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(e) Compute
Z
1
.
u
.x; t /
v
.x; t //
2
dx
0
for t
D
10
4
and t
D
10
2
.
(f) Define the set S by
S
Df
x
2
Œ0; 1
Ij
u
.x; 10
2
/
v
.x; 10
2
/
j
0:213
g
:
Show that
j
S
j
0:046:
˘
8.3
Projects
8.3.1
Neumann Boundary Conditions
In this chapter we have so far only considered problems with Dirichlet boundary
conditions. That is, the value of the unknown function in the PDE is specified at the
boundary, see (
8.2
).
In Chap. 7 we encountered a second type of boundary conditions, namely, Neu-
mann conditions. In this case the derivative, or flux, of the unknown function
u
in
the differential equation is prescribed at the boundary; that is, assuming that the
solution domain is the unit interval .0; 1/,
u
x
.0; t / and
u
x
.1; t / for t>0
are given. The purpose of this project is to reconsider the theory developed above
for this kind of boundary conditions. Let us therefore consider the following model
problem:
u
t
D
u
xx
for x
2
.0; 1/; t > 0;
(8.99)
u
x
.0; t /
D
u
x
.1; t /
D
0
for t>0;
(8.100)
u
.x; 0/
D
f.x/
for x
2
.0; 1/;
(8.101)
where f is a given initial condition.
For given positive integers m and n we define
T
m
;
t
D
1
n
1
;
x
D
x
i
D
.i
1/x
for i
D
1;:::;n;
t
`
D
`t
for `
D
0;:::;m;
