Existence and asymptotic analysis of solutions of singularly perturbed boundary value problems

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 ε 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) = e^y and A = 1. The second boundary value problem that we consider is ε² y' = y(q(x, ε) -y), y(-1)= α_-, y(1)=α_+,where q(x, ε) is a smooth function with uniformly bounded derivatives and is uniformly bounded from below by a positive constant q_∗ for ε sufficiently small. The boundary values α_± 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