| We study existence and uniform asymptotic expansions of solutions of two different classes of singularly perturbed boundary value problems. The first boundary value problem that we consider is epsilon y'' + 2y' + |f(y) = 0, y(0) = y(A) = 0; where f is a smooth, positive increasing function satisfying certain properties and A > 0. We will show that the problem has two solutions for certain values of A. We will also derive and prove a uniform asymptotic expansion of the smaller solution when f(y) = ey and A = 1. The second boundary value problem that we consider is epsilon2y'' = y( q(x, epsilon) - y), y(-1) = alpha-, y(1) = alpha+, where q(x, epsilon) is a smooth function with uniformly bounded derivatives and is uniformly bounded from below by a positive constant q☆ for epsilon sufficiently small. The boundary values alpha+/- are specified positive numbers bounded from above by q☆. We will derive uniform asymptotic expansion of solutions to this problem that have 3 or fewer critical points. |
