Introduction

The diffusion equation

$$\frac{\partial T({\bf r},t)}{\partial t} = D\,\nabla^2 T({\bf r},t),$$ (187)

where $D>0$ is the (uniform) coefficient of diffusion, describes many interesting physical phenomena. For instance, in heat conduction we can write

$${\bf q} = - \kappa\,\nabla T,$$ (188)

where ${\bf q}$ is the heat flux, $T$ the temperature, and $\kappa$ the coefficient of thermal conductivity. The above equation merely states that heat flows down a temperature gradient. In the absence of sinks or sources of heat, energy conservation requires that

$$-\frac{\partial Q}{\partial t} = \int {\bf q}\cdot d{\bf S},$$ (189)

where $Q$ is the thermal energy contained in some volume $V$ bounded by a closed surface $S$. The above equation states that the rate of decrease of the thermal energy content of volume $V$ equals the instantaneous heat flux flowing across its boundary. However, $Q=\int c\,T\,dV$, where $c$ is the heat capacity per unit volume. Making use of the previous equations, as well as the divergence theorem, we obtain the following diffusion equation for the temperature:

$$\frac{\partial T}{\partial t} = D\,\nabla^2 T,$$ (190)

where $D=\kappa/c$. In a typical heat conduction problem, we are given the temperature $T({\bf r}, t_0)$ at some initial time $t_0$, and then asked to evaluate $T({\bf r},t)$ at all subsequent times. Such a problem is known as an initial value problem. The spatial boundary conditions can be either of type Dirichlet (i.e., $T$ specified on the boundary), type Neumann (i.e., $\nabla T$ specified on the boundary), or some combination.