( see Diffusion Equation @ Wikipedia )
Second order partial differential equation of parabolic type on the space-time field variable U(t, {\bf r}):
| (1) | \beta({\bf r}, U) \cdot \frac{\partial U}{\partial t} = \nabla \left( \alpha({\bf r}, U) \nabla U \right) + f({\bf r}) |
where
U(t, {\bf r}) | space-time field variable |
|---|---|
t | time |
{\bf r} = (x,y,z) | Position vector |
f({\bf r}) | density of external forces |
\alpha({\bf r}, U) | mobility kinetic coefficient |
\beta({\bf r}, U) | capacitance kinetic coefficient |
In the absence of external forces and constant kinetic coefficients the diffusion equation takes form:
| (2) | \frac{\partial U}{\partial t} = D \cdot \Delta U + f({\bf r}) |
where D is called the diffusion coefficient.