The Overlapping Layer Solution to Nonlinear Singularly Perturbed Boundary Value Problem

A singularly perturbed problem of the nonlinear equation is considered.Under suitable conditions,firstly,using the expansion method of power series,the reduced solution and formal outer solution are constructed.Secondly,using the transformation of the stretched variable,the first boundary layer corrective term near the left endpoint is constructed.And then,using the stronger transformation of the stretched variable,the second boundary layer corrective term near the left endpoint is constructed.Finally,using the theory of differential inequalities the existence,uniform validity in the whole interval and asymptotic behavior of solution for the original boundary value problem are proved.And using the theory of differential inequalities,the uniformly valid behavior of its expansion is proved.The method and the result are meaningful for the study of nonlinear singularly perturbed boundary value problems.