We just saw in Section 6.4 how to approximate the solution of a second-order initial-value problem y″ = f(x, y, y′), y(x0) = y0, y′(x0) = u0. In this section we are going to examine two methods for approximating a solution of a second-order boundary-value problem y″ = f(x, y, y′), y(a) = α, y(b) = β. Unlike the procedures used with second-order initial-value problems, the methods of second-order boundary-value problems do not require rewriting the second-order DE as a system of first-order DEs.
Finite Difference Approximations
The Taylor series expansion, centered at a point a, of a function ...