Information Technology Reference
In-Depth Information
Chapter 8
Analysis of the Diffusion Equation
In Chap. 7 we studied several aspects of the theory of diffusion processes. We
saw how these equations arise in models of several physical phenomena and how
they can be approximately solved by suitable numerical methods. The analysis of
diffusion equations is a classic subject of applied mathematics and of scientific com-
puting. Its impact on the field of partial differential equations (PDEs) has been
very important, both from a theoretical and practical point of view. The purpose
of this chapter is to dive somewhat deeper into this field and thereby increase our
understanding of this important topic.
Our aim is to study several mathematical properties of the diffusion problem
u
t
D
u
xx
for x
2
.0; 1/; t > 0;
(8.1)
u
.0; t /
D
u
.1; t /
D
0
for t>0;
(8.2)
u
.x; 0/
D
f.x/
for x
2
.0; 1/;
(8.3)
where f is a given initial condition defined on the unit interval .0; 1/.Herewehave
introduced a different notation for the derivative than what was used in Chap. 7:
u
x
is
short-hand for @
u
=@x,
u
xx
is short-hand for @
2
u
=@x
2
,and
u
t
is short-hand for @
u
=@t.
The reason for introducing this notation is twofold: we save some writing space
when writing the derivatives, and it is a widely used notation in the mathematical
analysis of PDEs. Note that, if not stated otherwise, we will consider a problem with
homogeneous Dirichlet boundary conditions, cf. (
8.2
).
The three main subjects of this chapter are
-
To investigate whether or not several physical properties of diffusion are satisfied
by the mathematical model (
8.1
)-(
8.3
)
-
To derive a procedure, commonly referred to as “separation of variables”, for
computing the analytical solution of the diffusion equation
-
To analyze the stability properties of an explicit numerical method, scheme (7.91)
in Chap. 7, for solving (
8.1
)-(
8.3
)
Prior to reading this chapter the reader should study Chapter 7 in detail.
