Math 257 – Assignment 9. Due: Friday, March 25
1. [Application of Sturm-Liouville theory to boundary value problems of ODE.]
Consider the following second order ODE for Y (x) with boundary conditions:
Y 00 + Y = x,
0
Y (0) = 0,
0<x<1
(1)
Y (1) = 0.
Find a (series) solution to this problem. You have to compute the solution Y (x) explicitly. You can
proceed as follows:
• First, consider the Strum-Liouville problem
− X 00 = σX,
0
X (0) = 0,
0<x<1
X(1) = 0.
Find all the eigenfunctions Xn (x) and corresponding eigenvalues σn , n = 1, 2, 3, · · · , to this
problem.
P
• Express the solution Y as Y (x) = ∞
n=1 an Xn (x). (This is possible by Sturm-Liouville theory.) Plug this into the ODE (1) for Y , find conditions for the coefficients an , and determine
an ’s.
P
Hint: You will need to express x as x = ∞
n=1 bn Xn (x) and need to find bn .
2. [Wave equation with Dirichlet boundary condition] Use eigenfunction expansion method as given
in the class to solve the following initial-boundary-value problem for wave equation.
utt = 4 uxx ,
0 < x < 1, t > 0
u(0, t) = 0,
u(1, t) = 0,
u(x, 0) = x,
ut (x, 0) = 1,
t>0
0 ≤ x ≤ 1.
You must compute u(x, t) explicitly.
3. [Wave equation with Neumann boundary condition] Solve the following initial-boundary-value
problem for wave equation:
utt = 4 uxx ,
ux (0, t) = 0,
0 < x < 1, t > 0
ux (1, t) = 0,
t>0
u(x, 0) = cos(2πx) + 2 cos(5πx),
ut (x, 0) = cos(3πx),
0≤x≤1
You must compute u(x, t) explicitly.
4. [Wave equation with mixed boundary condition] Solve the following initial-boundary-value problem for wave equation:
utt = 4 uxx ,
0 < x < 1, t > 0
ux (0, t) = 0,
u(1, t) = 0,
u(x, 0) = x,
ut (x, 0) = 1,
You must compute u(x, t) explicitly.
Hint: Use Sturm-Liouville theory: eigenfunction expansion.
t>0
0≤x≤1