Singularly Perturbed Spline Difference Schemes on a Non‐Equidistant Grid

An exponentially fitted spline difference scheme on a non‐equidistant grid is formed for solving the singularly perturbed two point boundary value problem: ϵy″ + p(x) y′ = f(x), 0 < x < 1,0 < ϵ ≪1, y(0) = α 0 , y(1) = α 1 , p(x) ≧ p > 0. Uniform convergence of first order is proved under the condition on the grid: 0 ≦ h i − h i−1 ≦ M i h i−1 max(x i , ϵ), h i =x i+1 − x i , 0 = x 0 < x 1 <… < x n+1 = 1, 0 ≦ M i ≦ M, M is a constant independent of h i and ϵ. Numerical examples are given, too.