一类奇摄动边值问题的边界层

Boundary Layer of a Class Singularly Perturbed Boundary-value Problem

讨论一类最高阶导数项带有小参数的二阶半线性方程奇摄动D irichlet边值问题.通过直接展开法,构造了问题解的外部展开式,并引用伸长变量分别在区域内部和边界层附近构造了内层解和边界层解.利用匹配原理将对应的外部解、内层解和边界层解分别进行匹配,构造解的合成展开式.得到了原奇摄动边值问题解在整个区间内一致有效的渐近展开式.

The singularly perturbed Dirichlet boundary value problem for the semilinear equation of second order with a small parameter at the highest derivative term is considered. Firstly, the outer expansion of the solution is constructed, with the direct expansion method. Then the solutions of interior and boundary layers of the solution are gained via introducing stretching variables near the interior and boundaries respectively.Finally, by means of the matching principle, the corresponding outer solution, interior solution and boundary solutions are respectively matched, so that the composite expansion is obtained. Thus the uniformly valid asymptotic expansion of solution for the original singularly perturbed boundary value problem in the entire interval is found.