Math 3363 – Spring 2016 Name:
Test #2
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)[5Pts] Compute the Fourier sine series of f
f(x) = (xif 0 ≤x≤1
2
1−xif 1
2< x ≤1,
valid in [0,1] and discuss its convergence, that is, indicate for which values
of x∈[0,1] the Fourier sine series of fconverges to f, where it does not and
which value it takes at those points. Sketch the Fourier sine series of f.
(2)[3Pts] Consider a damped vibrating string that satisfies the equation
ρ∂2u
∂t2=T∂2u
∂x2−β∂u
∂t ,0< x < L,
Apply the method of separation of variables to derive two ordinary differential
equations with respect to time and space variables.
(3)[8Pts] Consider a vibrating string of uniform density and tension, with
length L= 1 and fixed ends. We found that vertical displacement is modelled
by the wave equation
∂2u
∂t2=c2∂2u
∂x2,0<x<1,
with B.C. u(0, t) = 0, u(1, t)=0.Compute and write explicitly the particu-
lar solution up(x, t) for the given I.C. below:
(a) u(x, 0) = 2 sin 3πx, ∂u
∂t (x, 0) = 0,0<x<1.
(b) u(x, 0) = 0,∂u
∂t (x, 0) = v0,0<x<1.
HINT: Use the calculation from the textbook (p.95-96) showing that
the Fourier sine series P∞
n=1 bnsin nπx of the function f(x) = 1, for
0≤x≤1, has coefficients bn=(0 if neven,
4/(nπ) if nodd.
