Green's Functions with Applications, 2nd Edition

Chapter 5

Green’s Functions for the Heat Equation

In this chapter, we present the Green’s function1 for the heat equation

$\begin{array}{cc}\frac{\partial u}{\partial t}-a^2{\nabla }^{2}u=q\left(\mathbf{\text{r,}}t\right),& \left(\mathbf{5.0.1}\right)\end{array}$

where ∇ is the three-dimensional gradient operator, t denotes time, r is the position vector, a2 is the diffusivity, and q(r, t) is the source density. In addition to Equation 5.0.1, boundary conditions must be specified to ensure the uniqueness of solution; the most common ones are Dirichlet, Neumann and Robin (a linear combination of the first two). An initial condition u(r,t = t0) is also needed.

The heat equation differs in many ways from the wave equation and the Green’s function must, of course, ...