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where Œ0; T is the time interval in which we want to discretize (
8.99
)-(
8.101
).
(a) Use the discretization technique presented in Chap. 7 for the boundary condition
(
8.100
) to derive an explicit scheme for this problem.
(b) Write a computer program that implements the scheme in (a).
(c) Recall the bound (8.7) valid in the case of homogeneous Dirichlet boundary
conditions. Explain why we can compare the size of the sum
x
1
2
.
u
n
/
2
!
2
.
u
1
/
2
C
n
X
i
D
2
/
2
C
1
.
u
i
(8.102)
with that of
x
1
!
(8.103)
2
.f .x
1
//
2
C
n
X
i
D
2
1
2
.f .x
n
//
2
.f .x
i
//
2
C
in order to test experimentally whether or not a similar inequality holds in the
case of homogeneous Neumann boundary conditions. Let
f.x/
D
e
x
x.1
x/
for x
2
.0; 1/:
and compute the sums (8.102)and(8.103) for a series of different discretization
parameters x and t . What are your experiments indicating?
(d) Show that
Z
1
u
2
.x; t /
dx
Z
1
0
u
2
.x; 0/
dx
D
Z
1
0
f
2
.x/
dx
for t
0:
0
(e) A physical interpretation of the homogeneous Neumann boundary condition
(
8.100
) is that heat can neither enter nor leave the body for which (
8.99
)-(
8.101
)
is modeling the heat evolution. Thus it seems reasonable that the “total” heat
present in the body will be constant with respect to time t .
Apply the definition of the initial condition f given in (c). For various values of
the discretization parameters x and t , compute the sums
x
1
!
C
n
X
i
D
2
1
2
u
n
2
u
1
u
i
C
and
x
1
2
f.x
n
/
!
:
2
f.x
1
/
C
n
X
i
D
2
1
f.x
i
/
C
What do you observe?
