Boundary Value Problems for Ordinary Differential Equations

Abstract

In this chapter we want to solve a differential equation of the form

$$y''(x) = g(x, y, y')$$

(18.1)

with boundary conditions

$$ \left\{ \begin{gathered} \alpha _1 y(a) + \alpha _2 y\prime(a) = A, \hfill \\ \beta _1 y(b) + \beta _2 y\prime(b) = B, \hfill \\ \end{gathered} \right. $$

(18.2)

where |α₁| + |β₁| > 0 (two-point boundary problem) and α _i ≠ 0 and β _i ≠ 0 for at least one index. The equations (18.2) are called linear boundary conditions, the differential equation (18.1) together with the boundary conditions (18.2) form a boundary value problem of second order. The boundary conditions (18.2) can be replaced by nonlinear conditions $r_1 (y(a),y\prime(a)) = 0$ and $r_2 (y(b),y\prime(b)) = 0$. Such conditions will not be treated explicitly here. The subroutine BVP in the program section can handle such conditions as well.