How does one motivates the method of separation of variables when teaching PDE's?

Question

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How to explain/motivate the method of separation of variables for PDEs to undergraduates? What's the real math behind it? It's not just because the guy who fancied it is very smart, right? (Although I feel like it does give students this impression...)

(Background: At Berkeley there is a course called Math 54, in which students learn linear algebra, linear ODEs and then 1 or 2 weeks of PDEs. Teaching Separation of variables is always my nightmare...)

Answer 1

First of all, separation of variables is mostly limited to linear PDEs. It is intimately related to group invariance. For instance, if the equation is invariant under time-translation, then you may look for solutions of the form $e^{-\lambda t}v(x)$. This suggest to have a complete theory of the Cauchy problem by means of the Laplace transform. If the PDE is translation-invariant in direction $x_1$, then search for solutions $e^{i\xi x_1}v(t,x_2,\ldots,x_n)$. If the equation is isotropic, then look for solutions $sin(m\theta)v(t,r)$ (in space dimension $2$) or $\phi(\omega)v(t,r)$ where $\phi$ is a spherical harmonic (dimension $3$).

Of course, some PDEs have a large invariance group, for instance $\Delta u=0$. Some have a smaller invariance group. For instance, the separation of variables is not as much efficient for Stokes equation ($\Delta u+\nabla p=0$ and ${\rm div} u=0$) as it is for the Laplacian.

An other important aspect is spectral analysis. If the domain, the differential operator $L$ and the boundary conditions are equivariant under the action of some group, then the function space splits naturally into invariant subspaces $W_j$ that are stable under $L$. The eigenvalue problem $Lu=\lambda u$ reduces to the case where $u$ is in some $W_j$. This simplifies the search of eigen-elements. For instance, one finds explicitly the spectrum of $\Delta$ with Dirichlet boundary condition over a sphere or a parallelotop. This aspect is also related to orthogonal polynomials. Edit. Inside an ellipse, one finds explicitly the spectrum of the Laplacian, but not that of the Stokes operator. The former splits in coordinates attached to the foci, but the latter don't.

Answer 2

Your course sounds very much like Harvard's 21b, which I've taught a few times. I actually really like the way we do the PDE's unit, although I think two weeks is not enough time and (consequently) the students don't like it so much. Perhaps you can adapt the following method to your satisfaction. Our textbook, by the way, is Otto Bretscher's "Linear Algebra with Applications", though the stuff on PDE's is not there (it's here instead).

Some pre-explanation: a few weeks earlier in the course we cover systems of first-order linear ODE's, namely,

$$\frac{d\vec{x}}{dt} = A \vec{x},$$

with A a matrix. We explain how this can be reduced to n simple integrations by diagonalizing A and then writing the general solution as a linear combination of eigenvectors scaled by the various resulting exponentials. This should impress on the students the importance of the idea of solving an equation of the form "derivative(x) = operator(x)" by diagonalizing "operator". (This part of the course IS in Bretscher's book.)

When we get to the PDE's unit, we have already talked about Fourier series a bit; no one quite gets why, but that's because we haven't done the application yet. Then we slap down the heat equation,

$$\frac{df}{dt} = \mu \frac{d^2f}{dx^2},$$

and observe that now "operator" is $d^2/dx^2$, but the same idea applies. We solve for its eigenvectors and lo! they are the trigonometric functions in x. Then we do the simple integrations in t and the resulting "linear combination" is a Fourier series (whose coefficients are decaying exponentials), which turn out to be useful after all. We even go on to do the wave equation in the same way. Note how the variables got separated so naturally.

Like I said, I love the way the process is reduced to an idea in linear algebra having nothing to do with sophisticated notions in analysis or tricks involving calculus. Those are for courses truly devoted to PDE's; in this course, it is enough to just explain why separating variables even happens.

EDIT: Upon rereading, I feel like I should preempt the inevitable objection that the hyperbolic trig functions (equivalently, exponential functions) are also eigenvectors of $d^2/dx^2$. Of course, we throw those out because they don't satisfy the boundary conditions we are using, which are generally that $f(0,t) = f(1,t) = 0$ or similar. Explaining this is usually the cause of moderate distress among the students, so recently, I noticed that we just ignore it.

Answer 3

Have a look at Chapter 1 of Fourier Analysis by Elias Stein. You start with the wave equation $\frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2}$ one of the simplest solutions is $u(x,t) = \sin m x \sin m t$. Then you can try seeing if other solutions $u(x,t) = \phi(x)\psi(t)$ work and this is your separation of variables. If you solve the two differential equations $\phi'' - \lambda \phi = 0$ and $\psi'' - \lambda \psi = 0$ you should get a combination of sines and cosines, rederiving Fourier series. Checking the boundary conditions only sines or only cosines will work and $\lambda = - m^2$ necessarily.

That's how I learned it but it was for a junior-level Fourier analysis class.

Answer 4

At the level of your Math 54 I suggest the following: By the time you get to a heat or wave equation, the students are used to seeing functions such as $ce^{at}$, $c_1\cos(at)+c_2\sin(at)$, as solutions to simple ODE. It is somewhat natural then to let the constants $c$ become functions of $x$. It becomes a reasonable thing to try, an extension of ``undetermined coefficients''. (The Notes on Differential Equations on my web page uses this description.)