18.336 spring 2009
lecture 2
02/05/09
Well-Posedness
Def.: A PDE is called well-posed (in the sense of Hadamard), if
(1) a solution exists
(2) the solution is unique
(3) the solution depends continuously on the data
(initial conditions, boundary conditions, right hand side)
Careful: Existence and uniqueness involves boundary conditions
Ex.: uxx + u = 0
a) u(0) = 0, u( π2 ) = 1 ⇒ unique solution u(x) = sin(x)
b) u(0) = 0, u(π) = 1 ⇒ no solution
c) u(0) = 0, u(π) = 0 ⇒ infinitely many solutions: u(x) = A sin(x)
Continuous dependence depends on considered metric/norm.
We typically consider || · ||L∞ , || · ||L2 , || · ||L1 .
Ex.:
⎧
heat equation
⎨
ut = uxx
u(0, t) = u(1, t) = 0 boundary conditions
⎩
u(x, 0) = u0 (x)
initial conditions
⎧
backwards heat equation
⎨
ut = −uxx
u(0, t) = u(1, t)
boundary conditions
⎩
u(x, 0) = u0 (x) initial conditions
Notions of Solutions
Classical solution
k th order PDE ⇒ u ∈ C k
Ex.: �2 u = 0 ⇒ u ∈ C ∞
�
�
ut + ux = 0
⇒ u(x, t) ∈ C 1
u(x, 0) ∈ C 1
Weak solution
k th order PDE, but u ∈
/ Ck.
1
⎫
⎬
well-posed
⎭
⎫
⎬
no continuous dependence
⎭
on initial data [later]
Ex.: Discontinuous coefficients
⎧
⎫
(b(x)u
)
=
0
⎪
⎪
x
x
⎪
⎪
⎪
⎪
⎪
⎪
� 4
u(0)
=
0
⎨
⎬
x
x < 12
3
u(1) = �
1
⇒
u(x)
=
2
�
⎪
x +
31 x ≥
21
⎪
1
3
⎪
⎪
1
x
<
⎪
⎪
2
⎪
⎪
⎩
b(x) =
⎭
2 x ≥ 12
Ex.: Conservation laws
ut + ( 12 u2 )x = 0 Burgers’ equation
Image by MIT OpenCourseWare.
Fourier Methods for Linear IVP
IVP = initial value problem
advection equation
ut = ux
ut = uxx
heat equation
ut = uxxx Airy’s equation
ut = uxxxx
�
w=+∞
a) on whole real axis: u(x, t) =
eiwx û(w, t)dw Fourier transform
w=−∞
+∞
�
b) periodic case x ∈ [−π, π[: u(x, t) =
ûk (t)eikx Fourier series (FS)
k=−∞
Here case b).
∂u
∂ nu
PDE:
(x, t) − n (x, t) = 0
∂t
∂x
�
+∞ �
�
dûk
n
insert FS:
(t) − (ik) ûk (t) eikx = 0
dt
k=−∞
Since (eikx )k∈Z linearly independent:
dûk
= (ik)n ûk (t) ODE for each Fourier coefficient
dt
2
n
ûk (t) = e(ik) t ûk (0)
� �� �
Solution:
Fourier coefficient of initial conditions: ûk (0)= 21π
+∞
�
⇒ u(x, t) =
�π
−π
u0 (x)e−ikx dx
nt
ûk (0)eikx e(ik)
k=−∞
n = 1: u(x, t) =
n = 2: u(x, t) =
n = 3: u(x, t) =
�
k
�
k
�
ûk (0)eik(x+t)
ûk (0)eikx e−k
ûk (0)eik(x−k
all waves travel to left with velocity 1
2t
2 t)
k
n = 4: u(x, t) =
�
ûk (0)eikx ek
4t
k
frequency k decays with e−k
2 t
frequency k travels to right
with velocity k 2 → dispersion
all frequencies are amplified
→ unstable
Message:
For linear PDE IVP, study behavior of waves eikx .
The ansatz u(x, t) = e−iwt eikx yields a dispersion relation of w to k.
The wave eikx is transformed by the growth factor e−iw(k)t .
Ex.:
wave equation:
heat equation:
conv.-diffusion:
Schrödinger:
Airy equation:
utt = c2 uxx
ut = duxx
ut = cux +duxx
iut = uxx
ut = uxxx
w
w
w
w
w
= ±ck
= −idk 2
= −ck−idk 2
= −k 2
= k3
3
conservative
dissipative
dissipative
dispersive
dispersive
|e±ickt | = 1
2
|e−dk t | → 0
2
|eickt e−dk t | → 0
2
|eik t | = 1
3
|e−ik t | = 1
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18.336 Numerical Methods for Partial Differential Equations
Spring 2009
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