   Next: 1-d problem with mixed Up: The diffusion equation Previous: The diffusion equation

## Introduction

The diffusion equation (187)

where is the (uniform) coefficient of diffusion, describes many interesting physical phenomena. For instance, in heat conduction we can write (188)

where is the heat flux, the temperature, and 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 (189)

where is the thermal energy contained in some volume bounded by a closed surface . The above equation states that the rate of decrease of the thermal energy content of volume equals the instantaneous heat flux flowing across its boundary. However, , where 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: (190)

where . In a typical heat conduction problem, we are given the temperature at some initial time , and then asked to evaluate 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., specified on the boundary), type Neumann (i.e., specified on the boundary), or some combination.   Next: 1-d problem with mixed Up: The diffusion equation Previous: The diffusion equation
Richard Fitzpatrick 2006-03-29