Let ∈ > 0.
(a) Find the solution u(x, s) to the boundary value problem -∈2u" + u = 1, u(0) = u(1) = 0.
(b) Show that as e -> 0+, the solution u(x, ∈) → u.(x) converges to the solution to u. = 1. What happened to the boundary conditions?
(c) Explain why the convergence of this singular perturbation is non-uniform.
The regions close to the end-points are boundary layers, and require a separate analysis. Boundary layers appear in fluid flows near solid boundaries, e.g., the flow of air past an airplane wing.