Perturbation Methods:
When do Small Parameters Have Large
Consequences?
Dave Goulet
Visiting Assistant Professor of Mathematics
Rose-Hulman Institute of Technology
Asymptotic Methods: Overview
• Boundary Layers: “shockingly” large changes over small distances
• Multiple Scales: processes occurring on different time and spatial scales
• Homogenization: how to smooth out the bumps
Example: secular terms
• Estimating approximation errors
| cos(1 + ✏)t
cos t| = | cos t(cos ✏t
1)
sin t sin ✏t|
• Some known inequalities
|a + b|  |a| + |b|
| sin x|  x
• An error bound
| cos(1 + ✏)t
| sin x|  1 | cos x|  1
1 2
| cos x 1|  x
2
1 2 2
cos t|  ✏t + ✏ t
2
• Asymptotic formula
cos(1 + ✏)t = cos t + O(✏t)
cos(1 + ✏)t ⇠ cos t
Should I worry about small things?
• Solutions can vaporize.
✏x2 + x = 1
x⇠1,x⇠
x=1
• Solutions can lose meaningful physical properties.
y 00 + ✏y = 0
y 0 = y(1
y 00 = 0
y0 = y
springs oscillate!
population stabilizes!
p
y = A cos( ✏t + )
y = ✏ + (1/y0
• Solutions may exist, but limits may not.
0
y = f (y/✏)
✏y)
✏)e
t
1
1/✏
Asymptotic Series
• Sometimes Taylor series:
y ⇠ y0
✓
0
y = ✏y
1 2 2
1 + ✏t + ✏ t
2
◆
y = y0 e✏t
• Sometimes not:
y 0 (t) = y(t
⌧)
✏xex = 1
x⇠
ln ✏
x = W(1/✏)
ln( ln ✏)
ln( ln ✏)
ln ✏
Boundary Layers
• enzyme kinetics
0
✏c = s
S+E $C !P +E
(s + k)c
s0 =
s + (s + k
s(0) = 1
c(0) = 0
• outer
s
c=
s+k
s
s =
s+k
0
k s
s e =e
• inner
⌧ = t/✏
c⌧ = s
s(s + k)c
e ⌧
c(⌧ ) =
1+k
1
s⌧ = 0
s(⌧ ) = 1
t
1)c
Boundary Layers:
staying warm in a wetsuit
• Fluid Motion:
X
ut + u · ru =
1
rp +
u
Re
r·u=0
• Steady, high Re flow over a flat plate
fast
slow
p
p
⇠ k x/ Re
cold
hot
p
p
3
⇠ c x/ P r
Boundary Layers:
staying warm in a wetsuit
fast
V
slow
T
V
p
p
⇠ k x/ Re
fast & hot
cold
T
hot
p
p
3
⇠ c x/ P r
V
v=0 & hot
Boundary Layers:
Multiple layers and Interior Layers
• quantum mechanics
✏y 00
(1
x2 )y =
O(✏
✏y 00 + 2(1
1
1/3
x2 )y = 1
y2
)
O(✏1/3 )
Bender & Orszag, 2010
Boundary Layers: shock layers
• Burgers’ equation
ut + uux = 0
ut + uux = ✏uxx
• solutions develop propagating discontinuities, shocks
• need viscosity to determine wave speed
• need viscosity to determine shock thickness
u(x, t) = f (⌘)
⌘=
x
cf 0 + f f 0 = (✏/ )f 00
• shock layer thickness is O(ℇ)
ct
(✏)
Boundary Layers: shock layers
• Look at the phase plane.
cf 0 + f f 0 = f 00
f0 = g
g0 =
cg + f g
• integrate across the shock
dg = ( c + f )df
g+
g =
c(f+
• find the shock speed
f+ + f
c=
2
1 2
f ) + (f+
2
f2 )
Boundary Layers:
infinitely many layers
• the delay recruitment model
✏y 0 (t) =
y(t) + F (y(t
1))
• lasers: Ikeda equation
• population biology
• white blood cell formation
Thomas Erneux, 2009
Summary:
Summary:
• Boundary layers are where interesting stuff occurs.
boring
interesting
• The outer solution can be close, but is often way off.
