THE MOST POWERFUL TOOL
AVAILABLE to the (APPLIED)
MATHEMATICIAN?
Robin Johnson
(Newcastle University)
Outline
 background ideas
 algebraic problems
 differential equations
 applications
Preamble
Mathematicians (but perhaps mainly applied)
– and physicists & engineers – use rather
specific families of skills, such as
• algebra, integration
• classical methods for solving DEs
• (also complex variables, group theory ...)
You will be familiar with those ideas & techniques
that are relevant to you.
3
Typically, physical systems are
represented by DEs
– but rarely standard ones;
– cannot use familiar solution-methods.
BUT not unusual for such problems to contain a
small parameter e.g.
celestial mechanics: small mass ratio
fluid mechanics: 1/(large Reynolds No.)
Can we take advantage of this?
YES!
4
Leads to the idea of asymptotic expansions
(a.e.s), based on the small parameter, and
to singular perturbation theory.
Not the forum for precise definitions and careful
developments, but
 can give an overview of the ideas
 show some techniques and properties
 discuss some elementary examples
 indicate what can be done
5
An Example (to set the scene)
Given
with
f ( x;  )  1  x    e
x 

x  0 and   0 .
Note that
f (0; )  2   


1
2 1    ... .
4
Approximate (asymptotic) representation
requires two ‘sizes’ of x :

x fixed (“= O(1)”) as   0 ;

x   X , X = O(1) as   0 .
6
To see this, we expand appropriately:
f ( x;  )  1  x    e
x 
 1  
x  O(1): f  1  x   ~ 1  x 1 
2 1 x 


x   X , X  O(1) :
f ( X ;  ) ~ 1  e
X 
Note: x = 0 in first gives
1 (1  X ) 
1  2 
X 
1 e

~ 1 1 
but X = 0 in second: ~ 2 (1 
2
1)
4

- wrong!
- correct.
7
Matching
N.B. Two expansions are required here to cover
the domain – a singular perturbation problem.
The two a.e.s are directly related:


1  
1
1  x 1 

1


X
1




2 1 x



2 1  X 


1
1
~ 1  X  
2
2
1 e
X 
1 (1  X ) 

1  2 

X

1 e

1 e
x  
1
1


2

1
~ 1  (  x)
2
x

x  
1 e

8
The two ‘expansions of expansions’
agree precisely (to this order);
they are said to ‘match’ – a fundamental
property of a.e.s with a parameter: the
matching principle.
Graph of our example:
f ( x;  )  1  x    e
x 
Plotted for
decreasing ε
9
Breakdown
Another important property of a singular
perturbation problem: breakdown of a.e.s.
E.g. f ( X ;  ) ~ 1  e
X
1 X 

1
1  2 
X 
1 e


which is valid for X = O(1), and correct on X = 0.
The expansion ‘breaks down’ (‘blows up’) where
two terms become the same size; here
 X  O(1) i.e. x  O(1) - the variable used in
the other a.e. !
(‘large’ X)
10
Introductory examples
Start with a simple exercise: quadratic equation
f ( x;  )   x  x  1  0.
2
Treat the expression as a function to be expanded:
x  O(1) : f ~ 1  x so (approx.) root x ~ 1 .

Seek better approx.: x ~ 1 

n
an
n 1
Second root? Can arise only for large x.
11
Breakdown of the ‘a.e.’:
f ( x;  )  x  1   x
2
2
is where  x  O( x) i.e. x  O(
1
)
so rescale: x  X  to give
f ( X  ; )  
X  O(1):
1
1
2
( X  X   )   F ( X ; )  0.
2
F~X X
so (approx.) roots X  0, 1 , but X = 0
corresponds to a breakdown: X  O( ) : X   x.
Roots are x ~ 1,  1  .
12
Another algebraic example
Consider f ( z;  )   z  z  1  0 , then
2 5
3
z  O(1) : f ~ z  1 : (approx.) roots z ~ 1, ,  .

2 5
3
Breakdown where  z  O( z ) : rescale z  
2
3
to give
3
f (  ;  )   F ( ;  )  
5
3

  
5
3
3
  0.
3
  O (1) : F ~    : relevant (approx.) roots
  i
Roots: z ~ 1,  ,  , i ,  i .


