Variable step size initial value algorithm for singular perturbation problems using locally exact integration

We consider quasilinear singular perturbation problems of the form ε y^″ + p (x) y^′ + q (x, y) = h (x), x ∈ [0, 1] ; y (0) = α, y (1) = β with a boundary layer at one end point. The original problem is reduced to an asymptotically equivalent linear first order initial-value problem (IVP). Then, a variable step size initial value algorithm is applied to solve this (IVP). The algorithm is based on the locally exact integration of quadratic linearized problem coefficients on a non-uniform mesh. Two term-recurrence relation with controlled step size is obtained. Several 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.