• Looking at the layer can give physical insight.
• Boundary layers don’t have to occur on boundaries or sit still.
• Not all asymptotic problems can be described with boundary layer theory...
Next Week:
part 2, the revenge
• Multiple Scales
• Homogenization
?
Perturbation Methods:
When do Small Parameters Have Large
Consequences?
Dave Goulet
Visiting Assistant Professor of Mathematics
Rose-Hulman Institute of Technology
Asymptotic Methods
• Boundary Layers: “shockingly” large changes over small distances
• Multiple Scales: processes occurring on different time and spatial scales
• Homogenization: how to smooth out the bumps
Summary
asymptotic methods
• Solutions can vaporize.
✏x2 + x = 1
x=1
x⇠1,x⇠
• Solutions can lose meaningful physical properties.
y 00 + ✏y = 0
y 00 = 0
p
y = A cos( ✏t + )
• Solutions may exist, but limits may not.
y 0 = f (y/✏)
1/✏
Multiple Scales:
climate vs. weather
Climate
Weather
ratio
Time
Scale
102-108 days 10-1-101 days
~10-6
Spatial
Scale
102-105 km
~10-3
100-102 km
Multiple Scales:
climate vs. weather
• don’t call them weathermen
⇢ut + ⇢u · ru =
rp + µ u + F
⇢t + r · (⇢u) = 0
st =
Q
u · rs +
T
Tt + u · rT = r · (D · rT ) + f (t, T, u)
Multiple Scales:
climate vs. weather
• Don’t call them weathermen!
⇢ut + ⇢u · ru =
rp + µ u + F
⇢t + r · (⇢u) = 0
st =
Q
u · rs +
T
Tt + u · rT = r · (D · rT ) + f (t, T, u)
Multiple Scales:
acoustic equations
• fluid motion: Navier-Stokes equations
⇢ut + ⇢u · ru =
rp + µ u
• disturb pressure slightly
⇢t + r · (⇢u) = 0
p = p0 + ✏P
• velocity and density also get perturbed
⇢ = ⇢0 + ✏D
u = ✏U
• get simpler equations
⇢0 U t =
rP + O(✏)
Dt + ⇢0 r · U = O(✏)
• add science
D = kP
}
kPtt
r2 P ⇠ 0
Multiple Scales:
recovering long term behavior
y 00 + ✏y = 0
00
y =0
• Is there a second time scale?
s=
p
yss + y = 0
✏t
• Just one scale; we we’re using the wrong one!
y 00 + y =
✏y 0
• trick fails
s = ✏t
y⇠e
• secular terms
2
s = t(✏ + a✏ + . . .)
✏t/2
cos t
y⇠e
✏t/2
cos t + O(✏2 t)
Multiple Scales:
effects of weak non-linearity
• Pendulum equation:
00
y + sin(y) = 0
sin(✏x)
x +
=0
✏
00
y(0) = ✏
x(0) = 1
0
y (0) = 0
y = ✏x
x0 (0) = 0
• approximate solution
x000 + x0 = 0
x0 = cos t
• correction
2
x ⇠ cos t + ✏ x1
x001
1
+ x1 = cos t
6
• secular terms --> new time scale
2
s=✏ t
resonance!
Multiple Scales:
effects of weak non-linearity
sin(✏x)
x +
=0
✏
00
x(0) = 1
0
x (0) = 0
s = ✏2 t
• get a PDE
@t2 x
2
+ 2✏ @t @s x +
✏4 @s2 x
sin(✏x)
+
=0
✏
• first approximation
@t2 x0 + x0 = 0
x0 = A(s) cos(t + (s))
• correction
resonance!
