Introduction to regular perturbation expansions for pdes:
First we will study asymptotic expansions of exact solutions (integrals involving Greens functions, Fourier transforms)
This approach is based on the assumption that we can nd an exact solution.
Ex

Real Ginzburg-Landau equation
Consider the scalar reaction-diusion equation:
ut = uxx + u + u3 + g(u) = 0
with conditions: u is spatially periodic, u(x, 0) = h(x)
Note that u = 0 is the basic solution. Here we have no boundary conditions to select particu

WKB Analysis
Slowly varying coecients/media:
We consider equations of the form
utt + 2 ( t)u =
f 0 (u, ut )
(1)
or
2
u + k 2 ( x)u = 0
(2)
The rst equation describes an oscillator with slowly varying frequency, with a possible weakly
nonlinear forcing.
Th

WKB Homework #2 for Math 605
Due: March 16, 2005
It may be helpful to read Zauderer, Chapter on Asymptotic Methods, on reserve.
1. Assume that the plane wave uI = eikx is incident on the parabola
C: x=
1 2 a
y
2a
2
, and that the total eld u = 0 on C. Fi

NM} NM PAR'HAL DIFFERENTIAL FGUATIOMS (9
MM
u II
WE CoNJ'DER SLATTIRWG BY A BODY OF RADIUJ L.
U”: c2 AU
C: co DIX) Co: nefoneuce Jpeed [OfJOuNd [E'bQPJ‘
DQC) NJQX o} RefrachroN.
5119me wt HAVE A rEmomc JOLUTION
not
ﬂow): e“ MIX).
mm w? an
ALI + K1 DIIX)

Quasi-steady approximations
Quasi-steady approximations:
k1
k
2
S + E C P + E
k3
substrate + enzyme = complex = further reaction to product + enzyme
k
1
S + E C
C = k1 SE k3 C k2 C
S = k1 SE + k3 C
E = k1 SE + k2 C + k3 C
P = k2 C
E + C = conserved quanti

Another place that free-boundary problems appear is in option pricing,
particularly American options.
An American call option is the right to buy a stock at the exercise price
K any time up to expiry date t = T . In contrast, a European call option is
the

Bifurcations
Nonlinear Boundary Value Problem
v + f (v) = 0, for v(x; )
v(0) = v() = 0
0<x<
(f (v) = v + av k )
Assume f (0) = 0, e.g.
v = 0 is a solution
v = 0 is the basic solution
Linearize about basic solution
v + f (0)v = 0
(expand f in a Taylor ser

Multiscale analysis
Example 1
Heat conduction + slow radiation
ut + u = uxx < x < t > 0
u(x, 0) = f (x)
Exact
u(x, t) = e t v(x, t)
vt = vxx
v=
G(x, , t) f ()d
(x)2
e 4t
4t
Compare to a regular perturbation expansion, for
1
u0 (x, 0) = f (x)
u0t = u0xx
u1

Freezing
liquid
u2
II
Melting
ice
u1
T = Ts
Tl
liquid
u1
I
ice
u2
I
II
Moving boundary problems
Heat equation (with convection) with phase change
For melting:
(I) Liquid:
(k1 u1x )x = 1 c1 u1t +
1 c1 u1x
possible convection
= bulk velocity
(II) Solid
(k2

Free Boundary Homework for Math 605
Due: April 12, 2005
1. Solve the problem
ut = uxx 2, 0 < x < X(t)
u(X(t), t) = 0, ux (X(t), t) = 0, u(0, t) = t, t > 0
X(0) = 0
a) Use the transformation u = Ctv(x, t) to nd an equation for v.
b) Find a similarity solut

Pattern Homework for Math 605
Due: March 31, 2005
1. Consider the problem
ut = uxx + (eu 1), 0 < x < , u(0, t) = u(, t) = 0
For the steady state problem (ut = 0), use perturbation expansions in
1 to determine
the solutions which bifurcate from u 0, as fol