This paper introduces an approach to decoupling singularly perturbed boundary value problems for fourth-order ordinary differential equations that feature a small positive parameter $\epsilon$ multiplying the highest derivative. We specifically examine Lidstone boundary conditions and demonstrate how to break down fourth-order differential equations into a system of second-order problems, with one lacking the parameter and the other featuring $\epsilon$ multiplying the highest derivative. To solve this system, we propose a mixed finite element algorithm and incorporate the Shishkin mesh scheme to capture the solution near boundary layers. Our solver is both direct and of high accuracy, with computation time that scales linearly with the number of grid points. We present numerical results to validate the theoretical results and the accuracy of our method.

翻译：本文提出了一种解耦奇异摄动四阶常微分方程边值问题的方法，此类方程最高阶导数项前带有小正参数$\epsilon$。我们特别研究了Lidstone边界条件，并展示了如何将四阶微分方程分解为二阶问题系统，其中一个系统不含参数，另一个系统的最高阶导数项前带有$\epsilon$。为求解该系统，我们提出了混合有限元算法，并引入Shishkin网格方案以捕捉边界层附近的解。该求解器既直接又具有高精度，计算时间与网格点数呈线性关系。我们给出数值结果以验证理论结果及方法的准确性。