# Chapter 11

# Ordinary Differential Equations: Boundary-Value Problems

**Core Topics**

The shooting method (11.2).

The finite difference method (11.3).

Use of MATLAB built-in functions for solving boundary value ODEs (11.4)

**Complementary Topics**

Error and stability in numerical solution of boundary value problems (11.5).

## 11.1 BACKGROUND

A specific solution of a differential equation can be determined if the constraints of the problem are known. A first-order ODE can be solved if one constraint, the value of the dependent variable (initial value) at one point is known. To solve an *n*th-order equation, *n* constraints must be known. The constraints can be the value of the dependent variable (solution) and its derivative(s) at certain values of the independent variable. When all the constraints are specified at one value of the independent variable, the problem is called an initial value problem (IVP). Solution of initial value problems is discussed in Chapter 10. In many cases there is a need to solve differential equations of second and higher order that have constraints specified at different values of the independent variable. These problems are called *boundary value problems* (BVP), and the constraints are called boundary conditions because the constraints are often specified at the endpoints or boundaries of the domain of the solution.

Figure 11-1: Heat flow in a pin fin.

As an example, ...