Higher-Order Finite-Difference Schemes for Nonlinear Two-Point Boundary Value Problems

We present a family of high-order multi-point finite difference methods for solving nonlinear two-point boundary value problems. The family involves some known methods as specific instances. We also introduce a highly efficient quintic B -spline method for solving nonlinear two-point boundary value problems, which yields an approximate solution in the form of a B -spline representation. This method successfully approximates the solutions to the one-dimensional Bratu, Troesch, and Lane–Emden problems without requirements of any information about the exact solutions.