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