# Chapter 2

# Solutions of the Heat Equation

In this chapter we use Fourier series and a technique called separation of variables to solve initial-boundary value problems involving the heat equation on a bounded interval or rectangle in the plane. The chapter concludes with proofs of properties of solutions of the heat equation, including a maximum principle and a measure of how sensitive solutions are to changes in initial and boundary conditions.

# 2.1 Solutions on an Interval [0, *L*]

We will solve the heat equation on an interval [0, *L*] for a variety of boundary conditions.

## 2.1.1 Ends Kept at Zero Temperature

The problem is

2.1)

This models the temperature distribution *u*(*x, t*) in a homogeneous bar of uniform cross section and length, *L*, if the ends are kept at temperature zero, and the initial temperature on the cross section of the bar at *x* is *f*(*x*).

The *method of separation of variables*, or the *Fourier method*, consists of looking for solutions of the form

Substitute this into the heat equation:

Then

The left side depends only on *t*, and the right side only on *x*, and *x* and *t* are independent. ...