Fall 2007

Math 587: Perturbation Theory

Hi All,

I will be teaching a graduate course on perturbation theory next Fall on Tuesdays and Thursdays from 11am-12.15pm in the Harvill building. All, from graduate students in Mathematics and Applied Mathematics, from Physics to Optics to Engineering are welcome.

The main goal in the course is to explore and present means of taking advantage of small parameters in the formulation of problems in order that the analysis can be simplified and understanding gained with the minimum amount of analytical work.

For examples: 1.Suppose you wanted to calculate the roots of ex^3-x+1=0 for 0

2.The flow of air around an aeroplane wing is, to a good approximation, inviscid in the sense that the ratio of viscous to inertial forces is almost everywhere very, very small. But of course, close to the wing itself, the

viscous forces are important because the relative velocity of the air and wing is zero (the air sticks to the wing surface). So how does one take advantage of the fact that away from the wing, the flow is harmonic (the velocity potential satisfies Laplace's equation) whereas near the wing the viscous forces come into play? In order to see how to do this, we study very simple but nontrivial models. For example, a simple model for this kind of boundary layer behavior is the following ode; ed2y/dx^2-y=0 with boundary conditions y(0)=A, y(1)=0, for e<<1. If you like this one, analyse what happens for the ode ed2y/dx^2 +y*dy/dx -y=0 with y(0)=A, y(1)=B, and 0

3.The waves on the sea surface have slopes which are certainly less than 10% so that one can approximate the surface to leading order as a linear combination of sinusoidal waves Acos(lx+my-w(l,m)t-c) for any choice of constants A,l,m,c. Under such an approximation, the sea surface is simply a superposition of plane waves. But the average wave slope e is not exactly zero and this leads to weakly nonlinear interactions between the waves. How does one describe the process by which waves of one wavelength share their energy with waves of another wavelength over time scales of 1/e*w where w is a typical wave frequency? The method we learn is called multiple scale analysis and it is good for all sorts of interactions between weakly coupled oscillators whether they be chemical or optical or geophysical or neurological.

4.A layer of fluid heated from below will first transport the heat across the layer by conduction. But when the temperature difference becomes big enough, the fluid starts to move and the heat is transported across the layer by warm parcels of fluid on the bottom rising to the top, carrying their heat with them and then depositing that excess heat at the upper plate before returning as heavier parcels down to the bottom plate. The heat is now both conducted and convected. We will learn how to describe the onset of this process in term of a small parameter e which measures the percentage amount by which the temperature difference exceeds that critical value where convection first sets in. What we learn will also enable us to understand the onset of laser action, the buckling of elastic surfaces, why leopards have spots, why flowers on plants are arranged in Fibonacci sequences and all sorts of behavior in pattern forming systems.

The course will not require you to know a priori anything about fluids or light or mechanics but will use simple models to learn how to exploit small parameters. Once one sees the idea, then one can apply it to more complicated situations. But students will be expected to choose one particular topic from the spectrum and present a project based upon a particular physical situation.

We will learn how to estimate orders of magnitude (you stir a cup of tea; how long does it take for the swirling motion to stop? do an experiment; can you explain the observed time?) and to estimate by how much diffraction broadens a beam of light. We will learn about asymptotic expansions which are series that are often divergent but from which much useful information can yet be gleaned. We will learn about such things as the method of steepest descent which is useful in figuring out the wave pattern behind a duck or a moving ship. We will meet all sorts of canonical equations with exotic sounding but familiar names, such as the complex Ginzburg Landau and Swift Hohenberg equations and see why it is that they are universal (i.e. tend to turn up all over the place).

It would be very helpful if I had an idea as to how many students might want to take the course either as enrolled students or as auditors. Please email me if you are interested, giving me your name and area of interest.

Alan Newell.