ON A CLASSICAL BOUNDARY-VALUE PROBLEM INVOLVING A SMALL-PARAMETER

Differential inequality techniques are used to provide accurate information throughout the interval [a, b] on boundary layer solutions of the problem epsilony'' = f (t, y)y' + g(t, y) for a less-than-or-equal-to t <less-than-or-equal-to b y(a) = A and y(b) = B, subject to weak regularity requirements on the data (epsilon > 0 is a small positive parameter). Such accurate information has been previously obtained by asymptotic expansion techniques coupled with the contraction mapping method, but only subject to more severe regularity requirements on the data, whereas the differential inequality technique has previously given such accurate information subject to weak regularity requirements, but only outside the boundary layer, with a loss of accuracy occurring inside the boundary layer. The detailed approximations of solutions obtained here may be very useful in studying solutions with other types of singularity perturbed behavior, such as shock or interior layer behavior. Problems of this type arise in fluid dynamics.