A uniformly accurate difference scheme for singular perturbation problem

Abstract

For the problem εy″ + p(χ) y′-d(χ) y = f(χ), y(0) = α 0 , y(1) = α 1 , p(χ) > 0 and d(χ) ≥ 0 a difference scheme is derived. It is proved that the errors at the grid points are bounded by Mh 4 /(ε 2 + h 2 ) where M is a constant independent of ε and step size h, for d(χ) = 0. The numerical results show that the estimate is valid when d(χ) ≠ 0. The scheme is derived via exponential spline from C 1 [0, 1]. A modification of the scheme giving better results for very small ε is also presented.