Singular Perturbation Theory for Algebraic Equations

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					Singular Perturbation Theory for Algebraic Equations
Monday, February 22, 2010
1:58 PM



             Readings: Holmes, Sec. 1.5
                       Hinch, Secs. 1.2 & 1.3

             Homework 1 due Thursday, February 25.

             Again we will consider some exactly solvable algebraic equations to see what can go wrong with
             regular perturbation theory and how to fix it. These techniques will have much more useful
             analogs that will carry over to the more interesting differential equation context.

             Consider trying to solve the following algebraic equation with regular perturbation theory:




                       Quadratic equation gives exact solution, but suppose we don't know
                       that.

                       Try regular perturbation theory to find the roots:




                 This looks like a fine solution, but we're supposed to get two for a
                 quadratic solution.

                 Metaprinciple (not necessarily rigorous but usually true): When
                 perturbation theory fails, you can often detect it from the
                 approximation it gives you.

                 Here we see that the perturbation theory only produced one solution
                 where there should have been two, so clearly one solution is not being
                 approximated.

                 Let's look at the exact solution to see what went wrong with the
                 perturbation method:




                  Let's expand this in a power series in  to see what a perturbation
                  theory might have been capable of doing.



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  theory might have been capable of doing.




 Well at least the regular perturbation theory did get one of the roots
 right.

 But the reason it couldn't find the second root is that the asymptotic
 expansion for the second root is inconsistent with the assumption of
 regular perturbation theory. In particular, the second root grows
 unboundedly as  becomes smaller, and this violates the implicit
 assumption that x0 , x1 are order unity quantities.

 How do we fix this deficiency of the regular perturbation expansion.
 Well we could have instead tried:




                        and this would work but not entirely satisfying because how do we
                        know how many negative powers of  we would need in a given
                        calculation?


A more efficient way to modify regular perturbation theory is to begin
with consideration of dominant balance.


This arises when we realize that a solution we're looking for is not actually order unity.
This gives a good reason for why regular perturbation theory doesn't find it because
regular perturbation theory is implicitly assuming the base solution (x0) is order unity.

A simple idea is to rescale the problem by an appropriate factor so that the missing
solution is order unity in terms of the new rescaled variable.




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Want the solution you're trying to approximate be order unity in terms
of X.

For our case, this would mean:




 And it's clear a regular perturbation theory could get this.

 But how would we have guessed this without knowing the exact
 solution?



  Make the rescaling in the equation, without presupposing the
  appropriate choice of .




    And remember that we want



   Principle of dominant balance: If you consider the magnitudes of each
   term in an equation, there cannot be one nontrivial term that has a
   larger order of magnitude than all other terms. So when each term is
   given an order of magnitude, there must be two or more nontrivial
   terms of the dominant (most important) order.

   So what one can do is to see what possible balances can be struck in
   the equation through appropriate choices of  and check whether
   those are dominant balances. If they're not dominant, forget it, if they
   are dominant, might be useful.


   In our equation, we have three terms, so have 3 possible pairwise
   balances to check.




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We've found two good dominant balances. The second one clearly
corresponds to the solution already found by regular perturbation
theory. The first one suggests how to find the missing root:

Try a regular perturbation theory on the rescaled variable:




         Clean up the negative powers:




                 The zero root corresponds to the one we already found, and to some
                 extent we are not correctly scaled to find it (though we could with
                 harder work). This hints that another rescaling would be appropriate to
                 approximate this small root, which is of course equivalent to what we
                 did before.

                 Continuing just with the new root for which we have properly rescaled.




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in this way, combining this result with the one from regular
perturbation theory, we obtain the approximations to the solutions of
the original quadratic equation:




Another singular perturbation problem for algebraic equations




          Try regular perturbation theory:




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These can be shown to be formally consistent with the exact solution.

But notice that the coefficients of the higher order terms are large. This may cause us to worry because a
fundamental premise of the perturbatoin theory is that these coefficients are order unity. Formally
speaking, 100 and 101 are order unity since they don't involve the small parameter. However, the utility of
the perturbation approximation will clearly degrade if higher order terms aren't getting smaller.
In particular, while this regular perturbation expansion will work very well for


     but would be very bad for


     And going to higher order wouldn't help, the terms would keep getting
     bigger.

     Saying this more generally, suppose we had the problem:




                                  where  is treated as an order unity constant.




   Repeating the regular perturbation expansion for fixed  meaning we
   assume it is order unity) and smal  would give the following:




 This is a good approximation when



        but becomes horrific when



  The point is that one way in which regular perturbation theory can give
  bad results is if the underlying problem has a hidden small parameter.
  That is, there are constants that are being treated as order unity but in
  fact are small enough to be comparable with the parameter that is


                                 Pertubation10 Page 6
  fact are small enough to be comparable with the parameter that is
  explicitly assumed to be small and this interferes with the perturbation
  expansion.

  what is the hidden small parameter here? It's not just seeing a .01 at
  the end of the number. The real hidden small parameter is the
  separation between the roots. We run into trouble when the small
  parameter is comparable to this separation between the roots because
  in some sense one is blurring over qualitatively distinct features.




 So we see that the perturbation can fundamentally change the nature
 of the problem (i.e. the base problem) if it is comparable in size to the
 separation of the roots. The perturbation could cause the roots to
 collide, bifurcate, etc. which really means the original base problem is
 not close to the perturbed problem in a computationally meaningful
 way.



How do we fix this?
 ○ If  is small but finite, then perhaps it should be linked to the explicit small
   parameter, i.e.,




   This gets complicated and we'll come back to it in a later lecture under
   the guise of distinguished limits. The metapriniciple is that it is usually
   impossible to develop a asymptotic expansion with respect to two
   independent small parameters. This is why one typically make linkages
   between small parameters and develop results separately for various
   asymptotic regimes of interest. More on this later in the course.
 • If  then we have a more fundamental problem. In this case, the
   base problem has multiple roots, and not coincidentally, the
   pertubation expansion produces infinities.


How does one deal with this problem? Rescaling the equation won't
help because clearly the roots should still be order unity. Let's look at
the exact solution to see what goes wrong:

Quadratic equation:




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So the equation was properly scaled, but the appropriate expansion
parameter was not in terms of integer powers of the small parameter.

This suggests modifying the asymptotic expansion in terms of
a more general gauge functions. Clearly looking at the exact solution
suggests we should have expanded in half-integral powres of the small
parameter but how could we have seen this without the exact solution?

First of all recognize that regular perturbation theory broke down, and
that the rescaling trick (finding another dominant balance) wouldn't
help.

Try a more general asymptotic expansion:




                                             are the "gauge functions"




        The gauge functions need to be determined on the fly…but let's sort
        this out as best we can without making special assumptoins yet.



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Hard to order these all at once but can attack the most important
piece.




If one wanted to estimate the actual size of the error, one would have
to go to higher order terms to see how big  should be to create a
balance with the most important uncancelled term.




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