Beginning Partial Differential Equations, 3rd Edition

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. ...