AI 中文总结
研究一维含两个小参数的四阶奇异摄动边值问题，通过解析输入数据，将解分解为光滑部分、边界层和余项，给出各部分任意阶导数估计，用于证明高阶数值方法收敛性，并给出解的解析性等相关结果。
AI 中文摘要
我们考虑一维情况下具有两个小参数的四阶奇异摄动边值问题，假设输入数据是解析的。我们表明解可分解为一个光滑部分、两个不同宽度的边界层和一个可忽略的余项。我们给出了分解中各项任意阶导数的估计，这些估计在微分阶数和奇异摄动参数方面是显式的，这对于证明高阶数值方法（如有限元方法的\(p/hp\)版本）的收敛性是必要的。我们还给出了经典的可微性结果，表明如果数据是解析的，解将是解析的，但一旦开始求导，奇异摄动参数的负幂就会出现。
英文摘要
We consider a fourth order singularly perturbed boundary value problem with two small parameters, in one dimension, under the assumption of analytic input data. We show that the solution may be decomposed into a smooth part, two different width boundary layers, and a negligible remainder. We provide estimates for arbitrary order derivatives of each term of the decomposition, which are explicit in the differentiation order and the singular perturbation parameters, and are needed for proving the convergence of high order numerical methods, such as the $p/hp$ versions of the Finite Element Method. We also provide classical differentiability results, which show that the solution will be analytic, if the data are analytic, but negative powers of the singular perturbation parameter(s) show up once we start differentiating.
