CAAM 336
·
DIFFERENTIAL EQUATIONS
Problem Set 10
·
Solutions
Posted Wednesday 3 November 2010. Due Wednesday 10 November 2010, 5pm.
1. [50 points: 8 points each for (a), (b), (d), (e); 4 points for (c); 14 points for (f)]
This problem and the next study the heat equation in two dimensions. We begin with the steadystate
problem. In place of the one dimensional equation,

u
00
=
f
, we now have

(
u
xx
(
x, y
) +
u
yy
(
x, y
)) =
f
(
x, y
)
,
0
≤
x
≤
1
,
0
≤
y
≤
1
,
with homogeneous Dirichlet boundary conditions
u
(
x,
0) =
u
(
x,
1) =
u
(0
, y
) =
u
(1
, y
) = 0 for all
0
≤
x
≤
1 and 0
≤
y
≤
1. The associated operator
L
is defined as
Lu
=

(
u
xx
+
u
yy
)
,
acting on the space
C
2
D
[0
,
1]
2
consisting of twice continuously differentiable functions on [0
,
1]
×
[0
,
1]
with homogeneous boundary conditions.
We can solve the differential equation
Lu
=
f
using the
spectral method just as we have seen in class before. This problem will walk you through the process;
you may consult Section 8.2 of the text for hints.
(a) Show that
L
is symmetric, given the inner product
(
v, w
) =
Z
1
0
Z
1
0
v
(
x, y
)
w
(
x, y
)
dxdy.
(b) Verify that the functions
ψ
j,k
(
x, y
) = 2 sin(
jπx
) sin(
kπy
)
are eigenfunctions of
L
for
j, k
= 1
,
2
,
. . . .
(To do this, you simply need to show that
Lψ
j,k
=
λ
j,k
ψ
j,k
for some scalar
λ
j,k
.)
(c) What is the eigenvalue
λ
j,k
associated with
ψ
j,k
?
(d) Compute the inner product (
ψ
j,k
, ψ
j,k
) =
k
ψ
j,k
k
2
.
(e) Let
f
(
x, y
) =
x
(1

y
). Compute the inner product (
f, ψ
j,k
).
(f) The solution to the diffusion equation is given by the spectral method, but now with a double
sum to account for all the eigenvalues:
u
(
x, y
) =
N
X
j
=1
N
X
k
=1
1
λ
j,k
(
f, ψ
j,k
)
(
ψ
j,k
, ψ
j,k
)
ψ
j,k
(
x, y
)
.
In MATLAB plot the partial sum
u
10
(
x, y
) =
10
X
j
=1
10
X
k
=1
1
λ
j,k
(
f, ψ
j,k
)
(
ψ
j,k
, ψ
j,k
)
ψ
j,k
(
x, y
)
.
Hint for 3d plots: To plot
ψ
1
,
1
(
x, y
) = 2 sin(
πx
) sin(
πy
), you could use
x = linspace(0,1,40); y = linspace(0,1,40);
[X,Y] = meshgrid(x,y);
Psi11 = 2*sin(pi*X).*sin(pi*Y);
surf(X,Y,Psi11)