Spline function methods for nonlinear boundary-value problems

Abstract

The solution of the nonlinear differential equation Y ″ = F ( x, Y, Y ′) with two-point boundary conditions is approximated by a quintic or cubic spline function y ( x ). The method is well suited to nonuniform mesh size and dynamic mesh size allocation. For uniform mesh size h , the error in the quintic spline y ( x ) is O ( h 4 ), with typical error one-third that from Numerov's method. Requiring the differential equation to be satisfied at the mesh points results in a set of difference equations, which are block tridiagonal and so are easily solved by relaxation or other standard methods.