ABSTRACT

All the problems discussed in Chapter 9 required conditions on y(t) and its derivatives at only one point that define the initial value. For that reason, such problems are termed initial-value problems. However, in many problems, conditions are specified at more than one point. Such problems are classified as boundary-value problems (BVP). These problems arise in the study of beam deflection, heat flow, and various physical problems. In this chapter we shall develop two principal classes of numerical methods used for solving the following boundary-value problems: The linear second-order boundary-value problem,

{ y′′(x) + p(x)y′ + q(x)y = r(x) y(a) = α, y(b) = β,

and the nonlinear second-order boundary-value problem, { y′′(x) = f(x, y, y′) y(a) = α, y(b) = β.