2
13
Ordinary differential equations
First example, to show ideas & methods:
y  2 y   xy  1  2 x ; x  1; y(1;  )  1.
2

ODE implies that y ( x;  ) ~

n
yn ( x) for
x  O(1),
n 0
so
with
y0  2 y0  1  2 x ; y1  2 y1 
2
xy0
 0, etc.,
y0 (1)  1; yn (1)  0, n  1.
Can now solve the sequence of problems.
14
This procedure gives
y( x;  ) ~ x 
1

8
3  6( x 2  x)  4 x3  e2(1 x) 


for x = O(1), but breaks down where  x  O( x)
3
so x  O(1
 ) and then y  O(1
) .
Rescale: x  X  , y  Y ( X ;  )  , to give the ODE
 Y   2Y  XY  2 X  
2
and no b.c.!
15

Seek a solution Y ( X ;  ) ~  
n 2
Yn ( X )
n 0
then
2
XY0  2Y0  2 X so
Y0  
1
X
1

2
1 2X .
X
Invoke the Matching Principle:
y( x;  ) ~
x 1
8
3  6( x 2  x)  4 x3  e2(1 x) 


gives Y ~ X  1 X 3 , and Y ( X ;  ) ~ Y ( X )
2
0
gives
1

2

2 4
1
y~
1  (1   x   x ) .
2
x
Matching accomplished with the positive sign.
16
Another type of ODE
A ‘boundary-layer’ problem:
2
 y  y  6 xy  2 x, 0  x  1,
with y(0;  )  2, y(1;  )  3.
N.B. Boundary layer – a scaling – is near x = 1.

n
For x away from x = 1 : y ( x;  ) ~   yn ( x),
n 0
with y0 (0)  2; yn (0)  0, n  1.
17
We obtain, for the first two terms in
the a.e. :
2 3

y ( x;  ) ~ 2  x   (2  x )  2 x  8 


2
which gives y ~ 1  9 on x = 1 – not correct.
Rescale: x  1   X with y(1   X ; )  Y ( X ;  ),
then Y   Y   6 2 (1   X )Y 2  2 (1   X )
with Y (0; )  3.
Write Y ( X ;  ) ~

n
  Yn ( X ); Y0 (0)  3; Yn (0)  0, n  1.
n 0
18
The first two terms, satisfying the
given boundary condition, are
Y ~ A  (3  A)e
X

  2 X  B(1  e
X

) .
(A and B arb. consts.)
Match :
2 3

y ( x;  ) ~ 2  x   (2  x )  2 x  8


gives Y ~ 1   (2 X  9),
2
and above gives y ~ A  2(1  x)   B;
a.e.s match with the choice A  1, B  9.
19
One further technique
Probably the most powerful & useful:
the method of multiple scales.
Describe idea by an example:
3
x  x   (2 x  x )  0, t  0
(a Duffing equation with damping; λ>0, constant)
with x(0;  )  1, x(0;  )  0.
Oscillation described by a ‘fast’ scale – carrier
wave, and a ‘slow’ scale – amplitude modulation.
20
An example of a modulated wave:
In this approach, we use both scales at the
same time!
We introduce



n
T   ( )t ~ 1    n  t


n2


(fast)
and    t (slow).
Impose periodicity in T, and uniformity in τ
(as   ).
21
Now seek a solution

x(t ;  )  X (T , ;  ) ~

n
X n (T , );
n 0
the equation for X becomes:
 X TT  2 X T   X  X  2 ( X T   X )   X  0.
2
Then
2
3
X 0TT  X 0  0;
X1TT  X1  2 X 0T  2 X 0T 
3
X0
 0,
and so on (together with the initial data).
22
Solving gives
X 0 (T , )  A0 ( ) cos T  0 ( )
and then periodicity of X1 (T , ) , satisfying the
initial data, requires
A0 ( )  e

and
3
0 ( )  16
(e
2
 1).
This leaves
X1 (T , )  A1 ( ) cos T  1 ( )  
1 3
e
32


2
3

cos 3 T 
(e
 1)  ,
16


and so on.
2
E.g. boundedness of X 2 requires  2   2.
23
Comment
Is it consistent to treat T and τ as independent
variables? (They are both proportional to t !)
If a uniformly valid
solution exists, then it
holds for T  0,   0 ;
thus it will be valid on
any line in the first
quadrant of (T,τ)space.

