A Numerical Scheme for Nonlinear Singularly Perturbed Two point Boundary Value Problems using Locally Exact Integration

We consider a class of nonlinear singular perturbation problems of the form [ ] β = α = ∈ = + ′ + ′ ′ ε ) ( , ) ( ]; , [ ), ( ) , ( )) ( ( ) ( b y a y b a x x r y x q x y p x y with a boundary layer at one end point. Using the theory of singular perturbations, the original problem is reduced to an asymptotically equivalent first order initial value problem. Then, a variable step size in itial value algorithm is applied to solve this initial value problem in a narrow region containing the layer region. The algorithm is based on the exact integration of a locally linearized problem (on a special non uniform mesh) exhibit ing uniform convergence in ε for any x. Some problems are solved to demonstrate the applicability and efficiency of the algorithm. It is observed that the present method approximates the exact solution very well.