May 29, 2007
MAE 105 Homework #8
Due: Tuesday
,06/05/07
Name:_________________________
NOTE:
To receive full credit, you must write down all steps neatly and draw neat and complete sketches.
PROBLEM 1:
Consider the nonhomogeneous heat equation
∂
u
∂
t
=
∂
2
u
∂
x
2
+
x
cos
xe
3
t
/2
,0
<
x
<
π
,
t
>0
,(
1)
with the boundary conditions
u
(0,
t
)
=
0,
u
(
,
t
)
=
0,
(2)
and the initial condition
u
(
x
,0)
=
x
sin
x
.(
3)
(a) (1 Point) Find the Fourier coefficients in the following Fourier series of
x
and cos
x
:
x
sin(
x
)
=
∞
n
=
1
Σ
a
n
sin
nx
4)
x
cos
x
=
∞
n
=
1
Σ
b
n
sin
nx
5)
for 0 <
x
<
.
(b) (0.5 Point) Express the solution of the original PDE (1),
i.e., u
(
x
,
t
), as an infinite series of sin
nx
,with time-
dependent coefficients, say,
A
n
(
t
). [Write down this series.]
(c) (1 Point) Substitute the infinite series expressions for
u
(
x
,
t
)and
x
cos
x
into the nonhomogeneous PDE (3), col-
lect coefficients of sin
nx
,use the orthogonality of these eigenfunctions, and find the necessary ODE’
sfor the
unknown coefficients,
A
n
(
t
).