Partial Differential Equations — Heat, Wave, and Laplace
Stub. Separation of variables for the three classical linear PDEs: the heat equation, the wave equation, and Laplace's equation, each reduced to a Sturm-Liouville eigenvalue problem plus a Fourier-series expansion to match boundary/initial data.
The payoff chapter: separation of variables turns each of the three archetypal PDEs — heat (diffusion, first-order in time), wave (second-order in time, finite propagation speed), and Laplace (steady-state, no time dependence) — into an ODE eigenvalue problem in space (solved via Sturm–Liouville theory) and a matching ODE in the remaining variable, with the boundary/initial data matched by expanding in the resulting eigenfunctions (a Fourier series). Same method, three different physical behaviors, depending on the order and sign of the time/normal derivative.
Outline (TODO — flesh out each)
- Begin heat equation — derivation, boundary/initial conditions
- Heat equation and separation of variables I
- Heat equation and separation of variables II
- Wave equation I — derivation, d’Alembert vs. series solution
- Wave equation II
- Laplace equation I — steady states, boundary value problems
- Laplace equation II