I'm not sure if this question is appropriate for MO. Add comments if it is not. Thanks.
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…)
Best Answer
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.