Abstract: The singularly perturbed Dirichlet boundary value problem for the semilinear equation of second order with a small parameter at the highest derivative term is considered. Firstly, the outer expansion of the solution is constructed, with the direct expansion method. Then the solutions of interior and boundary layers of the solution are gained via introducing stretching variables near the interior and boundaries respectively. Finally, by means of the matching principle, the corresponding outer solution, interior solution and boundary solutions are respectively matched, so that the composite expansion is obtained. Thus the uniformly valid asymptotic expansion of solution for the original singularly perturbed boundary value problem in the entire interval is found.

Key words: singular perturbation, boundary layer, interior layer, matching method