Numerics of Ordinary Differential Equations: Boundary Value Problems

Abstract

The numerical solution of linear boundary value problems is studied within this chapter. This requires first of all a classification of the various types of boundary conditions. In a first step the finite difference approximation of derivatives is applied to transform the ordinary differential equation together with its boundary conditions into a system of inhomogeneous linear equations. This system can then be solved using standard numerical methods. The second approach is defined by the so-called shooting methods. They try to treat the boundary value problem like an initial value problem. The initial values are then modified iteratively until the original boundary values are fulfilled. These methods are particularly effective whenever eigenvalue problems are to be studied. A particular flavor of shooting methods is the Numerov method to solve Schrodinger-equation-like eigenvalue problems with homogeneous boundary conditions.