Information Technology Reference
In-Depth Information
7.6
Exercises
Exercise 7.1.
Suppose we have a long tube, modeled as the interval Œ0; L, filled
with water. We inject droplets of ink in the middle of the tube at certain discrete
time events t
i
D
iT, i
D
1;2;:::. Argue why the following source function is a
reasonable model for such a type of injection:
f.x;t/
D
(
K; L=2
L
x
L=2
C
L; iT
t
t
iT
C
t; i
D
1;2;:::
0
otherwise
Here, K is a constant reflecting the mount of mass injected. Hint: Make sketches of
(o, better yet, animate) the function f.x;t/.
Set up a one-dimensional PDE model with appropriate boundary and initial
conditions
for
this
problem.
How
do
you
expect
the
solution
to
develop
in time?
˘
Exercise 7.2.
What kind of temperature problem can be described by the PDE
@t
D
@
2
T
@T
;
@x
2
with boundary conditions k
@
@x
D
h
T
.T
T
0
A sin.!t // at x
D
0 and T .1000; t/
D
T
0
, and initial condition T.x;0/
D
T
0
?
˘
Exercise 7.3.
What kind of flow can be described by the PDE
@t
D
@
2
v
@
v
;
@x
2
with boundary conditions
v
.0; t /
D
0 and
v
.H; t /
D
A sin.!t /, and initial condition
v
.x; 0/
D
0?
˘
Exercise 7.4.
Suppose we have a diffusion problem on Œ0; L:
@t
D
k
@
2
u
C
A sin
2
.!t / exp
.x
x
0
/
2
;
@
u
@x
2
u
.x; t /
D
c
0
;
x
D
0; L;
u
.x; 0/
D
c
0
:
Use c
0
as scale for
u
and a time scale as in the previous example (just omit % and c
v
from the t
c
expression). Show that the scaled version of the PDE may be written as
(dropping bars)
t/exp
L.x
x
0
/
2
;
@t
D
@
2
u
C
ı sin
2
.!
L
2
k
@
u
@x
2
