The Heat Equation

The purpose of this chapter is to study the one dimensional heat equation

ut(x; t) = c 2uxx(x; t) + F (x; t); x 2 (a; b) R; t > 0;

also known as the diusion equation, and its higher dimensional version

ut(x; t) = a 2u(x; t) + F (x; t); x 2 Rn; n = 2; 3:

In the rst section we will nd the fundamental solution of the initial value problem for the heat equation. In the next sections we will apply the separation of variables method for constructing the solution of the one and higher dimensional heat equation in rectangular, polar and spherical coordinates. In the last section of this chapter we will apply the Laplace and Fourier transforms to solve the heat equation.