Fourier Series
1. Are the following functions even, odd, or neither?
(a) sin 3
(b) 1 + sin 3
(c) sin 3 cos 2
(d) 10 + cos 10 + sin2
2. Let () = cos + sin 2 with its fundamental period being 2 and () =
0 + ∑∞
=1[ cos() + sin()].
1
1
– Compute 1 and 2 . Hint: use cos 2 = (1 + cos 2) and sin2 2 = (1 −
2
2
cos 4)
– Show that all other coefficients are zero. Hint: use orthogonality
3. Define a periodic function () as follows:
1 + −1 < < 0
() = {
() = ( + 2) ∀
1 − 0 < < 1.
(a) Solve for the Fourier series coefficient 0 .
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(b) Show that = ()2 (1 − cos ). Hint: you need to use integration by part
∫ ′()() = ()() − ∫ ()′ (). Check pg. 35-36 of the Ch. 11
reading material for examples.
(c) Solve for the Fourier series coefficients . Hint: you may want to check if
() is even so that the computation can be simplified.
(d) Use Matlab to plot () and the Fourier series for () with the first 2, 5, and
10 terms respectively (i.e., 0 + ∑
=1[ cos(… ) + sin(… )] with =
2, 5, 10). Print three figures: figure 1 with the plot of () and the Fourier series
for () with the first 2 terms; figure 2 with the plot of () and the Fourier series
for () with the first 5 terms; figure 3 with the plot of () and the Fourier series
for () with the first 10 terms
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4. Define a periodic function () as follows:
3
−3 < < 0
() = {
() = ( + 6) ∀
3 − 0 < < 3.
Let the Fourier Series of () be given by () = 0 + ∑∞
=1[ cos (
sin(
3
2
) +
)]. Compute the value of (3). Hint: () is discontinuous at = 3.
5. Is (, , ) = cos(5)sin(12)sin(26) a solution to the PDE
2
3
2
2
= 4(
2
2
+
)? Justify your answer.
6. Use the results from section 12.3 to find a solution (, ) to the wave equation
described by the following conditions: length = /4, 2 = 10,000, initial
velocity () = 0, and initial displacement () = 13sin(4) – 10sin(12) +
2sin(16).
Write out all terms in (, ) since there are only a few terms which are non-zero.
(The Fourier coefficients can be found by inspection.)
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7. Consider the PDE
2
2
= 100
. Using the method of separating variables, the
PDE decomposes into the following two coupled ODEs:
2 ()
()
−
100()
=
− () = 0
2
(Note that in section 12.3 it is the boundary conditions that generate a discrete set
of solutions depending upon = 1, 2, …
In this homework problem no boundary conditions are imposed, so you will not
have to introduce )
Let = 0. Find a solution (, ) to the PDE.