Fourier Series and Partial Differential Equations Lecture Notes

Chapter 5. Laplace’s equation in the plane 56
Then
∂T
∂r =
∞
nr n−1 [Cncos(nθ)+Dnsin(nθ)], (5.80)
n=1
and the boundary condition gives
∞
na n−1 [Cncos(nθ)+Dnsin(nθ)] = g(θ), 0 ≤ θ ≤ 2. (5.81)
n=1
We conclude immediately that the condition
2π
is necessary for a solution to exist.
If this condition is satisfied then there are solutions
where
T(r,θ) = A+
Cn =
Dn =
0
g(θ)dθ = 0, (5.82)
∞
r n [Cncos(nθ)+Dnsin(nθ)], (5.83)
n=1
1
nπan−1 2π
1
nπa n−1
0
2π
0
g(θ)cos(nθ) dθ, (5.84)
g(θ)sin(nθ) dθ, (5.85)
and A is an arbitrary constant, i.e. solutions are non-unique, if they exist.
Example 5.7 Find T so as to satisfy Laplace’s equation in the disc 0 ≤ r < a and the
boundary condition
∂T
∂n (a,θ) = sin3 θ, 0 ≤ θ ≤ 2. (5.86)
Here
sin 3 θ = 3 1
sinθ− sin(3θ), (5.87)
4 4
and 2π
sin 3 θdθ = 0, (5.88)
and the solutions are
where A is arbitrary.
5.4 Well-posedness
0
T(r,θ) = A+ 3 1
rsinθ −
4 12
r3 sin(3θ), (5.89)
a2 PDE problems often arise from modelling a particular physical system. In this case we
could like to be able to make predictions as to the behaviour of the system based on our
analysis of the PDE under consideration.