On a boundary-layer problem

摘要

This is a continuation of our earlier article concerning the boundary-value problem [GRAPHICS] where A, B are prescribed constants, and 0 < ε 1 is a small positive parameter. In that article, we assumed the coefficients a(x) and b(x) are sufficiently smooth functions with the behavior given by a(x) &SIM; αx and b(x) &SIM; β as x --> 0, where alpha > 0 and beta/alpha not equal 1, 2, 3..... In the present article, we are concerned with the case alpha < 0 and β/α &NOTEQUAL; 0, -1, -2..... An asymptotic solution is obtained for the problem, which holds uniformly for all x in [x(-), x(+)]. Our result is proved rigorously, and shows that a previous result in the literature is incorrect. [References: 11]