The Diffusion Equation - Elements of Scientific Computing - page 304

7.3.4 Summarizing the Models

We have seen how the diffusion equation arises in three different physical prob-
lems. The PDEs, in one-dimensional form, with appropriate boundary conditions
are reviewed here for easy reference. The initial condition is always that the primary
unknown u .x; t / must be known at t D 0. The governing PDEs are to be solved in
an interval .a; b/ for time t>0. One boundary condition must be applied at x D a
and one at x D b. Below, we have used a slightly different notation than we did
when deriving the different diffusion equations, because now we want to use a com-
mon set of symbols for the various physical applications (e.g., the unknown from
now on reads u .x; t /). This common set of symbols emphasizes that the mathemat-
ics are the same, a fact that is advantageous to explore when developing algorithms
and implementations; the code and its methodology are then obviously applicable
in different physical contexts.
We remark that for all the models below we need to prescribe an initial condition.
This condition takes the form

u .x; 0/ D I.x/

for all three physical applications, where I.x/ is a known function. Physically
it means that the concentration, the temperature, or the velocity must be known
everywhere in space when the process starts.

Diffusive Transport

Transport by molecular diffusion is governed by the PDE

@t D k @ 2 u

@ u

C f.x;t/: (7.39)

@x 2

The parameters are
- u .x; t /: the concentration of a substance in a fluid
- k: the diffusive transport coefficient
- f.x;t/: the external supply of the substance
Common boundary conditions for diffusive transport are as follows:
(a) Controlled concentration,

u D U 0 .t /; U 0 .t / is prescribed. (7.40)

(b) Impermeable boundary for the substance (“wall”), and

@ u
@n D 0:

(7.41)