bounded (and uniformly valid)
periodic (and bounded)
     ( )  T
T
24
And ever onwards
1. These ideas go over, directly, to PDEs.
Asymptotic expansions take the same form,
but now with coefficients that depend on more
than one variable cf. multiple scales for ODEs.
Breakdown (scaling) occurs, typically, in one
variable, as all the others remain O(1).
2. Relevant scalings are usually deduced directly
from the governing differential equation(s).
25
Some Applications – a small selection
1. Gas-lubricated slider bearing
Based on Reynolds’ thin-layer equations,
this describes the pressure (p) in a thin film
of gas between two (non-parallel) surfaces:



 h pp   hp  , 0  x  1, p(0;  )  p(1;  )  1
3
where h(x) is the gap between the surfaces,
and ε is the (small) inverse bearing number.
This is a boundary-layer problem, with the
boundary layer near x = 1.
26
2. Restricted 3-body problem
The ‘restricted’ problem is one for which one of
the masses is far smaller than the other two.
(It was for this type of problem that Poincaré
first developed his asymptotic methods.)
In a frame centred on one of the larger masses,
we obtain


xy
x
y





2
3
3
3
3


dt
x
x

y
x
y


2
d x
x
2
and
d y
dt
2

y
y
3
(position of small mass: x, of second large mass: y).
27
Solution for small μ is a singular
perturbation problem if
x  y  O(  )
i.e. the small mass is close to the second large mass.
Introduce x  y   X to give


y


X
y

 2X  
  (1   ) 

3
3
3

X
y 
 y  X
X
and
y
y
y
3
and then t  t0  T near to the time of
close encounter. Expand each and match.
28
3. Michaelis-Menten kinetics
This is a model for the kinetics of enzymes,
describing the conversion of a substrate (x) into a
product, via a substrate-enzyme complex (y) :
dx
dt
 x  (x     ) y
and

dy
 x  (x   ) y
dt
with x = 1 and y = 0 at t = 0, for   0 .
Equations exhibit a boundary-layer structure in
y but not in x ;
a convenient approach is to use multiple scales.
29
Introduce T  t and    1g (t )
and seek an asymptotic solution for
x  X (T , ;  )
and
y  Y (T , ;  ).
Problem now becomes
1
 g X  X T   X  ( X     )Y ; g Y   YT  X  ( X   )Y
with X (0,0; )  1, Y (0,0;  )  0.
Obtain, for example,
X 0 (t )    dt

0
g (T )  
T
where X ~ X 0, the solution of X 0   ln X 0  1   T .
30
4. Josephson junction
This junction, between two superconductors
which are separated by a thin insulator, can
produce an AC current when a DC voltage is
applied – by the tunnelling effect.
An equation that models an aspect of this is
u   (1  a cos u)u  sin u   b
with u (0; )  u(0;  )  0 , for the voltage u(t;ε).
Relevant solution is u = εU(T,τ;ε), using
multiple scales.
31
Introduce
τ = εt and T  t ,  ~ 1 

n
  n
n2
and then we find that


u ~  b 1  (cos T ) exp  1 (1  a) 
2


for a  1.
Higher-order terms can be found directly,
and in the process we determine each n .
32
5. Fluid mechanics I: water waves
The equations for the classical (1-D) inviscid
water-wave problem:
u   (u  uu  wuz )   p ;
 2 [ w   ( w  uw  wwz )]   pz ;
u  wz  0,
with
p  h & w  h   (h  uh )
and
w  0 on z  b.
on
z  h( , )
These are written in suitable variables, with
two parameters: δ and ε.
33
The governing equations are essentially an
elliptic system, but the surface b.c.s
produce a hyperbolic problem for the
surface profile, z = h.
This problem can be analysed, for example, for
  0 (small amplitude waves) with δ fixed
 0
(long waves) with ε fixed
  0 and   0 (small amplitude, long waves)
34
6. Fluid mechanics II:
viscous boundary layer
The appropriate form of the Navier-Stokes
equation, mass conservation, etc., is
uu x  vu y   px  R
uvx  vv y   p y  R
1
1
(u xx  u yy );
(vxx  v yy );
u x  v y  0,
2
2
with u  U ( x), v  0, p  0 as x  y  
and
u  v  0 on the body,
for 2D, incompressible, steady flow.
35
The classical boundary layer, of
thickness O( R1 2 ) as R   , is
represented schematically as:
However, at the trailing edge, where
there is the necessary adjustment to the
wake, we have a ‘triple-deck’ structure:
all described by matched asymptotics.
37
Conclusions
We have
 outlined the ideas and methods that
underpin the use of asymptotic
expansions with parameters;
 described, in particular, their rôle in the
solution of differential equations;
 mentioned a few classical examples.
38
The End