Hydrobates

If the inner and outer solutions which we introduced previously are to define a good approximation then their difference should tend to zero as tends to zero. Their derivatives should also do so. This idea is implemented by looking at some intermediate point which is not too close to the origin but not too far away so that it can be hoped that both approximations are valid at that point. In the book the choice is made and this is another case where intuition is used instead of a clear algorithm. This leads to the choice and and in that case the difference vanishes faster than any power of as . Since the equation is linear we can add the inner and outer solutions to get a new solution. Here the lowest order term is present in both summands and so we need to subtract it off. The result is . It satisfies the boundary condition at exactly but it satisfies the boundary condition at only approximately, whereby the error is exponentially small.

What has been achieved here? The method of matched asymptotic expansions has been applied to identify a candidate approximation to the solution of a certain boundary value problem. Since in this example the solution can be computed explicitly it could be checked that this candidate actually does satisfy criteria which makes it a good solution. On the other hand if the exact solution is not used it gives us no independent argument that the candidate is a good approximation. An interesting question is how it is possible to obtain a rigorous proof that a function obtained by this method satisfies the conditions which it was designed to satisfy.

This problem can be related to the technique of geometric singular perturbation theory (GSPT). Let . Then the original ODE can be written in the form , . This is the canonical form of a system with two time scales in GSPT. We have a slow variable and a fast variable . The critical manifold is given by . This problem is normally hyperbolic with an eigenvalue . Intuitively, the solution approaches the slow manifold very fast and then moves slowly along it. It is not possible to immediately apply the most basic techniques of GSPT to this problem because the solution on the slow manifold goes to infinity – it does not remain in a compact set.

In a previous post I wrote about matched asymptotic expansions, concentrating on a classical example, the Kaplun-Lagerstrom model. Now I want to go into some aspects of this subject more deeply. My main source is once again the book of Hastings and McLeod. In this post I discuss the ODE on the interval with boundary conditions and . The equation contains a parameter . We would like to understand the behaviour of solutions of this problem for small. The equation is linear and looks quite simple. That this problem is not straightforward can be seen if we set . A naive hope would be that the solution for small and positive can be approximated uniformly by a solution of the limiting equation with . In fact the equation with has the general solution and this does not satisfy the desired boundary conditions for any value of . This has to do with the fact that the equation for , being second order, has a two-parameter family of solutions while that for , being first order, has only a one-parameter family. This shows that this is what is called a singular perturbation problem, where the nature of the equation changes essentially as a certain parameter value is reached. To obtain more information we expand the solution in powers of , and substitute this into the equation. This means that we try the simplest expansion imaginable. Then satisfies the equation for and we are confronted with the problem which is already familiar. If we ignore this we find that it is consistent to set all with to zero. There is no clear preferred choice for the constant . In fact the ODE and thus the boundary value problem can be solved explicitly. The strategy here is to ignore that fact as much as possible and gain information in other ways. The hope is that these other methods will apply in cases where no explicit solution is available. Looking at the explicit solution reveals that the choice gives a function which solves the boundary condition and is a good approximation to the true solution on most of the interval . When is approached the exact solution swerves off from the approximate solution so as to head for the point corresponding to the boundary condition . The derivative becomes very large when is very small. It looks like the solution could be got by piecing together an outer solution, the explicit one we already saw, with an inner solution defined in a small region near .

How can we get information about the inner solution? One strategy is to use a magnifying glass to see more clearly what is happening in the small region of interest. This is done by introducing a new coordinate . This is a bit arbitrary. In principle we could have used a different power of in the rescaling. It turns out that this choice is useful in this example. Call the transformed solution . It satisfies . We can now do an expansion of this equation in powers of . Substituting this in the original ODE gives a sequence of equations for the expansion coefficients . The case can be solved to give . This is the general solution satisfying . In the case the general solution with is of the form . It is consistent to set for and we do so. Thus we have two free constants and . We now want to combine these inner and outer solutions to approximate the solution of the original problem as well as possible. We try to match them as well as possible by choosing the parameters and . An account of how this is done will be postponed to a later post.