Math 257/316 Assignment 10
Due: Fri. Apr. 10
1. Find the eigenvalues and eigenfunctions of the following Sturm-Liouville problem on
[0, 1]
X 00 (x) + λX(x) = 0, X 0 (0) = 0, X(1) = 0,
and use them to solve the following heat equation with mixed BCs and a source term:
ut = uxx + cos(5πx/2)t,
ux (0, t) = 0,
0 < x < 1, t > 0,
u(1, t) = 0
u(x, 0) = cos(7πx/2)
2. Consider the problem of steady heat conduction in a semi-circular annulus
∆u =
∂ 2 u 1 ∂u
1 ∂2u
+
+
= 0,
∂r2
r ∂r
r2 ∂θ2
u(1, θ) = u(2, θ) = 0,
1 < r < 2,
u(r, 0) = 0,
0 < θ < π,
u(r, π) = 1.
Separate variables, u(r, θ) = R(r)Θ(θ), and find the Euler equation to be solved for
R(r). Show that it can also be converted to the Sturm-Liouville form
1
(rR0 )0 + λ R = 0,
r
R(1) = R(2) = 0.
Find the eigenfunctions Rn and eigenvalues λn , and check the orthogonality condition
Z 2
1
0
m 6= n
Rn (r)Rm (r) dr =
,
C
m=n
1 r
where you should determine the constant C.
Hence, find the solution u(r, θ) by eigenfunction expansion.
Hint: remember to include the weighting function 1/r when calculating the generalized
Fourier coefficients.
1