@t2 x1 + x1 = A3 /8 + 2A
0
cos t + 2A0 sin t + (A3 /24) cos 3t
• suppress secular terms by avoiding resonance
0
A =0
0
=
2
A /16
frequency correction
x ⇠ cos(1
2
✏ /16)t
Multiple Scales:
effects of weak non-linearity
x ⇠ cos(1
x ⇠ cos t
2
✏ /16)t
Multiple Scales:
destabilizing a swing
• manipulate length of pendulum
0
L
y 00 + 2 y 0 + sin y = 0
L
L = ✏ cos !t
• look for an asymptotic approximation
y = y0 + ✏y1 + . . .
y000 + sin y0 = 0
y100 + sin y1 = 2! sin !t
• choose forcing frequency to destabilize equilibrium
! = 1 + b✏ + . . .
s = t(✏ + a✏2 + . . .)
Homogenization:
repetitive arrays of resistors
• series
Rtotal = nR1 + nR2 = 2nRef f
Ref f
R 1 + R2
=
2
• parallel
Rtotal =
n R11
Ref f
1
1 = 2n
+ n R2
Ref f =
1
R1
2
+
1
R2
Homogenization:
propagation through a complex environment
• velocity changes over short time scale periodically
⇣
⌘
x
dx
= f x,
dt
✏
x = ✏y
✏y 0 = f (✏y, y)
• separate & integrate
Z
x
0
ds
=t
f (s, s/✏)
• Riemann-Lebesgue Lemma
Z
b
g(s/✏)ds = (b
a
Z
b
g(s, s/✏)ds =
a
a)
Z
b
a
Z
Z
1
g(z)dz + O(✏)
0
1
g(s, z)dzds + O(✏)
0
0 = f (0, y)
Homogenization:
propagation through a complex environment
Z
x
0
ds
=t
f (s, s/✏)
Z
b
g(s, s/✏)ds =
a
Z
b
a
Z
1
g(s, z)dzds + O(✏)
0
• homogenize!
Z
x
0
Z
1
0
dz
ds = t + O(✏)
f (s, z)
• differentiate!
0
x (t)
Z
1
0
dz
=1
f (x, z)
• algebra-tize!
x0 (t) = R 1
1
dz
0 f (x,z)
effective wave speed
⇣
x0 (t) = f x,
⌘
x
✏
Homogenization:
effective diffusivity
• heat conduction in 1-D, with small-scale periodic changes in conductivity
d
dx
✓
du
a(x/✏)
dx
◆
d
dx
=0
• can solve this exactly
du
aef f
dx
◆
=0
Rx
u(x) = u(0) + (u(1)
• compute the flux
du
u(1)
a(x/✏)
= R1
dx
✓
ds
0 a(s/✏)
u(0)) R 1
ds
0 a(s/✏)
u(0)
dz
0 a(s/✏)
u(1)
⇠ R1
u(0)
dz
0 a(z)
du
aef f
= aef f (u(1)
dx
• effective heat transfer constant?
aef f ⇠ R 1
1
dz
0 a(z)
u(0))
Homogenization:
effective diffusion tensor
r · (a(y/✏)ru) = 0
• solve two problems on a square of this material
no flux
u=1
no flux
no flux
u=0
â =
Z
r·
â
0
u=1
no flux
1
a(z)dz
0
• homogenized equation
✓✓
u=0
◆
0
ru
ā
◆
=0
ā = R 1
1
dz
0 a(z)
âuxx + āuyy = 0
Summary:
• Small scales can be the hideout for interesting physics.
2
r P ⇠0
kPtt
• Analysis of space and time scales improves understanding.
• Secular terms can give you control.
• Effective (homogenized) equations give insight.
y100 + sin y1 = 2! sin !t
r·
✓✓
• Smudging out details instead of brute force computation.
â
0
◆
0
ru
ā
◆
=0