A new fourth-order cubic spline method for second-order nonlinear two-point boundary-value problems

Bickley 5] had suggested the use of cubic splines for the solution of general linear two-point boundary-value problems. It is well known since then that this method gives only order h² uniformly convergent approximations. But cubic spline interpolation itself is a fourth-order process. We present a new fourth-order cubic spline method for second-order nonlinear two-point boundary-value problems: y″ = f(x, y, y′), a < x < b, α₀y(a) − α₀y′(a) = A, β₀y(b) + β₁y′(b) = B. We generate the solution at the nodal points by a fourth-order method and then use ‘conditions of continuity’ to obtain smoothed approximations for the second derivatives of the solution needed for the construction of the cubic spline solution. We show that our method provides order h⁴ uniformly convergent approximations over a, b]. The fourth order of the presented method is demonstrated computationally by two examples.