Pertubation10 Page 1

Transcription

1 Pertubation10 Page 1 Singular Perturbation Theory for Algebraic Equations Monday, February 22, :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

2 Pertubation10 Page 2 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 x 0, x 1 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 (x 0) 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.

3 Pertubation10 Page 3 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.

4 Pertubation10 Page 4 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.

5 Pertubation10 Page 5 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:

6 Pertubation10 Page 6 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

7 Pertubation10 Page 7 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:

8 Pertubation10 Page 8 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.

9 Pertubation10 Page 9 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.