Fourier Series and Partial Differential Equations Lecture Notes

Chapter 2
Fourier series
In the following chapters, we will look at methods for solving the PDEs described in
Chapter1. Inordertoincorporategeneralinitialorboundaryconditionsintooursolutions,
it will be necessary to have some understanding of Fourier series.
For example, we can see that the series
∞
nπx
y(x,t) = sin
L
nπct nπct
Ancos +Bnsin , (2.1)
L L
n=1
is a solution of the wave equation
∂2y ∂t2 = c2∂2 y
∂x2, x ∈ [0,L], t ≥ 0, (2.2)
which satisfies the boundary conditions
y(0,t) = 0 = y(L,t). (2.3)
We may view y(x,t) as the solution of the problem which models a vibrating string of
length L pinned at both ends, e.g. a guitar string.
y
0
We would like to find a solution with initial conditions
y(x,0) = αsin
πx
L
,
l
∂y
(x,0) = 0, (2.4)
∂t
and we do this by calculating An and Bn as follows: from equation (2.1) we have
y(x,0) =
∞
nπx
Ansin , (2.5)
L
n=1
9
x