Global (finite-difference) methods for boundary-value problems

In the initial-value methods of the last chapter at least some of the boundary conditions were not directly involved in the numerical solution of the derived initial-value problems./In global methods all the boundary conditions play equal parts from the beginning and are involved in all parts of the computation. There are essentially two main types of method. The first, considered in this chapter, is the class of finite-difference methods, in which the finite-difference equations and boundary-condition equations are essentially solved simultaneously rather than in step-by-step succession. The second, considered in Chapter 7, is the class of expansion methods, in which the solution is represented in the approximate form (6.1) $$y\, = \,\sum\limits_{r = 0}^N {a_r \phi _r (x)} $$ , where the φ r (x) are a preselected set of functions of which the most obvious is the sequence of polynomials (6.2) $$\phi _r (x)\, = \,x^r ,\,\,r = 0,1,2, \ldots $$