A new spline method for singular two-point boundary value problems

We describe a new spline method for the solution of the class of singular two-point boundary value problems: . We construct our spline approximation s(x) for the solution y(x) of the two-point boundary value problem such that, while ,with the uniform mesh , in each subinterval , our spline approximation s(x) linearly spans a certain set of (non-polynomial) basis functions in the representation of the solution y(x) of the two-point boundary value problem and satisfies the interpolation conditions: where .However, in the first interval [0,x 1], our spline approximation s(x) is constructed satisfying the interpolation conditions: and the global spline approximation s(x) is uniquely determined under the additional interpolation condition s(0) =y(0). We show that our spline s(x) provides order h 2 uniformly convergent approximations over [0,1] for the solution y{x) of the two-point boundary value problem. The second order of the spline approximation is illustrated computationally.