Information Technology Reference
In-Depth Information
Summary and Concluding Remarks
We shall in this chapter deal with one-dimensional models only, because this
keeps the amount of mathematical and numerical details at a modest level. One-
dimensional models are relevant in a number of contexts. Of course, reducing any
three-dimensional phenomena to one dimension implies some approximation. This
approximation can be rough, as in the refrigerator example, or it can be very accu-
rate, as when one models diffusion of some substance in a long, thin tube filled with,
e.g., water. The results may be qualitatively or quantitatively correct, depending on
whether the approximation is rough or very good, respectively.
It requires quite some experience with physical and mathematical modeling to
judge the quality of reducing the number of space dimensions from three to one
or from three to two. From now on, we just assume that it is meaningful to work
with a one-dimensional diffusion model in a physical problem of interest. Our focus
will be on how one builds such a model, how one can solve the involved equations,
mainly by means of a computer, and the graphical display (visualization) of the
solutions. The latter topic will contribute to increasing our understanding of what
a diffusion process is, at least from a mathematical point of view. (Most diffusion
processes arise from the motion of and interaction between molecules. Our mod-
els can only predict the average (macroscopic) result of very detailed microscopic
molecular dynamics. If you want to learn about the underlying physics of diffusion,
you should consult an introductory topic on physics.)
7.2
The Mathematical Model of Diffusion
Let
u
.x; t / be the temperature or the concentration function in a diffusion process
that can be considered one dimensional. At each point x in the one-dimensional
space, and at each point t in time, we associate a value of
u
. The purpose of the
present chapter is to compute
u
, and occasionally quantities that can be derived from
a knowledge of
u
. Computing
u
means solving equations entering a mathematical
model for diffusion. We will briefly present the different type of equations in the
following.
7.2.1
The Diffusion Equation
The function
u
.x; t / is governed by the
diffusion
equation
@t
D
k
@
2
u
@
u
C
f.x;t/:
(7.1)
@x
2
