3 - Boundary Value Problems

Summary

Introduction

By itself, a system of ordinary differential equations has many solutions. Commonly a solution of interest is determined by specifying the values of all its components at a single point x = a. This point and a direction of integration define an initial value problem. In many applications the solution of interest is determined in a more complicated way. A boundary value problem specifies values or equations for solution components at more than one point in the range of the independent variable x. Generally IVPs have a unique solution, but this is not true of BVPs. Like a system of linear algebraic equations, a BVP may not have a solution at all, or may have a unique solution, or may have more than one solution. Because there might be more than one solution, BVP solvers require an estimate (guess) for the solution of interest. Often there are parameters that must be determined in order for the BVP to have a solution. Associated with a solution there might be just one set of parameters, a finite number of possible sets, or an infinite number of possible sets. As with the solution itself, BVP solvers require an estimate for the set of parameters of interest. Examples of the possibilities were given in Chapter 1, and in this chapter others are used to penetrate further into the matter.