinear problems. Error estimation and convergence analysis show that the approximate results have uniform convergence to the continuous solutions.

本文旨在为解决一类具有二阶时滞微分方程的奇摄动边值问题提供一种新的方案。通过解决方案，我们使用了再生核希尔伯特空间（RKHS）方法作为一种有效的方法来获得出现在科学和工程学广泛领域的常微分方程或偏微分方程的解析解。该方法的关键是保持问题的连续形式。确实，在不离散化连续性问题的情况下，我们将其更改为等效的迭代形式并证明其收敛性。同样，我们将提出希尔伯特空间中满足所考虑问题的齐次非线性边界条件的复制核的构造。通过解决几个难于解决的非线性问题，研究了针对奇异性不同参数的绝对误差的准确度，以求该方法的性能。误差估计和收敛分析表明，近似结果与连续解具有一致的收敛性。