Fourier Series and Partial Differential Equations Lecture Notes

Chapter 5. Laplace’s equation in the plane 57
Definition A problem is said to be well-posed (well-set) if the following three conditions
are satisfied:
1. EXISTENCE—there is a solution;
2. UNIQUENESS—there is no more than one solution;
3. CONTINUOUS DEPENDENCE—the solution depends continuously on the data.
Example 5.8 As an illustration consider the IVP
∂2y ∂t2 = c2∂2 y
∂x2 −∞ < x < ∞, t > 0, (5.90)
y(x,0) = f(x),
∂y
(x,0) = g(x), −∞ < x < ∞,
∂t
(5.91)
where f and g are the initial data. We know that there is exactly one solution, given by
D’Alembert’s formula:
y(x,t) = 1
x+ct 1
[f(x−ct)+f(x+ct)]+ g(s)ds.
2 2c
(5.92)
Thus 1. and 2. hold.
Suppose we are interested in making predictions in the time interval 0 < t < T for
time T. Consider a similar problem
∂2y ∂t2 = c2∂2 y
∂x2 −∞ < x < ∞, t > 0, (5.93)
y(x,0) = F(x),
∂y
(x,0) = G(x), −∞ < x < ∞,
∂t
(5.94)
whereF andGaredifferent initial data. Again, weknow that thereis exactly onesolution:
Y(x,t) = 1
x+ct 1
[F(x−ct)+F(x+ct)]+ G(s)ds,
2 2c
(5.95)
and
Y(x,t)−y(x,t) = 1
[(F(x−ct)−f(x−ct))+(F(x+ct)−f(x+ct))] (5.96)
2
+ 1
x+ct
[G(s)−g(s)] ds. (5.97)
2c
x−ct
Now let ǫ > 0 be arbitrary and suppose that
Then
x−ct
x−ct
|F(x)−f(x)| < δ and |G(x)−g(x)| < δ for −∞ < x < ∞. (5.98)
|Y(x,t)−y(x,t)| ≤ 1
2 |F(x−ct)−f(x−ct)|
+ 1
2 |F(x+ct)−f(x+ct)|
+ 1
2c
x+ct
|G(s)−g(s)|ds, (5.99)
x−ct
x+ct
< 1 1 1
δ + δ + δds,
2 2 2c x−ct
(5.100)
= 1 1 1
δ + δ + ·2ctδ,
2 2 2c
(5.101)
= (1+t)δ < (1+T)δ. (5.102)