Global convergence method for singularly perturbed boundary value problems

Abstract

We give uniformly convergent splines difference scheme for singularly perturbed boundary value problems (1) - ε u ″ + p ( x ) u ′ + q ( x ) u = f ( x ) , u ( a ) = α 0 , u ( b ) = α 1 , by using splines fitted with delta sequence due to the very stiff nature of the problem under consideration. We prove the O ( min ( h 2 , ε 2 ) ) order of uniform convergence with respect to small parameter ε at nodes on uniform mesh and O ( min ( h , ε ) ) order of uniform global convergence with respect to the approximate solution given by S ( x ) = ∑ i = 1 N S Δ i ( x ) H ( x i - x ) where H is the Heaviside function, which is the approximation for the closed form of the exact solution.