Boundary layer solutions to singularly perturbed problems via the implicit function theorem
Boundary layer solutions to singularly perturbed problems via the implicit function theorem
复制标题
通过隐函数定理求解奇扰动问题的边界层
作者：
O. Omel’chenko;L. Recke
本文证明了具有Dirichlet和Neumann边界条件的e2 u = f（x，u，eu，e），0 <x< 1型奇摄动问题边界层解的存在性、局部唯一性和渐近估计.为此，我们假设存在一族满足微分方程和边界条件的近似解，其精度较低。此外，我们表明，如果这个精度是高的，那么近似解的精确解的接近度是相应的高。证明的主要工具是一个广义隐函数定理，这是接近那些法夫和格林利（Uspechi垫。Nauk 24（1974），103-130）和Magnus（Proc.皇家学会爱丁堡136 A（2006），559-583）。最后，我们展示了如何在一定的自然条件下构造近似解。
We prove existence, local uniqueness and asymptotic estimates for boundary layer solutions to singularly perturbed problems of the type e 2 u = f (x, u, eu, e), 0 <x< 1, with Dirichlet and Neumann boundary conditions. For that we assume that there is given a family of approximate solutions which satisfy the differential equation and the boundary conditions with certain low accuracy. Moreover, we show that, if this accuracy is high, then the closeness of the approximate solution to the exact solution is correspondingly high. The main tool of the proofs is a generalized implicit function theorem which is close to those of Fife and Greenlee (Uspechi Mat. Nauk 24 (1974), 103-130) and of Magnus (Proc. Royal Soc. Edinburgh 136A (2006), 559-583). Finally we show how to construct approximate solutions under certain natural conditions.