Fourier Series and Partial Differential Equations Lecture Notes

Chapter 5. Laplace’s equation in the plane 58
Thus if the new data (F,G) are close to the original data (f,g) in the sense that
|F(x)−f(x)| < ǫ
1+T
and |G(x)−g(x)| < ǫ
1+T
then the corresponding solutions are close together in the sense that
for −∞ < x < ∞, (5.103)
|Y(x,t)−y(x,t)| < ǫ for −∞ < x < ∞ and 0 < t < T. (5.104)
In this sense 3. holds and the IVP is well-posed.
Example 5.9 BycontrastthecorrespondingIVPforLaplace’s equation isnot well-posed.
If y(x,t) = 0, f(x) = 0, g(x) = 0 then y is a solution of the IVP
If
Y(x,t) = δ 2 cos
∂2y ∂x2 + ∂2y = 0, −∞ < x < ∞, t > 0, (5.105)
∂t2 y(x,0) = f(x),
x
sinh
δ
Then Y(x,t) is a solution of the IVP
∂y
(x,0) = 0, −∞ < x < ∞. (5.106)
∂t
t
, F(x) = 0, G(x) = δcos
δ
x
, (5.107)
δ
∂2Y ∂x2 + ∂2Y = 0, −∞ < x < ∞, t > 0, (5.108)
∂t2 Y(x,0) = F(x),
∂Y
(x,0) = G(x), −∞ < x < ∞.
∂t
(5.109)
Again suppose we want to make predictions in 0 < t < T. Then
x
|F(x)−f(x)| = 0 < δ, |G(x)−g(x)| = δcos
< δ,
δ
(5.110)
and
But
and we cannot make
by making δ suitably small.
|Y(0,T)−y(0,t)| = δ 2 sinh
t
< δ
δ
2
T
sinh . (5.111)
δ
δ 2
T
sinh =
δ
1
2 δ2e
T/δ −e −T/δ
→ ∞ as δ → 0, (5.112)
|Y(0,t)−y(0,t)| < ǫ for 0 < t < T, (5.113)