Fourier Series and Partial Differential Equations Lecture Notes

Chapter 2. Fourier series 10
and
∂y
(x,0) =
∂t
Hence, we want An, Bn such that
αsin
πx
=
L
n=1
∞
n=1
Bn
∞
nπx
Ansin , 0 =
L
nπc
nπx
sin . (2.6)
L L
∞
n=1
Bn
nπc
nπx
sin . (2.7)
L L
By inspection we see that A1 = α, An = 0 for n = 1 and Bn = 0 ∀n. Thus, for these
initial conditions, the solution is
πx
πct
y(x,t) = αsin cos . (2.8)
L L
If we would like to take more general initial conditions
y(x,0) = f(x),
we need to find {An,Bn} such that
f(x) =
∞
nπx
Ansin , g(x) =
L
n=1
∂y
(x,0) = g(x), (2.9)
∂t
∞
n=1
These are known the Fourier sine series of the functions f and g.
2.1 Periodic, even and odd functions
Definition f is a periodic function if there is an a > 0 such that
Bn
nπc
nπx
sin . (2.10)
L L
f(x+a) = f(x), ∀x ∈ R. (2.11)
If this is the case a is called a period for f. Note that the period is not unique, but if there
is a smallest such a, it is called the prime period of f.
Notes.
1. Observe that this means that f(x) = c for c constant does not have a prime period.
2. Examples of periodic functions are sinx with prime period 2π and cos(2πx/a) with
prime period a. Examples of non-periodic functions are x and x 2 .
3. Observe that if f is defined on the half-open interval (α,α +a] we can extend it to
be a periodic function by demanding it is periodic with period a. This is called a
periodic extension.
α−a
y
α α+a
x