In Eq. 6, is a positive quantity
representing the flux of material
**M** in response to a unit gradient in the field variable **u**.
Time dependent conservation
law PDEs with flux ``down gradient'' are usually parabolic while steady
state conservation laws with such flux are generally elliptic.
These parabolic and elliptic conservation
law equations will be the vehicles by which a number of important
computational issues surrounding PDE problems will be introduced.

If we put the constitutive relationships Eq. 4 and Eq. 6 into the conservation law , the result is the one dimensional parabolic conservation law PDE

where **c** denotes .
If density is a linear function of **u**,

and if is a constant, then Eq. 7 reduces to the constant coefficient diffusion equation

where .