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：本文提出一种解耦奇异摄动四阶常微分方程边值问题的方法，其中最高阶导数项乘以小正参数ε。我们专门研究了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，其中一个不包含该参数，另一个则包含乘以最高阶导数的ε。为求解该系统，我们提出一种混合有限元算法，并引入Shishkin网格格式以捕捉边界层附近的解。我们的求解器直接且具有高精度，计算时间随网格点数量线性增长。我们通过数值结果验证了理论结果的正确性及方法的精确性。