Information Technology Reference
In-Depth Information
7.2.3
The One-Dimensional Initial-Boundary Value Problem
The collection of a PDE together with initial and boundary conditions is called an
initial-boundary value problem
. An example of an initial-boundary value problem
for diffusion is
@t
D
k
@
2
u
@
u
C
f.x;t/;
x
2
˝;t > 0;
@x
2
u
.a; t /
D
U
a
;
t > ;
u
.b; t /
D
U
b
;
t > ;
u
.x; 0/
D
I.x/;
x
2
˝:
These equations constitute the complete set of information for determining the
unknown function
u
.x; t /. For this to be true, the
u
values at the boundary of ˝,
U
a
and U
b
, must be known functions of time or constants. Similarly, the initial
value of
u
in ˝ must be known as a function I.x/. The parameter k in the PDE
must also be known in advance.
3
7.2.4
The Three-Dimensional Diffusion Equation
Mathematical models for three-dimensional diffusion are quite similar to one-
dimensional models. The function
u
now depends on three spatial coordinates x, y,
and
z
,aswellastimet ,soweseekafunction
u
.x; y;
z
;t/ (describing temperature
or concentration). The governing PDE for
u
.x; y;
z
;t/ reads
@t
D
k
@
2
u
C
f.x;y;
z
;t/:
C
@
2
u
@y
2
C
@
2
u
@
z
2
@
u
(7.2)
@x
2
This PDE is defined in a three-dimensional domain ˝ and for times t
0.The
boundary conditions can be that
u
is known at the boundary @˝ of ˝. The ini-
tial condition is, as in the one-dimensional case, that
u
is known as some function
I.x; y;
z
/ for t
D
0:
u
.x; y;
z
;0/
D
I.x; y;
z
/.
The sum of second-order derivatives on the right-hand side of (7.2) appears in
numerous applications, not only diffusion. The sum has therefore received its own
notation,
r
2
u
:
3
We should add that in real life, some of the input data U
a
, U
b
, k, f.x;t/,andI.x/ may be
unknown, seemingly making the mathematical model inapplicable. Nevertheless, one can have
some other kind of information, e.g., some measurements of
u
at some points in space and time.
One can then use the PDE to estimate the unknown parameters. Such situations are referred to as
inverse problems
. The PDE model can then, in some cases, be combined with other mathematical
techniques such that
u
can be computed (Chap. 9 gives a brief introduction to the topic), but this is
a much more complicated topic than calculating
u
from a PDE with standard boundary and initial
conditions.
