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

Abstract

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.

Keywords:

• Singular two-point boundary value problems
• spline solution
• basis functions
• interpolation conditions
• continuity conditions

C.R. Categories:

• G.1.7