MATH 152, FALL 2004: FINAL
There are five problems. Do all of them. Total score: 160 points.
For both of the following functions
f on [0, 11, state whether the Fourier cosine series
on [0, I] converges in each of the following senses: uniformly, pointwise, in L2. If
the Fourier series converges pointwise, state what it converges to for each x E [0,
Make sure that you give the reasoning that led you to the conclusions.
0, for 0
1 for 112
(25 points total)
Find the general solution of the PDE
on R, x Ry.
Now impose in addition that u(0, y)
y. Find u explicitly.
Consider the PDE
Find its general solution.
(35 points total)
Consider the differential operator
on twice differentiable functions
f on [0, I] which satisfy Dirichlet boundary con-
0. That is, for these functions
f, A f
-(x2 fl(x))'. Let
f (x)g(x) dx denote the standard inner product on functions.
1. (10 points)