Sat 04 January 2014
Somehow I escaped undergrad and grad school without ever formally learning partial differential equations (PDEs), or much of any mathematical physics. I identify more with the discrete, algorithmic, data-structures side of the applied math universe -- beyond basic undergraduate physics, I was largely PDEngorant (sorry).
My reasoning always went along the lines of
Most real physical systems I ever build will be of geometry sufficiently complex to need numeric solvers
Modern numeric solvers are compelx enough that I probably won't learn much by implementing a trivial one myself
Manipulating symbols is hard.
It turns out these things may all be true, but that's not really the point -- the amazing things about fluency with PDEs is that it makes learning all of physics much easier. By understanding simpler geometries and problems without getting held up by the math, we can build physical intuition.
Now that I'm an adult (ugh) I also really appreciate some of the beauty present in things like Sturm-Liouville theory.
So if you want to quickly grok PDEs, there are a few things you should know.
First, there are basically three "canonical" types of PDEs that show up in introductory mathematical physics, many of which you may have seen before:
The heat equation:
This says, roughly, that the rate of change in time at a point is proportional to how much the point differs from the average for its neighborhood. PDEs that can be transformed into equations analogous to the heat/diffusion equation are called parabolic PDEs.
The wave equation:
PDEs that look like the wave function are called hyperbolic, and retain discontinuities present in their initial conditions -- waves!
Laplace / Poisson's equation
Laplace's equation (Poisson's equation is Laplace's equation with an inhomogeneous term) says a function must be "maximally boring". The second derivative everywhere must be zero! For the 1-d case this is a line; for higher-dimensions solutions 1. can only take their maximum on the boundary of the problem domain or, equivalently, 2. any point that's a solution must be the average of nearby points. These are called harmonic functions, and are the source of a lot of the "special" functions in mathematical physics.
These are linear and second-order and capture a tremendous amount of the intuition behind both a wide variety of physical systems and how to solve a large number of PDEs.
Methods of solution
Then there are a couple of methods for "solving" the PDEs above that arise from simple, convenient boundary boundary conditions including:
- Separation of variables -- for PDEs involving time and space, you can often separate out the time part and the space part and then solve them with regular ODE techniques. Yes, this means you need to remember some ODEs.
- Eigenfunction solutions -- a VERY LARGE number of the PDEs that show up in the real world are linear, and via methods like separation of variables you essentially get eigenfunction relationships, and can find a nice orthonomal basis and get a series solution.
- Green's functions -- for some driven systems, compute the impulse response of the system, and just use that to find the solution for the driven system.
I actually started down this path by trying to better understand green's solutions in electrostatics problems.
Other Neat Facts
First-order PDEs can be just as complex to solve than some second-order PDEs, which is both surprising and neat. Traffic flow is often modeled via first-order PDEs. Finding good material to study first-order PDEs can be tricky, as there's a lot of sloppyness with symbols in most texts. I highly recommend watching the videos from Chris below.
Sturm Liouville theory
Many PDEs reduce (via separation of variables) to a class of ODEs described by Sturm-Liouville Theory. Now, this is really neat -- when combined with the right boundary conditions, these ODEs constitute a Hermitian differential operator which is guaranteed to have eigenvalues and eigenfunction solutions with very elegant properties. Bonus: You get to learn about weighted \(L^p\) spaces. I highly recommend chapter 10 of the Olver text.
Euler-Bernoulli Beam Theory is one of the canonical examples of a fourth-order PDE!
At the time, science and engineering were generally seen as very distinct fields, and there was considerable doubt that a mathematical product of academia could be trusted for practical safety applications. Bridges and buildings continued to be designed by precedent until the late 19th century, when the Eiffel Tower and Ferris wheel demonstrated the validity of the theory on large scales.
Admittedly this is from wikipedia, but if true than this neat little PDE helped convince the world that science could inform engineering!
One of the great things about studying as well-worn a path as PDEs is that there is almost certainly a textbook out there that speaks directly to your preferred learning style and background. Four sources of material I found exceptionally helpful were:
PDE Tutorial Videos from Dr Chris Tisdale were quite fantastic, especially given the paucity of material out there about the mechanics of first-order equations.
Peter Olver's Introduction to Partial Differential Equations was by far my favorite of the books, hitting the right blend of pure and applied math. I used the notes that turned into this book. Highly recommended.
Farlo's Partial Differential Equations for Scientists and Engineers is a great "I have a flight to the east coast and back and want to learn PDEs in 12 hours" text. It's light on the math, focusing mostly on the 1D case and working hard to build up physical intuition. I highly recommend it for the busy engineer, or someone not quite ready for the beauty that is Olver's more theoretical treatment.
Also, if you want to continue to learn more about differential equations (both PDEs and ODEs) I recommend @diff_eq on Twitter.