Math 3363 – Spring 2016
Name:
Test #1
Please, write clearly and justify all your steps, to get proper credit for your work. You can cite
general results from the book, but no examples or exercises.
(1)[8 Pts] (a) Solve the heat equation
∂ 2u
∂u
= k 2,
∂t
∂x
with
∂u
∂u
(B.C.)
(0, t) = 0,
(L, t) = 0,
∂x
∂x
3πx
(I.C.)
u(x, 0) = 2 − cos
.
L
(b) Find the solution of the problem above if the boundary condition is
changed to
∂u
∂u
(B.C.) u(−L, t) = u(L, t), (−L, t) =
(L, t).
∂x
∂x
(c) Compute the steady state solutions for the problems in (a-b).
(2)[8 Pts] For the following functions, state whether or not the corresponding Fourier series in the interval [−1, 1] converges to the function in [−1, 1].
If it does not, indicate where it does not converge.
(a) f (x) = 1 − x2
(b) f (x) = x − x2
(c) f (x) = cos(2πx)
(d) f (x) = sin x
(3)[8 Pts] Compute the Fourier (
series of f
0 |x| ≤ L/2
f (x) =
1 |x| > L/2
valid in [−L, L] and discuss its convergence, that is, indicate for which values
of x ∈ [−L, L] the Fouries series of f converges to f , where it does not and
which value it takes at those points.
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