an eigenfunction expansion
An eigenfunction expansion is the act of rewriting a function — and more importantly the whole solution of a PDE — as a sum over the natural modes of the problem. It is the same move as the generalized Fourier series, but the name emphasizes the strategy: change to the coordinate system in which the operator becomes diagonal.
Here is the method in plain steps. Take a PDE like u_t = k u_xx on an interval with homogeneous boundary conditions. The Sturm-Liouville eigenvalue problem -X'' = lambda X with those conditions yields modes X_n(x) and eigenvalues lambda_n. Write the unknown solution as u(x, t) = sum of T_n(t) X_n(x). Substitute: each X_n is just multiplied by lambda_n, so the PDE collapses, mode by mode, into a simple ODE T_n' = -k lambda_n T_n, with solution T_n(t) = T_n(0) e^(-k lambda_n t). The initial coefficients T_n(0) are the expansion coefficients of the initial data. You have turned a PDE into infinitely many uncoupled, trivial ODEs.
This is the diagonalization idea from linear algebra carried into function space. In the eigenbasis a symmetric matrix becomes diagonal and you solve each component separately; in the eigenfunction basis the differential operator becomes 'multiply mode n by lambda_n', and you solve each amplitude separately. Every modal solution method — heat decay, vibrating modes of a drum, quantum stationary states — is an eigenfunction expansion. It is the single most reusable tool for linear PDEs on bounded domains.
Heat in a rod with u(0,t) = u(L,t) = 0 and u(x,0) = f(x): expand f = sum of b_n sin(n*pi*x/L), then u(x,t) = sum of b_n e^(-k (n*pi/L)^2 t) sin(n*pi*x/L). Each mode simply decays at its own rate lambda_n = (n*pi/L)^2 — high modes vanish fast, which is why the heat equation smooths data.
Expand the data in eigenfunctions, evolve each mode by its own ODE, sum back up.
The method requires the boundary conditions to be homogeneous (zero) so that each mode satisfies them separately; inhomogeneous boundary data must first be removed by subtracting a steady state. The eigenfunction expansion is a finite-domain tool — on infinite domains the discrete sum becomes a Fourier-transform integral.